Positive Energy Representations of the Poincare...

Gregory Naber

Positive Energy Representationsof the Poincare Group

A Sketch of the Positive Mass Case and itsPhysical Background

December 7, 2016

Springer

This is for my grandchildren Amber, Emily,Garrett, and Lacey and my buddy Vinnie.

Preface

During the period in which the foundations of quantum mechanics were being laidit was clear that there were two issues that demanded attention. On the one hand,there were classical physical systems, such as the electromagnetic field, that couldnot be understood from the perspective of classical mechanics and so were not di-rectly amenable to the techniques that were evolving for the quantization of me-chanical systems. On the other hand, at the time this development was taking placethe special theory of relativity was a firmly established part of theoretical physicsand yet quantum mechanics took no account of it and it was clear that a really satis-factory quantum theory must be “relativistic.” The problem of reconciling quantummechanics and special relativity is formidable and, indeed, they are essentially ir-reconcilable if one insists on retaining also the classical notion of a point materialparticle. Naively, this is due to the fact that special relativity requires that any suchparticle can be regarded as at rest in some inertial frame of reference and, in such aframe, it would have a well-defined position (the spot where it rests) and momentum(namely, zero) and this would be forbidden by the uncertainty principle of quantummechanics if, as relativity demands, all such frames of reference must be regardedas physically equivalent.

In a proposed sequel to the manuscript [Nab5] on the Foundations of QuantumMechanics we will attempt to sort out, for those whose background is in mathemat-ics rather than physics, how such a reconciliation might be carried out and how thenotion of a quantum field emerges as a result. The brief manuscript before you nowis an excerpt from this sequel that, we hope, may be of some independent interest.It centers around the problem of building the machinery required to define what itmeans to say that a quantum theory is “relativistic”.

Chapter 1 is a synopsis of the material on Lie groups, Lie algebras and their rep-resentations that we will require. Sections 1.1 and 1.2 contain the basic definitionsand examples, while Section 1.3 introduces the important notion of a projective rep-resentation of a Lie group. In Sections 1.4 and 1.5 we describe the so-called Mackeymachine for constructing all of the irreducible, unitary representations of certainsemi-direct products of Lie groups. There are some rather deep theorems here that

vii

viii Preface

we will state, but not prove. We will, however, make every e↵ort to provide amplereferences for all of the details we do not include.

In Chapter 2 we turn our attention special relativity. The first three sections moti-vate and describe the basic geometrical structure of Minkowski spacetime M and aredrawn largely from [Nab4]. Section 2.4 contains the construction of the 2-fold uni-versal covering groups SL(2,C) and ISL(2,C) of the Lorentz and Poincare groupsL"+ and P"+, respectively. In Section 2.5 we study the structure of the Lie algebrasof L"+ and P"+ (and therefore of SL(2,C) and ISL(2,C)). Here we also introduce theuniversal enveloping algebra U(g) of a Lie algebra g since this is where one finds theso-called Casimir invariants. The application of the Mackey machine to ISL(2,C)requires some rather detailed information about the algebraic dual of Minkowskispacetime, called momentum space, and all of this will be derived in Section 2.6.In Section 2.7 we identify the relativistic invariance of a quantum system with theexistence of a projective representation of P"+ on the Hilbert space H of the sys-tem and indicate how the problem of finding all of these can be reduced, via a deeptheorem of Bargmann, to that of finding the unitary representations of the doublecover ISL(2,C) of P"+. Finally, we pull all of this information together in Section2.8 to enumerate those particular irreducible, unitary representations of ISL(2,C)that, from the perspective of the physical applications we have in mind, are of inter-est to us, namely, those of positive mass and positive energy. We will not considerthe massless case, but will supply references.

The source of our interest in the irreducible, projective representations of thePoincare group resides in physics. Specifically, these are precisely the objects re-quired for a rigorous definition of the relativistic invariance of a quantum theory.One need not understand the physical motivation in order to appreciate the math-ematics, but one then knows only half of the story. For those inclined toward fulldisclosure we have included Appendix A with brief discussions of some physi-cal background material on classical and quantum mechanics, taken largely from[Nab5]. For our purposes here the central notion is that of a symmetry or, more gen-erally, a symmetry group of a physical system. In Sections A.2 and A.3 we will try todescribe enough of the formalism of classical Lagrangian and Hamiltonian mechan-ics to understand what it means to say that a classical mechanical system admits asymmetry group and what can be inferred from this. In Section A.4 we will attemptthe same thing for quantum mechanics and this will lead us to the projective repre-sentations of P"+. The Appendix concludes with Section A.5 in which we provide abrief introduction to the quantum mechanical phenomenon of spin.

A few editorial comments are worth making at the outset. The subtitle we havechosen contains the word “sketch” and this is entirely appropriate. The rather modestgoal here is to provide something of a global picture of the ideas, both mathematicaland physical, that are involved in understanding the construction and significanceof one particular class of irreducible, unitary representations of the Poincare groupand its universal cover. We will not pretend to have o↵ered an exhaustive treatment.We will, however, make a concerted e↵ort to direct those who need the full story toappropriate sources for all of the details we have not included here.

Preface ix

We must also confess that the manuscript assumes a fairly substantial mathe-matical background. We have provided brief synopses of some of the mathematicalmachinery required to carry out our task, but generally this has been limited to topicsthat were not discussed in some detail in [Nab5]. Naturally, it does not matter wherethis background has been acquired and when it comes time to provide references wewill include some readily accessible sources in addition to [Nab5].

We will consistently employ the Einstein summation convention according towhich a repeated index, one subscript and one superscript, indicates a sum over therange of values that the index can assume. For example, if i and j are indices thatrange over 1, . . . , n, then

Xi @L@qi =

nX

i=1

Xi @L@qi = X1 @L

@q1 + · · · + Xn @L@qn

(a superscript in the denominator counts as a subscript), whereas, if ↵ and � take thevalues 0, 1, 2, 3, then

⌘↵�v↵w� =

3X

↵,�=0

⌘↵�v↵w� = ⌘00v0w0 + · · · + ⌘03v0w3 + · · · + ⌘30v3w0 + · · · + ⌘33v3w3,

and so on.We should also say a few words about the signature of the quadratic form of

Minkowski spacetime. There was a time when the world was evenly divided be-tween (� + ++) and (+ � ��) and heartfelt arguments were put forth in favor ofeach. However, there is some statistical evidence that (+ � ��) has won the day. Ifso, this would put our reference [Nab4] on the losing side. In any case we have optedfor (+ � ��) here. This changes nothing essential beyond a few minus signs andthe sense of a few inequalities in the definitions. Even so, considering the numberof references we will make to [Nab4], we thought it only fair to alert the reader tothese adjustments. Forewarned is forearmed.

Contents

1 Lie Groups and Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Unitary Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Projective Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.4 Induced Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.5 Representations of Semi-Direct Products . . . . . . . . . . . . . . . . . . . . . . . 33

2 Minkowski Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.3 Geometrical Structure of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.4 Lorentz and Poincare Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.5 Poincare Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.5.2 Lie Algebra of L"+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.5.3 Lie Algebra of P"+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.5.4 Universal Enveloping Algebra and Casimir Invariants . . . . . . 85

2.6 Momentum Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922.6.1 Orbits and Isotropy Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922.6.2 Invariant Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992.6.3 Momentum Space and the Character Group . . . . . . . . . . . . . . 102

2.7 Spacetime Symmetries and Projective Representations . . . . . . . . . . . 1052.8 Positive Energy Representations of P"+ with m > 0 . . . . . . . . . . . . . . . 107

A Physical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123A.2 Finite-Dimensional Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . 123A.3 Finite-Dimensional Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . 133A.4 Postulates of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147A.5 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

xi

xii Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Chapter 1Lie Groups and Representations

1.1 Lie Groups

We will record here only those specific items that we require in what follows. Essen-tially everything we need is treated concisely, but in detail in Chapter 3 of [Warn].One can also consult Chapter 10, Volume I, of [Sp2], Section 5.8 of [Nab2], or, fora much more comprehensive treatment, [Knapp].

A Lie group is a group G that is also a (C1) di↵erentiable manifold for whichthe operations of multiplication (x, y) 7! xy : G ⇥ G ! G and inversion x 7!x�1 : G ! G are smooth (C1). If G1 and G2 are Lie groups, then a di↵eomorphism' : G1 ! G2 of G1 onto G2 that is also a group isomorphism is an isomorphismof Lie groups and, if such a thing exists, G1 and G2 are said to be isomorphic. Anisomorphism of G onto itself is called an automorphism of G and the set of all such,denoted Aut(G), is a group under composition, called the automorphism group ofG.

Note: Manifolds are assumed to be Hausdor↵ and second countable and they arealways locally compact so the same is true of Lie groups. In particular, every Liegroup G admits a Haar measure, that is, a nonzero Radon measure µG on G that isleft-invariant in the sense that µG(gB) = µG(B) for every g 2 G and every Borel setB in G, where gB = {gb : b 2 B} (see Section 2.2 of [Fol2]).

The requirement that inversion is a smooth map is actually superfluous. The follow-ing is Theorem 5.8.1 of [Nab2].

Theorem 1.1.1. Let G be a group that is also a di↵erentiable manifold for whichgroup multiplication (x, y) 7! xy : G ⇥G ! G is smooth. Then G is a Lie group.

Some obvious examples are R with real number addition, C with complex addi-tion,Rn andCn with coordinatewise addition, the nonzero real numbersR⇤ with real

1

2 1 Lie Groups and Representations

number multiplication, the nonzero complex numbers C⇤ with complex multiplica-tion, and the complex numbers of modulus one S 1 (also denoted T in this context)with complex multiplication. The general linear groups GL(n,R) and GL(n,C) con-sisting of all nonsingular n ⇥ n matrices with real and complex entries, respectively,are also Lie groups under matrix multiplication. The product of two Lie groups is aLie group with the product manifold structure and the direct product group structure.We will see more examples shortly. The following is Theorem 3.21 of [Warn].

Theorem 1.1.2. (Closed Subgroup Theorem) A closed subgroup H of a Lie groupG is an embedded submanifold of G and a Lie group with respect to the inducedmanifold structure and the group operations inherited from G.

The proof of the following Corollary is a nice application of the Closed SubgroupTheorem so we will include the argument.

Corollary 1.1.3. Let G and H be Lie groups and � : G ! H a continuous grouphomomorphism. Then � is smooth.

Proof. Let �� : G ! G ⇥ H be the graph map defined by

��(g) = (g, �(g)).

Let ⇡G and ⇡H be the projections of G ⇥ H onto G and H, respectively. �� is con-tinuous, injective, and maps onto the graph ��(G) of �. Its inverse is the restrictionof ⇡G to ��(G) and this is also continuous. Consequently, �� is a homeomorphismof G onto ��(G). Since � is a homomorphism, ��(G) is a subgroup of the Lie groupG ⇥ H. It is closed because it is the inverse image of the diagonal in H ⇥ H underthe continuous map G ⇥ H ! H ⇥ H defined by (g, h) 7! (�(g), h).

According to the Closed Subgroup Theorem, ��(G) is an embedded submanifoldof the Lie group G ⇥H and is itself a Lie group under the group operation on G ⇥Hand the submanifold structure. Now notice that

� = (⇡H |��(G)) � (⇡G |��(G))�1.

The projections ⇡G : G ⇥ H ! G and ⇡H : G ⇥ H ! H are smooth and, because��(G) is an embedded submanifold of G ⇥ H, the inclusion map ◆ : ��(G) ,!G ⇥ H is smooth. Since ⇡H |��(G) = ⇡H � ◆, it is also smooth. Similarly, ⇡G |��(G) =⇡G � ◆ is smooth. To show that � is smooth it will therefore su�ce to show thatthe homeomorphism ⇡G |��(G) is a di↵eomorphism. Moreover, since ��(G) is a Liegroup, left translation by any element is a di↵eomorphism so it will be enoughto show that ⇡G |��(G) is a local di↵eomorphism near some point. Since ⇡G |��(G) issmooth this follows from Sard’s Theorem which asserts that ⇡G |��(G) must have aregular value.

1.1 Lie Groups 3

Remark 1.1.1. If you would prefer to avoid an application of Sard’s Theorem thereis a direct proof of the last assertion in Lemma 9.4 of [VDB].

ut

Example 1.1.1. A closed subgroup of some general linear group GL(n,R) orGL(n,C) is called a matrix Lie group. We enumerate a few of these.

1. The special linear groups SL(n,R) and SL(n,C) consist of all of those elementsof GL(n,R) and GL(n,C), respectively, that have determinant one. As manifolds,SL(n,R) and SL(n,C) have dimension n2�1. When n = 2 the special linear groupSL(2,C) is closely related to the Lorentz group (see Section 2.4).

2. The orthogonal group O(n) consists of all n⇥n real matrices A that satisfy AAT =AT A = idn⇥n. These are precisely the matrices which, when identified with lineartransformations on Rn, preserve the standard inner product hx, yi = x1y1 + · · · +xnyn, that is, which satisfy hAx, Ayi = hx, yi for all x, y 2 Rn. These all havedeterminant ±1. As a manifold, O(n) has dimension n(n�1)/2 and two connectedcomponents corresponding to det A = 1 and det A = �1.

3. The special orthogonal group SO(n) is the subgroup of O(n) consisting of thoseelements that have determinant 1. It is an open subset of O(n) so the dimensionof SO(n) is also n(n� 1)/2 and it is the connected component of O(n) containingthe identity. When n = 3 we will refer to the special orthogonal group SO(3)as the rotation group. We will see in Theorem A.2.2 below that the elements ofSO(3) are the matrices, with respect to orthonormal bases for R3, of rotationsabout some axis.

4. The unitary group U(n) consists of all n⇥n complex matrices A that satisfy AAT=

AT

A = idn⇥n, where AT

is the conjugate transpose of A. These are precisely thematrices which, when identified with linear transformations on Cn, preserves thestandard Hermitian inner product hx, yi = x1y1 + · · · + xnyn, that is, which satisfyhAx, Ayi = hx, yi for all x, y 2 Cn. The determinant of any element of U(n) is acomplex number of modulus one. As a manifold, U(n) has dimension n2. Notethat we have taken the Hermitian inner product to be complex-linear in the secondslot and conjugate linear in the first.

5. The special unitary group SU(n) is the subgroup of U(n) consisting of thoseelements that have determinant 1. As a manifold, SU(n) has dimension n2 � 1.When n = 2 the special unitary group SU(2) is closely related to the rotationgroup SO(3) (see Section 2.4).

6. If n is a positive integer and p and q are non-negative integers with n = p+q, thenthe semi-orthogonal group O(p, q) consists of all n⇥ n real matrices A satisfyingAT⌘A = ⌘, where


⌘ = diag (1, p. . . , 1, �1, q. . . ,�1).

These are precisely the matrices which, when identified with linear transforma-tions on Rn, preserve the standard indefinite inner product

hx, yi = x1y1 + · · · + xpyp � xp+1yp+1 � · · · � xnyn

of index q on Rn, that is, which satisfy hAx, Ayi = hx, yi for all x, y 2 Rn.These all have determinant ±1. As a manifold, O(p, q) has dimension n(n� 1)/2.When p = 1 and q = 3, the semi-orthogonal group O(1, 3) is the generalLorentz group (see Section 2.4). The elements of O(1, 3) are generally denoted⇤ = (⇤↵�)↵,�=0,1,2,3 and they all satisfy det ⇤ = ±1 and either ⇤0

0 � 1 or⇤0

0 �1. These four possibilities determine the four connected componentsof O(1, 3).

7. The special semi-orthogonal group SO(p, q) is the subgroup of O(p, q) consist-ing of those elements with determinant 1. It is an open subset of O(p, q) so thedimension of SO(p, q) is also n(n � 1)/2. When p = 1 and q = 3, the specialsemi-orthogonal group SO(1, 3) is the proper Lorentz group (see Section 2.4). Itis the union of two of the four connected components of O(1, 3), one of whichcontains the identity matrix. The component of O(1, 3) containing the identity isdenoted SO+(1, 3) and is just the proper, orthochronous, Lorentz group L"+ (seeSection 2.4).

8. Lie groups that are not given directly as closed subgroups of some GL(n,R) orGL(n,C) can often be shown to be isomorphic to matrix Lie groups. For ourpurposes the most important examples will arise as “semi-direct products” ofthe matrix Lie groups described above. Semi-direct products are discussed ingeneral in Section 1.5. There we will illustrate the process of re-interpreting themas matrix Lie groups for the inhomogeneous rotation group ISO(3). The mostimportant example, however, is the inhomogeneous Lorentz group, or Poincaregroup, which will be described in detail in Section 2.4.

A real Lie algebra is a finite-dimensional, real vector space A on which is de-fined a (real) bilinear operation [ , ] : A ⇥ A ! A, called bracket, which is skew-symmetric

[Y, X] = �[X,Y]8X,Y 2 A

and satisfies the Jacobi identity

[[X,Y],Z] + [[Z, X],Y] + [[Y,Z], X] = 08X,Y,Z 2 A.

A complex Lie algebra is defined in exactly the same way, but with A a complexvector space and [ , ] complex-bilinear.

1.1 Lie Groups 5

Remark 1.1.2. Any real Lie algebra A has a Lie algebra complexification AC de-fined in the following way. The points of AC are ordered pairs (X1, X2) of vec-tors in A and one defines addition and complex scalar multiplication in AC by(X1, X2)+ (Y1,Y2) = (X1+Y1, X2+Y2) and (a+bi)(X1, X2) = (aX1�bX2, aX2+bX1).This gives AC the structure of a complex vector space. It is customary to write(X1, X2) as X1 + iX2 so that these operations take the same form as those of C, thatis,

(X1 + iX2) + (Y1 + iY2) = (X1 + Y1) + i(X2 + Y2)

and

(a + bi)(X1 + iX2) = (aX1 � bX2) + i(bX1 + aX2).

The bracket [ , ]C on AC is then defined by just “multiplying out”, that is,

[X1 + iX2,Y1 + iY2]C =�

[X1,Y1] � [X2,Y2]�+ i

�[X1,Y2] + [X2,Y1]

�.

If A1 and A2 are two Lie algebras with brackets [ , ]1 and [ , ]2, respectively,then a linear map T : A1 ! A2 that satisfies T ([X,Y]1) = [T (X),T (Y)]2 for allX,Y 2 A1 is called a Lie algebra homomorphism. If A2 is the algebra End(V)of endomorphisms of some complex vector space V with the commutator bracket,then T is called a Lie algebra representation of A1. If T : A1 ! A2 is a linearisomorphism then it is called a Lie algebra isomorphism and, if such a thing exists,we say that A1 and A2 are isomorphic. A Lie algebra isomorphism of A onto itselfis called an automorphism of A. A Lie subalgebra of a Lie algebra A is a linearsubspace B of A that is closed under the bracket [ , ] of A and therefore is a Liealgebra in its own right with this same bracket.

Every Lie group G has associated with it a Lie algebra g that can be defined intwo equivalent ways. A vector field X on G is said to be left invariant if, for eachg 2 G, (Lg)⇤ � X = X � Lg, where Lg : G ! G is the left translation di↵eomorphismLg(g0) = gg0 for every g0 2 G and (Lg)⇤ is its derivative. Left invariant vector fieldsare necessarily smooth (Theorem 5.8.2 of [Nab2]). Every tangent vector at the iden-tity in G gives rise, via left translation, to a unique left-invariant vector field on G.One can think of g as the real linear space of left invariant vector fields on G with[X,Y] taken to be the Lie bracket of the vector fields X and Y (see pages 263-264of [Nab2] for Lie brackets). Equivalently, g can be identified with the tangent spaceTe(G) at the identity e in G with the bracket of two tangent vectors x and y in Te(G)defined by writing x = X(e) and y = Y(e) for left invariant vector fields X and Y andsetting [x, y] = [X,Y](e). In particular, the linear dimension of g is the same as themanifold dimension of G.

Exercise 1.1.1. Work all of this out for the additive group Rn. Specifically, proveeach of the following.

1. Vector addition is a smooth map from Rn ⇥Rn to Rn so Rn is a Lie group.


2. The tangent space at any point in Rn can be canonically identified with Rn.3. If La is the left translation map La(b) = a + b on Rn and tangent spaces are

identified with Rn, then the derivative (La)⇤ is the identity at each point.4. The left-invariant vector fields on Rn are precisely the constant vector fields and

the Lie bracket of any two left-invariant vector fields is the zero vector field.Conclude that the Lie algebra of Rn is isomorphic to Rn with the trivial bracket([ , ] ⌘ 0) and therefore to T0(Rn) with the trivial bracket.

The general linear group G = GL(n,R) is an open submanifold of Rn2 so itstangent space at the identity matrix is linearly isomorphic to Rn2 , that is, to thespace of all real n⇥n matrices. Thought of in this way the bracket in the Lie algebragl(n,R) of GL(n,R) is just the matrix commutator (see pages 278-279 of [Nab2]).More generally, the Lie algebra of any matrix Lie group is some set of real matriceswith the bracket given by matrix commutator (see page 279 of [Nab2]). Thus, todetermine the Lie algebra of a matrix Lie group G one need only identify it with aset of matrices and this one can do by computing velocity vectors to smooth curvesin G through e. We will work out one simple example here and then just list somemore. The example of real interest to us is in Section 2.5.

Example 1.1.2. We will identify the Lie algebra o(n) of the orthogonal groupO(n). Any element of the tangent space Tid(O(n)) is A0(0) for some smooth curveA : (�✏, ✏) ! O(n) in O(n) with A0(0) = id. Since O(n) is a submanifold of Rn2 wecan regard A as a smooth curve in Rn2 and use standard coordinates to di↵erentiateentrywise. Denote the entries of A by Ai j, i, j = 1, . . . , n, and the standard coordi-nates on Rn2 by xi j, i, j = 1, . . . , n. Thus, the components of A0(0) relative to @

@xi j |idare ((Ai j)0(0). Since A(t) 2 O(n) for each t 2 (�✏, ✏), A(t)A(t)T = id, that is,

nX

k=1

Aik(t)Ajk(t) = �i j

for each t. Di↵erentiating at t = 0 gives

nX

k=1

[(Aik)0(0)� jk + �ik(Ajk)0(0)] = 0

(Ai j)0(0) + (Aji)0(0) = 0

(Aji)0(0) = �(Ai j)0(0)

so (Ai j)0(0) is a real, n ⇥ n, skew-symmetric matrix. Thus, Tid(O(n)) is contained inthe linear subspace of gl(n,R) consisting of the skew-symmetric matrices. But thedimension of this subspace is n(n � 1)/2 and this is precisely the dimension of O(n)so o(n) is precisely the set of real, n⇥n, skew-symmetic matrices under commutator.

Notice that, since SO(n) is an open submanifold of O(n), its tangent space at theidentity is the same as that of O(n) so

1.1 Lie Groups 7

so(n) = o(n).

O(n) and SO(n) are not isomorphic as Lie groups, but they have the same Lie al-gebras. Two Lie groups are said to be locally isomorphic if their Lie algebras areisomorphic.

Once the Lie algebra g of a matrix Lie group G is identified with a particularvector space of matrices it is often convenient to have in hand a more explicit de-scription of the commutator bracket. For this one can select a basis {Xi}i=1,...,n forg. The bracket on g is then completely determined by the commutators [Xi, Xj] fori, j = 1, . . . , n. But each [Xi, Xj] is a linear combination of the basis elements sothere exist constants Ci jk, k = 1, . . . , n, such that

[Xi, Xj] =nX

k=1

Ci jkXk.

These are called the structure constants of g relative to the chosen basis and theydetermine the bracket completely.

Exercise 1.1.2. Show that

X1 =

0BBBBBBBB@

0 0 00 0 �10 1 0

1CCCCCCCCA , X2 =

0BBBBBBBB@

0 0 10 0 0�1 0 0

1CCCCCCCCA , X3 =

0BBBBBBBB@

0 �1 01 0 00 0 0

1CCCCCCCCA

is a basis for o(3) (and so(3)) and compute the commutators to show that

[Xi, Xj] = ✏i jkXk, i, j = 1, 2, 3,

where ✏i jk is the Levi-Civita symbol (1 if i jk is an even permutation of 123, -1 if i jkis an odd permutation of 123, and 0 otherwise).

Example 1.1.3. It is not a trivial matter to come up with Lie groups that are notisomorphic to matrix Lie groups and all of the examples of interest to us are, in fact,isomorphic to matrix Lie groups (although perhaps not obviously so). For each ofthe following matrix Lie groups we will specify the set of matrices that constituteits Lie algebra; the bracket is always matrix commutator.

1. The Lie algebra gl(n,R) of GL(n,R) is the linear space of all real, n⇥n matrices.

2. The Lie algebra gl(n,C) of GL(n,C) is the real linear space of all complex, n ⇥ nmatrices.

3. The Lie algebras o(n) and so(n) of O(n) and SO(n), respectively, both consist ofall real, n ⇥ n, skew-symmetric matrices.


4. The Lie algebra u(n) of U(n) is the real linear space of all complex, n⇥n matricesX that are skew-Hermitian (X

T= �X).

5. The Lie algebra su(n) of SU(n) is the real linear space of all complex, n ⇥ nmatrices X that are skew-Hermitian (X

T= �X) and tracefree (Trace (X) = 0).

Remark 1.1.3. In Exercise 1.2.6 (9) we will exhibit a basis for su(2) in terms ofthe Pauli spin matrices and you will use it to show that su(2) is isomorphic toso(3) so that SU(2) and SO(3) are locally isomorphic.

6. The Lie algebras o(p, q) and so(p, q) of O(p, q) and SO(p, q), respectively, bothconsist of all real, n ⇥ n matrices X that satisfy XT = �⌘X⌘, where

⌘ = diag (1, p. . . , 1, �1, q. . . ,�1).

When p = 1 and q = 3 this implies that o(1, 3) = so(1, 3) consists of all realmatrices of the form

X =

0BBBBBBBBBBBB@

0 X01 X0

2 X03

X01 0 �X1

2 X13

X02 X1

2 0 �X23

X03 �X1

3 X23 0

1CCCCCCCCCCCCA.

This is also the Lie algebra so+(1, 3) of the proper, orthochronous Lorentz groupSO+(1, 3) = L"+ (see Section 2.4).

7. For Lie groups such as the Poincare group (see Section 2.4) which are given assemi-direct products one can define the general notion of a semi-direct product ofLie algebras and prove that the Lie algebra of a semi-direct product of Lie groupsis the semi-direct product of the Lie algebras of these groups (see pages 301-306of [Nab5] for a brief description and an example or Section I.4 of [Knapp] forthe details). One can often avoid this abstract description of the Lie algebra byfinding an explicit representation of the Lie group semi-direct product as a matrixLie group; such a representation of the Poincare group is described in Section 2.4.

Example 1.1.4. An associative algebra over a fieldK is a vector space A overK onwhich is defined a K-bilinear map B : A ⇥A! A, written simply B(a, b) = ab forall a, b 2 A and called multiplication, that satisfies a(bc) = (ab)c for all a, b, c 2 A.A is said to be unital if there exists an element 1 2 A, called the unit, satisfying1a = a1 = a for all a 2 A. A subalgebra of A is a linear subspace B of A thatis closed under multiplication and is therefore an algebra in its own right with thesame operations as A. If A is unital, then B is also required to contain the unit 1.On the other hand, a (two-sided) ideal in A is an additive subgroup J of A with theproperty that x 2 J implies ax 2 J and xa 2 J for all a 2 A. If A is unital and J

1.1 Lie Groups 9

contains the unit, then, in fact, J = A. If S is an arbitrary subset of A, then the idealgenerated by S is the intersection of all ideals in A that contain S .

Any associative algebra A overR orC can be given the structure of a Lie algebraby defining on it the commutator bracket

[a, b] = ab � ba.

We call this the commutator Lie algebra structure of A. In Section 2.5.4 we will dis-cuss the problem of representing a given Lie algebra as the commutator Lie algebraof some associative algebra.

If G is a Lie group with Lie algebra g we define the exponential map

exp : g! G

from g to G as follows. Regard g as the tangent space to the identity e in G. For any⇠ 2 g there exists a unique homomorphism �⇠ : R ! G satisfying �⇠(0) = e and�⇠(0) = ⇠ (see page 102 of [Warn]). We define

exp (⇠) = �⇠(1).

The terminology and notation are motivated by the fact that, for the general lineargroup GL(n,R),

exp : gl(n,R)! GL(n,R)

is precisely matrix exponentiation (Example 3.35 of [Warn]). The same is true ofany matrix group so in this case we will often use exp (⇠) and e⇠ interchangeably.

Exercise 1.1.3. Let G be the additive group Rn. Then the Lie algebra can be canon-ically identified with Rn (Exercise 1.1.1). Show that, with this identification, theexponential map is given by

exp (⇠) = ⇠.

We will have occasion to need a more geometrical description of the rotationgroup SO(3). The following is Lemma A.1 of [Nab2].

Theorem 1.1.4. Let N be an element of so(3). Then the matrix exponential etN isin SO(3) for every t 2 R. Conversely, if A is any element of SO(3), then there is aunique t 2 [0, ⇡] and a unit vector n = (n1, n2, n3) in R3 for which

A = etN = id3⇥3 + (sin t)N + (1 � cos t)N2,

where N is the element of so(3) given by


N =

0BBBBBBBB@

0 �n3 n2

n3 0 �n1

�n2 n1 0

1CCCCCCCCA .

In particular, the exponential map on so(3) is surjective.

Geometrically, one thinks of A = etN as the rotation of R3 through t radians aboutan axis along n in a sense determined by the right-hand rule from the direction of n.

If G is a group and M is a set, then a left action of G on M is a map � : G⇥M !M, generally written �(g, x) = g · x, satisfying e · x = x for every x 2 M, where e 2 Gis the identity element, and g1 · (g2 · x) = (g1g2) · x for all g1, g2 2 G and all x 2 M.For each g 2 G the map �g : M ! M defined by �g(x) = g · x is then a bijection ofM onto itself with inverse �g�1 . The orbit Ox0 of x0 2 M under this action is definedby

Ox0 = {g · x0 : g 2 G}

and the isotropy subgroup Hx0 if x0 is

Hx0 = {g 2 G : g · x0 = x0}.

The action is said to be transitive if Ox0 = M for every x0 2 M and free if Hx0 = {e}for every x0 2 M. If G is a topological group and M is a topological space, then theaction � : G ⇥ M ! M is required to be continuous and it follows that each �g is ahomeomorphism. If G is a Lie group and M is a smooth manifold, then � : G⇥M !M is required to be smooth and it follows that each �g is a di↵eomorphism.

Similarly, a right action of G on M is a map ⌧ : M ⇥G ! M, denoted ⌧(x, g) =x ·g, that satisfies x ·e = x and (x ·g1) ·g2 = x · (g1g2) for all x 2 M and all g1, g2 2 G.Given a right action ⌧ one can define a left action � by �(g, x) = ⌧(x, g�1). Similarly,every left action gives rise to a right action. All of the definitions we have introducedfor left actions have obvious analogues for right actions. We will see an abundanceof examples of such group actions as we proceed so for the moment we will contentourselves with just the following.

Example 1.1.5. Let G be a matrix Lie group and identify its Lie algebra g with thetangent space Tid(G) at the identity. G acts on itself by conjugation, that is, the map� : G⇥G ! G defined by �(g, h) = g · h = ghg�1 is a smooth left action of G on G.

Remark 1.1.4. The corresponding right action ⌧ : G ⇥G ! G is given by ⌧(h, g) =h · g = g�1hg. Everything that follows has an obvious analogue for this right actionwhich we will leave it to you to write out.

For each fixed g 2 G, the map �g : G ! G defined by

�g(h) = ghg�1

1.1 Lie Groups 11

for all h 2 G is a di↵eomorphism. Its derivative at the identity is denoted Adg.

Adg = (�g)⇤id : g! g

We claim that, for every X 2 g,

Adg(X) = gXg�1.

Indeed, since t 7! etX is a smooth curve in G through id with velocity vector X att = 0,

Adg(X) = (�g)⇤id(X) =ddt

(getXg�1)��t=0 = g

✓ ddt

etX��t=0

◆g�1 = gXg�1.

Thus, G acts on its Lie algebra g by conjugation.

Exercise 1.1.4. Show that, for any g 2 G and any X,Y 2 g,

Adg([X,Y]) = [Adg(X), Adg(Y)].

Each Adg is therefore a Lie algebra homomorphism from g to g. Since Adg isclearly bijective, it is a Lie algebra isomorphism of g onto itself, that is, an auto-morphism of g. The set Aut(g) of all automorphisms of g is a closed subgroup of thegeneral linear group GL(g) of g and is therefore a Lie group. The map

Ad : G ! Aut(g)

that sends g 2 G to Adg 2 Aut(g) is smooth and its derivative at the identity is alinear map from g to the Lie algebra of Aut(g). This map is denoted

ad = Ad⇤id

and its value at any X 2 g is denoted adX = Ad⇤id(X). It follows from the Jacobiidentity that each adX is a derivation of g, that is, a linear map that satisfies theLeibniz rule

adX[Y,Z] = [Y, adXZ] + [adXY,Z]

for all Y,Z 2 g. Thus,

ad : g! Der(g) ✓ End(g),

where Der(g) is the linear space of all derivations of g (which is, in fact, the Liealgebra of Aut(g)). We claim that, for any Y 2 g, the value of adX at Y is given by

adXY = [X,Y].


Indeed, since t 7! etX is a smooth curve in G through id with velocity vector X att = 0,

adXY = [Ad⇤id(X)](Y) =ddt

✓AdetX (Y)

◆ ��t=0=

ddt

✓etXYe�tX

◆ ��t=0

= etX��t=0

ddt

(Ye�tX)��t=0 +

ddt

(etX)��t=0(Ye�tX)

��t=0

= �YX + XY= [X,Y].

The map ad is called the adjoint representation of g and adXY = [X,Y] is the adjointaction of X on Y . The Lie algebra homomorphism ad : g! End(g) is a Lie algebrarepresentation of g on g.

Now let G be a Lie group, H a closed (not necessarily normal) subgroup of G,and ⇡ : G ! G/H the natural projection of G onto the set of left cosets of H in G.

⇡(g0) = [g0] = g0H

Supply G/H with the quotient topology determined by ⇡. Then it is not di�cult toshow that ⇡ is a continuous, open map and the left action of G on G/H defined by

(g, [g0]) 2 G ⇥G/H 7! g · [g0] = [gg0] 2 G/H

is continuous. The action is also transitive on G/H, that is, for any two points [g0]and [g1] in G/H there is a g 2 G for which [g1] = g · [g0], namely, g = g1g�1

0 . Forthis reason G/H is called a homogeneous space. The corresponding statements inthe di↵erentiable category require considerably more work and are contained in thefollowing result which combines Theorem 3.58 and Theorem 3.64 of [Warn].

Theorem 1.1.5. Let G be a Lie group, H a closed subgroup of G and ⇡ : G ! G/Hthe natural projection of G onto the space of left cosets of H in G. Then G/H admitsa unique di↵erentiable structure for which the left action (g, [g0]) 2 G ⇥ G/H 7!g · [g0] = [gg0] 2 G/H is smooth. Relative to this di↵erentiable structure all of thefollowing are satisfied.

1. ⇡ : G ! G/H is smooth.2. If H is a normal subgroup of G, then G/H is a Lie group.3. ⇡ : G ! G/H admits smooth local sections, that is, for each point [g0] 2 G/H

there exists an open neighborhood U of [g0] in G/H and a smooth map s : U !⇡�1(U) ✓ G such that ⇡ � s = idU.

The third apart of Theorem 1.1.5 asserts that one can locally make a smoothselection of a representative from each equivalence class in G/H. We will put Theo-rem 1.1.5 to good use in Section 1.4 when we describe Mackey’s theory of inducedrepresentations. The same is true of the following, which is Theorem 3.62 of [Warn].

1.1 Lie Groups 13

Theorem 1.1.6. Let � : G ⇥ M ! M, (g,m) 7! g · m, be a smooth, transitive leftaction of the Lie group G on the manifold M. Let m0 be an arbitrary point of M andHm0 = {g 2 G : g ·m0 = m0} the isotropy subgroup of m0 in G. Then Hm0 is a closedsubgroup of G and the map

�m0 : G/Hm0 ! M�m0 ([g]) = g · m0

is a di↵eomorphism of G/Hm0 onto M.

Notice that g ·m0 is independent of the representative g chosen from [g] 2 G/Hm0

since Hm0 is the isotropy group of m0.

Example 1.1.6. The rotation group SO(3) acts smoothly and transitively on the 2-sphere S 2 in the following way. Identify each matrix A 2 SO(3) with the matrix,relative to the standard basis for R3, of an orthogonal linear transformation of R3,which we will also denote A. Define � : SO(3)⇥S 2 ! S 2 by �(A, x) = A · x = A(x).Then � is clearly a smooth left action of SO(3) on S 2. This action is transitive on S 2

and the isotropy subgroup of the north pole in S 2 is isomorphic to SO(2) (see pages90-91 of [Nab2]). We conclude from Theorem 1.1.6 that S 2 is di↵eomorphic to thehomogeneous manifold SO(3)/SO(2). All of this generalizes at once to prove

S n�1 � SO(n)/SO(n � 1)

for any n � 2 (SO(1) is the trivial group).

Finally, we will need (in Section 2.4) a result on covering spaces. Recall that ifM and N are smooth manifolds and F : M ! N is a smooth mapping of M ontoN, then F is said to be a (smooth) covering map and M is said to be a (smooth)covering space of N if each point x 2 N has an open neighborhood U in N withthe property that F�1(U) is a disjoint union of open sets in M each of which ismapped di↵eomorphically onto U by F. If M is simply connected, then it is calledthe universal covering space of N because it covers any other covering space of N.The following is Proposition 9.30 of [Lee].

Theorem 1.1.7. Let G and H be connected Lie groups and F : G ! H a smoothhomomorphism. Then the following are equivalent.

1. F is surjective and has discrete kernel.2. F is a smooth covering map.

In this case we refer to G as a covering group of H. Since G and H are locallydi↵eomorphic, they have the same Lie algebras (see Exercise 1.2.6 (5) for a simpleexample). In Section 2.4 we will construct two important examples of universalcovering groups.


1.2 Unitary Group Representations

Although our interest is almost exclusively in unitary representations of Lie groupson Hilbert spaces it will be convenient to begin in a more general context. We letG denote an arbitrary group and V a real or complex vector space. For the momentwe impose no topological structure on either of these. Denote by GL(V) the group,under composition, of all invertible linear transformations of V onto itself. Then ahomomorphism � : G ! GL(V) of G into GL(V) is called a representation of Gon V. The dimension of V (whether finite or infinite) is called the dimension of therepresentation and the elements of V are called carriers of the representation. Noticethat a representation of G on V determines a left action of G on V defined by g · v =�(g)(v) and that, conversely, any left action of G on V by linear transformationsdetermines a representation of G on V by the same equation. A subspace S of V withthe property that S is invariant under every �(g) (meaning �(g)(S) ✓ S8g 2 G), issaid to be an invariant subspace for �. If G and V are endowed with topologies,then some sort of continuity requirement is imposed on the representations of G onV. We turn now to the case of most interest to us.

We let G be a Lie group and denote its identity element eG or simply e if noconfusion will arise. Let H be a separable, complex Hilbert space (either finite-or infinite-dimensional) and U(H) the group of unitary operators on H. A unitaryrepresentation of G on H is a group homomorphism

� : G ! U(H).

� is strongly continuous if, for each fixed v 2 H, the map

g! �(g)v : G ! H

is continuous in the norm topology of H, that is,

g! g0 in G ) k�(g)v � �(g0)vk ! 0 in R. (1.1)

The representation is said to be trivial if it sends every g 2 G to the identity operatoridH = I on H.

Exercise 1.2.1. Let G be a matrix Lie group, H a separable, complex Hilbert spaceand � : G ! U(H) a unitary representation. Show that the following are equivalent.

1. � is strongly continuous, that is, satisfies (1.1).2. � is weakly continuous, that is, for all u, v 2 H,

g! g0 in G ) h�(g)u, vi ! h�(g0)u, vi in C.

3. For each u 2 H, the map g 2 G ! h�(g)u, ui 2 C is continuous at e.

Show also that if H is finite-dimensional, then all of these are equivalent to thecontinuity of � : G ! U(H) when U(H) is given the operator norm topology.

1.2 Unitary Group Representations 15

Hint: For (3) ) (1) show that k�(g)u � �(g0)u k2 = 2kuk2 � 2Re h�(g�10 g)u, ui �� kuk2 � h�(g�1

0 g)u, ui��.

A linear subspace H0 of H is said to be invariant under � : G ! U(H) if�(g)(H0) ✓ H0 for every g 2 G. The zero subspace 0 and H itself are alwaysinvariant. If � : G ! U(H) is nontrivial and if 0 and H are the only closed invariantsubspaces, then � : G ! U(H) is said to be irreducible. If there are closed invariantsubspaces other than 0 and H, then the representation is said to be reducible. Twounitary representations �1 : G ! U(H1) and �2 : G ! U(H2) of G are said to beunitarily equivalent if there exists a unitary operator U : H1 ! H2 of H1 onto H2such that

U�1(g) = �2(g)U 8g 2 G,

that is,

�2(g) = U�1(g)U�1 8g 2 G.

In this case, U is said to intertwine the representations �1 and �2.Another item we will need is the following infinite-dimensional version of

Schur’s Lemma . The proof is a nice application of the Spectral Theorem (see Sec-tion 5.5 of [Nab5]) and is not so readily available in the literature so we will providethe details.

Remark 1.2.1. A more general version of Schur’s Lemma is proved in Appendix 1of [Lang4].

Theorem 1.2.1. (Schur’s Lemma) Let G be a Lie group, H a separable, complexHilbert space, and � : G ! U(H) a strongly continuous unitary representation ofG. Then � : G ! U(H) is irreducible if and only if the only bounded operatorsA : H ! H that commute with every �(g)

�(g)A = A�(g) 8g 2 G

are those of the form A = cI, where c is a complex number of modulus one and I isthe identity operator on H.

Proof. Suppose first that the only bounded operators that commute with every �(g)are constant multiples of the identity. We will show that the representation is irre-ducible. Let H0 be a closed subspace of H that is invariant under every �(g).

Exercise 1.2.2. Show that the orthogonal complement H?0 of H0 is also invariantunder every �(g).

Now, let P : H ! H0 be the orthogonal projection onto H0.


Exercise 1.2.3. Show that �(g)P = P�(g) for every g 2 G.

According to our assumption, P is a constant multiple of the identity. Being a pro-jection, P2 = P so the constant is either 0 or 1. Thus, H0 = P(H) is either 0 or H,as required.

Now, for the converse we will assume that � : G ! U(H) is irreducible and thatA : H ! H is a bounded operator that commutes with every �(g). We must showthat A is a constant multiple of the identity operator on H. Let A⇤ : H ! H denotethe adjoint of A (also a bounded operator on H).

Exercise 1.2.4. Show that A⇤ also commutes with every �(g).

Notice that 12 (A + A⇤) and i

2 (A � A⇤) are both self-adjoint and both commute withevery �(g). Moreover,

A =12

(A + A⇤) +1i

i2

(A � A⇤)�.

Consequently, it will be enough to prove that bounded self-adjoint operators thatcommute with every �(g) must be constant multiples of the identity. Accordingly,we may assume that A is self-adjoint. Then, by the Spectral Theorem, A has associ-ated with it a unique spectral measure EA. Moreover, since A commutes with every�(g), so does EA(S ) for any Borel set S ✓ R. From this it follows that each closedlinear subspace EA(S )(H) is invariant under � : G ! U(H). But, by irreducibility,this means that

EA(S )(H) = 0 or EA(S )(H) = H

for every Borel set S in R.Since A is bounded there exist a1 < b1 in R such that, if S \ [a1, b1] = ;, then

EA(S ) = 0. In particular, EA([a1, b1]) = I. Write

[a1, b1] =a1,

a1 + b1

2

�[

a1 + b1

2, b1

�.

Now notice that, if EA�� a1+b12

�= I, then the Spectral Theorem gives A = a1+b1

2 Iand we are done. Otherwise, EA must be I on one of the intervals and 0 on theother. Denote by [a2, b2] the interval on which it is I. Applying the same argu-ment to [a2, b2] we either prove the result (at the midpoint) or we obtain an interval[a3, b3] of half the length of [a2, b2] on which EA is I. Continuing inductively, weeither prove the result in a finite number of steps or we obtain a nested sequence[a1, b1] ◆ [a2, b2] ◆ [a3, b3] ◆ · · · of intervals whose lengths approach zero andfor which EA([ai, bi]) = I for every i = 1, 2, 3, . . .. By the Cantor Intersection The-orem (Theorem C, page 73, of [Simm1]), \1i=1[ai, bi] = {c} for some c 2 R. SinceEA(R� [ai, bi] ) = 0 for each i, 0 = EA([1i=1(R� [ai, bi]) ) = EA(R�\1i=1[ai, bi] ) =EA(R � {c}). Thus, EA({c}) = I so again we have A = cI. ut


Corollary 1.2.2. Let A be an Abelian Lie group. Then every irreducible, unitaryrepresentation of A on a complex, separable Hilbert space H is 1-dimensional.

Exercise 1.2.5. Prove Corollary 1.2.2.

It is the unitary representations of the Poincare group and its universal cover(Section 2.4) that are of most interest to us, but to describe these we will need notonly all of the machinery of the next three sections, but also an explicit descriptionof the irreducible, unitary representations of SU(2). We will conclude this sectionwith the latter.

Example 1.2.1. The special unitary group SU(2) consists of all 2 ⇥ 2 complex ma-trices U that are unitary (UU

T= U

TU = id2⇥2) and have determinant det (U) = 1.

Every U 2 SU(2) can be written uniquely in the form

U = ↵ �� ↵

!,

where ↵, � 2 C and |↵|2 + |�|2 = 1 (Lemma 1.1.3 of [Nab2]). The inverse of U isgiven by

U�1 =

↵ �� ↵

!.

SU(2) is a group under matrix multiplication. Indeed, it is a closed subgroup of theLie group GL(2,C) of invertible 2 ⇥ 2 complex matrices and is therefore a matrixLie group. The map

↵ �� ↵

!2 SU(2) 7! (↵, �) 2 C2 = R4

identifies SU(2) topologically with the 3-sphere S 3 (see Section 1.1 of [Nab2]). Inparticular, SU(2) is compact (being closed and bounded in R4) and simply con-nected (see pages 118-119 of [Nab2]). It follows from compactness that every con-tinuous, irreducible representation of SU(2) is finite-dimensional (see Theorem 5.2of [Fol2]). Another consequence of compactness is that every continuous, finite-dimensional representation � : S U(2)! GL(V) of SU(2) on a complex vector spaceV is “unitarizable” in the sense that there exists a Hermitian inner product on V withrespect to which � is unitary (see page 128 of [Fol2]). As a result it will be enoughto consider the continuous, finite-dimensional, irreducible, unitary representationsof SU(2). It is our good fortune that all of these are known.

We will begin by just writing out a few nontrivial, finite-dimensional representa-tions of SU(2). The most obvious of these is the standard, or defining representation⌧ of SU(2) on C2 that simply identifies any U 2 SU(2) with a linear transformationon C2 by matrix multiplication, that is,


⌧(U)

z1z2

!= U

z1z2

!=

↵ �� ↵

! z1z2

!=

↵z1 + �z2

��z1 + ↵z2

!.

Remark 1.2.2. We will feel free to regard the elements of Cn as either n-tuplesor column vectors of complex numbers and will write them as Z = (z1, . . . , zn) or

Z =

0BBBBBBBBBB@

z1...

zn

1CCCCCCCCCCA. In any case, the standard Hermitian inner product hZ,Wi of Z and W in

Cn is z1w1 + · · · + znwn. Naturally, this turns Cn into a complex Hilbert space. Withthe topology determined by the corresponding norm, Cn is homeomorphic to R2n

and ⌧ is continuous.

There is another obvious representation of SU(2) on C2 called the conjugationrepresentation which matrix multiplies by U rather than U. We will denote this ⌧ sothat

⌧(U)

z1z2

!= U

z1z2

!=

↵ �� ↵

! z1z2

!=

↵z1 + �z2��z1 + ↵z2

!.

We would like to show, however, that ⌧ is not really anything new since it is unitarilyequivalent to ⌧. To prove this we will use some properties of the so-called “Pauli spinmatrices” which we introduce in the following Exercise. We will try to give somesense of where they come from and why they are interesting.

Exercise 1.2.6. Denote by R3 the set of all 2 ⇥ 2 complex, Hermitian matrices X(X

T= X) with trace zero (Trace (X) = 0) .

1. Show that every X 2 R3 can be uniquely written as

X =

x3 x1 � ix2

x1 + ix2 �x3

!= x1�1 + x2�2 + x3�3,

where x1, x2 and x3 are real numbers and

�1 =

0 11 0

!,�2 =

0 �ii 0

!,�3 =

1 00 �1

!

are the so-called Pauli spin matrices. Show that these are all unitary and equal totheir own inverses.

2. Show that, with the operations of matrix addition and (real) scalar multiplication,R3 is a 3-dimensional, real vector space and

��1,�2,�3

is a basis. Consequently,

R3 is linearly isomorphic to R3. Furthermore, defining an orientation on R3 bydecreeing that

��1,�2,�3

is an oriented basis, the map X ! (x1, x2, x3) is an

orientation preserving isomorphism when R3 is given its usual orientation.


3. Show that

�1�2 = i�3, �2�3 = i�1, �3�1 = i�2, �1�2�3 = iI,

where I is the 2 ⇥ 2 identity matrix.4. Show that �1,�2,�3 satisfy the commutation relations

[�1,�2]� = 2i�3, [�2,�3]� = 2i�1, [�3,�1]� = 2i�2,

where [ , ]� denotes the matrix commutator ([A, B]� = AB � BA).5. Show that �1,�2,�3 satisfy the anticommutation relations.

[�i,� j]+ = 2�i jI, i, j = 1, 2, 3,

where [ , ]+ denotes the matrix anticommutator ([A, B]+ = AB + BA) and �i j isthe Kronecker delta. In particular, each �i squares to the identity matrix.

6. Show that, if X = x1�1 + x2�2 + x3�3 and Y = y1�1 + y2�2 + y3�3, then

12

[X,Y]+ = (x1y1 + x2y2 + x3y3)I.

Conclude that, if one defines an inner product hX,YiR3 on R3 by

12

[X,Y]+ = hX,YiR3 I,

then��1,�2,�3

is an oriented, orthonormal basis for R3 and R3 is isometric to

R3. We will refer to R3 as the spin model of R3.7. Regard the matrices �1,�2,�3 as linear operators on C2 (as a 2-dimensional,

complex vector space with its standard Hermitian inner product) and show thateach of these operators has eigenvalues ±1 with normalized, orthogonal eigen-vectors given as follows.

�1 :1p2

11

!,

1p2

1�1

!

�2 :1p2

1i

!,

1p2

1�i

!

�3 : 10

!,

01

!

8. Show that, for any U 2 SU(2),

U = �2U��12

and conclude that the conjugation representation ⌧ and the standard representa-tion ⌧ are unitarily equivalent.


9. Show that u j = � i2� j, j = 1, 2, 3, is a basis for the Lie algebra su(2) of SU(2)

relative to which the structure constants are given by

[u j, uk] = ✏ jklul, j, k = 1, 2, 3

and that the linear map from so(3) to su(2) defined by Xj 7! u j, j = 1, 2, 3, is anisomorphism of Lie algebras (see Example 1.1.2).

10. Define the standard representation ⌧ and the conjugation representation ⌧ ofSL(2,C) in exactly the same way as for SU(2) and show that these are not uni-tarily equivalent. Hint: Let

U0 =

i 00 �i

!.

Then U0 2 SU(2) ✓ SL(2,C) so �2U0��12 = U0. Show that �2 and its nonzero,

real scalar multiples are the only complex, 2 ⇥ 2 matrices for which this is true.Now find an element A of SL(2,C) for which �2A��1

2 , A.

Now let’s make a general observation. Suppose we have a representation � :G ! GL(V) of G on some finite-dimensional complex vector space V. Let F(V)be the vector space of all complex-valued functions f : V ! C on V. Now define� : G ! GL(F(V)) by

(�(g) f )(v) = f (�(g)�1(v))

for all g 2 G and all v 2 V. Then � is also a representation of G because

(�(g1g2) f )(v) = f (�(g1g2)�1(v)) = f (�(g�12 g�1

1 )(v))

= f (�(g2)�1�(g1)�1(v)) = (�(g2) f )(�(g1)�1(v))= (�(g1)�(g2) f )(v)

and so �(g1g2) = �(g1)�(g2). The same is true if F(V) is replaced by any vector spaceof complex-valued functions on V with the property that it contains f (�(g)�1(v))whenever it contains f (v). This suggests a means of producing new representationsfrom given representations that we will now apply to SU(2).

We begin with the standard representation ⌧ : SU(2)! GL(C2) of SU(2) on C2.For any integer j � 0 we let V j denote the complex vector space of all homogeneouspolynomial functions of degree j in the two complex variables z1 and z2 (V0 consistsof the constant functions of z1 and z2 so one can identify it with C). Thus, anyelement of V j is of the form

f (z1, z2) = a0z j1 + a1z j�1

1 z2 + a2z j�21 z2

2 + · · · + a jzj2.

The complex dimension of V j is j + 1. Define ⌧ j : SU(2)! GL(V j) by


(⌧ j(U) f )(z1, z2) = f (⌧(U)�1(z1, z2)) = f (↵z1 � �z2, �z1 + ↵z2)

= a0(↵z1 � �z2) j + a1(↵z1 � �z2) j�1(�z1 + ↵z2)+

a2(↵z1 � �z2) j�2(�z1 + ↵z2)2 + · · · + a j(�z1 + ↵z2) j

(⌧0(U) is the identity on V0 = C for every U 2 SU(2)). Expanding the terms on theright-hand side one sees that (⌧ j(U) f )(z1, z2) is also a homogeneous polynomial ofdegree j in z1 and z2 so ⌧ j(U) carries V j into itself and it is certainly linear in f .

Notice that

{z j1, z

j�11 z2, z

j�21 z2

2, . . . , zj2} (1.2)

forms a basis for the linear space V j so the map that sends a0z j1+a1z j�1

1 z2+a2z j�21 z2

2+

· · · + a jzj2 onto (a0, a1, a2, . . . , a j) is an isomorphism of V j onto C j+1. One can now

move the topology and Hilbert space structure of C j+1 back to V j by this isomor-phism. In particular, the basis (1.2) is orthonormal and the representation ⌧ j is uni-tary. Not at all so clear, but true nonetheless, is that each ⌧ j is irreducible and that,up to unitary equivalence, these are all of the irreducible, unitary representations ofSU(2). The following result combines Theorems 5.37 and 5.39 of [Fol2].

Theorem 1.2.3. Each ⌧ j : SU(2) ! GL(V j), j � 0, is an irreducible, unitaryrepresentation of SU(2) and every irreducible, unitary representation of SU(2) isunitarily equivalent to one and only one ⌧ j.

Remark 1.2.3. ⌧0 : SU(2) ! GL(C) is the trivial representation that sends everyU 2 SU(2) to the identity idC on V0 = C.

Generally we will find it more convenient to simply identify

V j = Cj+1

via the isomorphism described above and regard ⌧ j as acting on the coe�cients(a0, a1, a2, . . . , a j). Doing this in the notation we have employed up to this point israther cumbersome so we will introduce a new way of writing all of this that ismore convenient and also more in line with what one is likely to see in the physicsliterature. We will describe this in the following example..

Example 1.2.2. A typical element of V1 is a homogeneous polynomial f (z1, z2) =a0z1 + a1z2 of degree 1 with complex coe�cients. We now prefer to write this in theform

f (z1, z2) =2X

A=1

⇠AzA,


where ⇠1 = a0 and ⇠2 = a1. With the summation convention this is just

⇠AzA.

We are interested in how the coe�cients ⇠A “transform” ⇠A ! ⇠A under the actionof ⌧1. For this we will write the entries of U 2 SU(2) as

U =

U11 U1

2U2

1 U22

!

so that

U�1 = UT=

0BBBBB@

U1

1 U2

1

U1

2 U2

2

1CCCCCA .

Now we compute

(⌧1(U) f )(z1, z2) = f (⌧(U)�1(z1, z2)) = f (U1

1z1 + U2

1z2, U1

2z1 + U2

2z2)

= ⇠1(U1

1z1 + U2

1z2) + ⇠2(U1

2z1 + U2

2z2)

= (U1

1⇠1 + U

12⇠

2)z1 + (U2

1⇠1 + U

22⇠

2)z2

= ⇠AzA,

where the transformed coordinates ⇠A are given by

⇠1

⇠2

!=

0BBBBB@

U1

1 U1

2

U2

1 U2

2

1CCCCCA

⇠1

⇠2

!.

This is just the conjugation representation of SU(2) on C2. As we have seen inExercise 1.2.6 (8), this is unitarily equivalent to the standard representation

⇠1

⇠2

!=

U1

1 U12

U21 U2

2

! ⇠1

⇠2

!

of SU(2) on C2 and it will simplify the notation if we deal with this equivalentrepresentation instead. We would now like to write this as

⇠A = UAB ⇠

B, A = 1, 2,

where B is summed over B = 1, 2.Now notice that a typical element of V2 is a homogeneous polynomial a0z2

1 +a1z1z2 + a2z2

2 of degree 2 with complex coe�cients and that this can be written inthe form

2X

A1,A2=1

⇠A1A2 zA1 zA2 ,


where the coe�cients ⇠A1A2 2 C are symmetric in A1 and A2 (specifically, ⇠11 =a0, ⇠22 = a2, ⇠12 = ⇠21 = a1/2). With the summation convention this is just

⇠A1A2 zA1 zA2 .

Exercise 1.2.7. Show that the action of ⌧2 on V2, when thought of as a transforma-tion of the coe�cients, is unitarily equivalent to the representation of SU(2) on thesubspace of C4 consisting of all 4-tuples (⇠A1A2 )A1,A2=1,2 that are symmetric in A1A2and given by

⇠A1A2 = UA1B1 UA2

B2⇠B1B2 ,

where B1, B2 are summed over B1, B2 = 1, 2.

In general, the elements of V j can be written in the form

⇠A1A2···A j zA1 zA2 · · · zAj ,

where the coe�cients ⇠A1A2···Aj 2 C, A1, A2, . . . , Aj = 1, 2, are symmetric under allpermutations of A1A2 . . . , Aj. The action of ⌧ j on V j, when thought of as a transfor-mation of the coe�cients, is unitarily equivalent to the representation of SU(2) onthe subspace of C2 j consisting of all 2 j-tuples (⇠A1A2···Aj )A1,...,Aj=1,2 that are invariantunder all permutations of A1A2 · · · Aj and given by

⇠A1A2···Aj = UA1B1 UA2

B2 · · ·UAjBj ⇠

B1B2···Bj , A1, A2, . . . , Aj = 1, 2, (1.3)

where B1, B2, . . . , Bj are summed over B1, B2, . . . , Bj = 1, 2.

Exercise 1.2.8. Show that when j = 2 the transformation law (1.3) can be writtenas 0

BBBBBBBBBBBBB@

⇠11

⇠12

⇠21

⇠22

1CCCCCCCCCCCCCA=

U1

1 U12

U21 U2

2

!⌦

U1

1 U12

U21 U2

2

!0BBBBBBBBBBBB@

⇠11

⇠12

⇠21

⇠22

1CCCCCCCCCCCCA,

where ⇠21 = ⇠12 and ⌦ means the matrix tensor product. Now generalize.

The transformation law (1.3) describes an irreducible, unitary representation ofSU(2). It is called the spinor representation of SU(2) of weight j

2 and is denotedD( j/2). The carriers of the representation are the 2 j-tuples (⇠A1A2···Aj )A1,...,A j=1,2 in C2 j

that are invariant under all permutations of A1A2 · · · Aj and are called the compo-nents of a (contravariant) SU(2)-spinor of rank j. . The dimension of the spaceC(D( j/2)) of carriers (that is, the dimension of the representation) is j + 1.

Every irreducible, unitary representation of SU(2) is unitarily equivalent to oneand only one of the spinor representations D( j/2), j = 0, 1, 2, . . ..


Remark 1.2.4. It is common to denote the spinor representations D(s), where s = j/2is in {0, 1

2 , 1,32 . . .} and refer to s as the spin of the representation.

Remark 1.2.5. One can think of an SU(2)-spinor intuitively in a way that is entirelyanalogous to the old fashioned way of thinking about a tensor. One has assigned toevery orthonormal basis forC2 a set of 2 j complex components ⇠A1···Aj , A1, . . . , Aj =1, 2, totally symmetric in A1 . . . Aj. If two such bases are related by U = (UA

B)A,B=1,2in SU(2), then the corresponding components are related by the transformation law(1.3). It is not unreasonable to ask if such things are of any use to physics. Now, oneis familiar with a great many physical quantities that are represented mathematicallyby tensors of various types (mass, velocity, momentum, elastic stress and so on) andthese tensors are simply the carriers of various representations of SO(3). But noticethat when j is even, that is, when the representation D( j/2) of SU(2) has integralspin, D( j/2)(�U) = D( j/2)(U) for every U 2 SU(2). We will see in Section 2.4 thatSU(2) is the universal double cover of the rotation group SO(3) and will concludefrom this that integral spin representations of SU(2) descend to representations ofSO(3) and so their carriers can be identified with tensors. Half-integral spin repre-sentations of SU(2) do not descend to representations of SO(3) so their carriers arenot tensors and represent something new. That there are, indeed, physical quantitiesthat require such spinors for their mathematical description and that these quanti-ties have unexpected and counterintuitive properties is the topic of Appendix B of[Nab4].

Remark 1.2.6. If U is assumed to be only in SL(2,C) rather than the subgroupSU(2), then (1.3) still determines an irreducible, unitary representation of SL(2,C)and the carriers are called the components of a (contravariant, undotted) SL(2,C)-spinor. In this case, however, these do not exhaust all of the irreducible, unitaryrepresentations of SL(2,C). The representations of SL(2,C) are discussed in muchmore detail in Chapter 3 and Appendix B of [Nab4].

1.3 Projective Representations

Let H be a complex, separable Hilbert space. For each v 2 H � {0}

[v] = Cv =��v : � 2 C

is the ray in H containing v. The projectivization of H is the set of all such rays andis denoted

P(H) =�[v] : v 2 H � {0} .

The map

1.3 Projective Representations 25

⇡P : H � {0}! P(H)⇡P(v) = [v]

is a surjection and we provideP(H) with the quotient topology it determines. Statedotherwise, P(H) is the quotient space of H � {0} by the equivalence relation ⇠defined by v1 ⇠ v2 if and only if v2 = �v1 for some � 2 C. The topology on P(H) isHausdor↵ and ⇡P is an open mapping. There are two other useful ways of viewingP(H).

1. Let S (H) =�e 2 H : kekH = 1

be the unit sphere in H with its relative topology.

Define an equivalence relation ⇠ on S (H) by e1 ⇠ e2 if and only if e2 = �e1 forsome � 2 C with |�| = 1. Then P(H) is homeomorphic to the quotient spaceS (H)/ ⇠ of S (H) by ⇠. In particular, one can think of P(H) as the space of unitrays [e] = {�e : � 2 C, |�| = 1} in H.

2. Let End(H) be the Banach space of all bounded linear operators on H withthe operator norm. For each v 2 H � {0}, let ⇡v 2 End(H) be the orthogonalprojection of H onto [v]. Then v 7! ⇡v is a continuous map of H�{0} into End(H)which depends only on [v] and therefore induces a continuous map of P(H) intoEnd(H). If the image of this map is given the relative topology from End(H),then this is a homeomorphism of P(H) onto the image so we can identify

P(H) =�⇡v 2 End(H) : v 2 H � {0} .

At this point we know only that P(H) is a Hausdor↵ topological space, but it is,in fact, a topological Hilbert manifold and, if H is infinite-dimensional, it is locallyhomeomorphic to H. One sees this in the following way. Let e be a unit vector inH and e? its orthogonal complement in H. Then e? is a closed linear subspace ofH and consequently it is a Hilbert space in its own right so P(e?) is defined. Theinclusion e? ,! H induces a continuous embedding ofP(e?) intoP(H) and we willidentify P(e?) with its image. Thus, Ue = P(H) � P(e?) is an open neighborhoodof ⇡P(e) in P(H) which we claim is homeomorphic to e? (and this is isometricallyisomorphic to H if H is infinite-dimensional). Each element of Ue is a ray [v] forwhich he, viH , 0 and we can define �([v]) 2 e? by

�([v]) =v � he, viHehe, viH

.

Note that �([v]) clearly depends only on [v] and is, indeed, in e?.

Exercise 1.3.1. Show that � is bijective with inverse given by

��1(w) = [e + w]

for every w 2 e?.

Both � and ��1 are continuous so � is a homeomorphism. Since the unit vector ewas arbitrary the result follows.


Next we need to introduce the automorphism group Aut(P(H)) of P(H). Moti-vated by the physical interpretation in quantum mechanics (see page 235 of [Nab5])we define, for any v,w 2 H � {0}, the transition probability ([v], [w]) from state [v]to state [w] by

([v], [w]) = ([v], [w])P(H) =| hv,wiH |2kvk2

Hkwk2

H

.

Notice that the right-hand side is independent of the representatives v and w cho-sen for the rays. Let H1 and H2 be two complex, separable Hilbert spaces. A mapT : P(H1) ! P(H2) is an isomorphism of their projectivizations if it is a home-omorphism and preserves transition probabilities in the sense that ([v], [w])P(H1) =(T ([v]),T ([w]))P(H2) for all v,w 2 H1 � {0}. An automorphism of P(H) is an iso-morphism of P(H) onto itself. The set of all such is denoted Aut(P(H)) and is agroup under composition. We will supply Aut(P(H)) with a topology shortly, butfirst we need to describe an important result due to Wigner [Wig1]. For this we recallthat a map U : H1 ! H2 is anti-unitary if it is additive (U(v+w) = Uv+Uw for allv,w 2 H1), anti-linear (U(�v) = �U(v) for all � 2 C and all v 2 H1), and satisfieshUv,Uwi2 = hv,wi1 for all v,w 2 H1. A map U that is either unitary or anti-unitaryinduces an isomorphism TU on the projectivizations via TU([v]) = [U(v)]. Wigner’sTheorem asserts that every isomorphism T : P(H1) ! P(H2) arises in this wayfrom a map from H1 to H2 that is either unitary or anti-unitary. In fact, the resultis more general in that it does not require the topological assumptions. For a proofof the following result one can consult [Barg]; [VDB] contains a shorter proof, butone that assumes continuity.

Remark 1.3.1. Wigner defined a symmetry of the quantum system whose Hilbertspace is H to be a bijection T : P(H) ! P(H) that preserves transition probabil-ities. Note that no continuity assumption is made. The following result implies, inparticular, that every symmetry of a quantum system arises from an operator on Hthat is either unitary or anti-unitary. One gets continuity (and much more) for free.We will return to the discussion of symmetries and their physical significance inSection 2.7.

Theorem 1.3.1. (Wigner’s Theorem on Symmetries) Let H1 and H2 be complex,separable Hilbert spaces and T : P(H1)! P(H2) a mapping ofP(H1) intoP(H2)satisfying

(T ([v]),T ([w]))P(H2) = ([v], [w])P(H1)

for all [v], [w] 2 P(H1). Then there exists a mapping U : H1 ! H2 satisfying

1. U(v) 2 T ([v]) for all v 2 H1 � {0},

2. U(v + w) = U(v) + U(w) for all v,w 2 H1, and

1.3 Projective Representations 27

3. either

a. U(�v) = �U(v) and hU(v),U(w)iH2 = hv,wiH1 , or

b. U(�v) = �U(v) and hU(v),U(w)iH2 = hv,wiH1

for all � 2 C and all v,w 2 H1.

Furthermore, if H1 and H2 are of dimension at least 2 and T ([e1]) = T ([e2]), wheree1 and e2 are unit vectors in H1, then there is a unique such mapping U : H1 ! H2for which U(e1) = U(e2).

In particular, every automorphism T of P(H) arises from an operator U on H thatis either unitary or anti-unitary via T ([v]) = [Uv].

We denote by U(H) the group of unitary operators on H with the strong operatortopology. The set U�(H) of anti-unitary operators, also with the strong operatortopology, is not a group, but the product (composition) of two anti-unitary operatorsis unitary so the disjoint union U(H) t U�(H) is a group under composition. Withthe strong operator topology, U(H) t U�(H) is a Hausdor↵ topological group andU(H) is a closed and open subgroup (the component containing the identity operatorI). We identify the group S 1 of complex numbers of modulus one with a subgroupof U(H) t U�(H) via the map z 7! zI for z 2 S 1. The following result gives anexplicit description of Aut(P(H)) and is the tool we need to provide Aut(P(H))with a natural topology.

Proposition 1.3.2. Let H be a complex, separable Hilbert space of dimension atleast two. Then the map

Q : U(H) t U�(H)! Aut(P(H))

defined by

Q(U)([v]) = U([v]) = C(Uv)

is a surjective group homomorphism with kernel S 1. Consequently, Aut(P(H)) isisomorphic as a group to the quotient [U(H) t U�(H)]/S 1.

Proof. Q is a homomorphism because

Q(U1U2)([v]) = [U1U2)(v)] = [U1(U2v)] = Q(U1)([U2v])= Q(U1)(Q(U2)([v])) = Q(U1) � Q(U2)([v]).

Q is surjective by Wigner’s Theorem 1.3.1. The kernel of Q contains S 1 because,for every z 2 S 1,

Q(zI)([v]) = [(zI)(v)] = [zv] = [v].

Finally, suppose that U is in the kernel of Q. Then Q(U) = idP(H). Now select anarbitrary unit vector e in H. Then [Ue] = Q(U)([e]) = [e]. Since U is either unitary


or anti-unitary, Ue is also a unit vector so there is a unique z 2 C with |z| = 1 suchthat Ue = ze, that is, z�1Ue = e. Consequently,

Q(z�1U)([e]) = [(z�1U)e] = [e].

Notice also that

Q(I)([e]) = [Ie] = [e].

The uniqueness assertion in Wigner’s Theorem then implies that z�1U = I so U = zIas required. ut

Identifying Aut(P(H)) with [U(H)tU�(H)]/S 1 we provide it with the quotienttopology determined by Q and one can show that it thereby acquires the structure ofa Hausdor↵ topological group.

Now let G be an arbitrary Hausdor↵ topological group. A projective represen-tation of G on the complex, separable Hilbert space H is a continuous group ho-momorphism ⇢ : G ! Aut(P(H)) of G into the automorphism group Aut(P(H)).Two projective representation ⇢ j : G ! Aut(P(H j)), j = 1, 2, of G are equiv-alent if there is an isomorphism T : P(H1) ! P(H2) of P(H1) onto P(H2)such that ⇢2(g) � T = T � ⇢1(g) for every g 2 G. Notice that any representation⇢ : G ! U(H) t U�(H) of G into U(H) t U�(H) gives rise to projective represen-tation of G by composing with Q

⇢ = Q � ⇢.

It is not the case, however, that every projective representation arises in this way. Wewill say that ⇢ : G ! Aut(P(H)) lifts if there exists a continuous homomorphism⇢ : G ! U(H) t U�(H) for which ⇢ = Q � ⇢; ⇢ is then called a lift of ⇢. As withany lifting problem for continuous maps the existence of such lifts is a topologicalissue. For connected, simply connected Lie groups G the obstruction is the secondLie algebra cohomology H2(g;R).

Remark 1.3.2. Lie algebra cohomology can be introduced in a variety of ways.A very thorough discussion together with a number of applications to physics isavailable in [deAI]. We will not pursue this here since our interest in the generalresult is limited to the fact that it implies a theorem of Bargmann (Theorem 2.7.1)that we will state, but not prove in Section 2.7. Nevertheless, we will, for the record,formulate the result precisely. The following is Theorem 4, Section 2, of [Simms]and Corollary 3.12 of [VDB].

Theorem 1.3.3. Let G be a connected, simply connected Lie group whose Lie alge-bra g satisfies H2(g;R) = 0 and let H be a complex, separable Hilbert space. Thenevery projective representation ⇢ : G ! Aut(P(H)) of G on P(H) lifts to a unitaryrepresentation ⇢ : G ! U(H) of G on H.

1.4 Induced Representations 29

Notice that, under the circumstances described in the Theorem, the lift ⇢ actuallymaps into the unitary subgroup U(H) of U(H) t U�(H).

1.4 Induced Representations

Let G be a Lie group, H a closed subgroup of G and ⇡ : G ! G/H the naturalprojection of G onto the set of left cosets of H in G. We supply G/H with the di↵er-entiable structure described in Theorem 1.1.5. Now notice that right multiplicationby elements of H defines a smooth right action of H on G that preserves the cosetsof H, that is, satisfies

⇡(g · h) = ⇡(gh) = [gh] = (gh)H = gH = [g] = ⇡(g)

for all g 2 G and all h 2 H.To put this in the proper context we will need to recall a few basic facts about

principal bundles and their associated vector bundles. The most authoritative sourcefor this material is [KN1], but one might also consult Chapter 4, Section 5.4, andSection 6.7 of [Nab2].

Let P and X be smooth manifolds, ⇡ : P! X a smooth map of P onto X, and H aLie group. We say that ⇡ : P! X has the structure of a smooth, principal H-bundleif there is a smooth right action (p, h) 2 P ⇥ H 7! p · h 2 P of H on P such that

1. ⇡(p · h) = ⇡(p) for all p 2 P and all h 2 H.2. (Local Triviality) For each x0 2 X there exists an open neighborhood U of x0 in

X and a di↵eomorphism : ⇡�1(U)! U ⇥ H of the form

(p) = (⇡(p), (p)),

where : ⇡�1(U)! H satisfies

(p · h) = (p)h

for all p 2 ⇡�1(U) and all h 2 H.

P is the bundle space or total space, X is the base space, H is the structure groupand ⇡ is the projection of the principal H-bundle ⇡ : P ! X. It follows from lo-cal triviality that, for each x 2 X, the fiber ⇡�1(x) above x is a submanifold of Pdi↵eomorphic to H. One thinks of P as a family of copies of H parametrized bythe points of X and glued together topologically in such a way as to achieve localtriviality. For a given X and H there are generally many ways to do this gluing andit is sometimes possible to classify them (see Section 6.4 of [Nab3]). If, in the localtriviality condition, one can take U to be all of X, then the principal H-bundle is saidto be trivial. One can show that any principal H-bundle over any Rn is necessarilytrivial. If U is an open subset of X, then a smooth map s : U ! ⇡�1(U) ✓ P for


which ⇡ � s = idU is called a (local) section of the principal H-bundle ⇡ : P ! X.If U = X, then s is called a global section of ⇡ : P ! X. By local triviality, localsections always exist, but global sections exist if and only if the principal H-bundleis trivial. Generally, when the context makes the rest clear, it is common to refer to⇡ : P! X, or even just to P, simply as a principal bundle over X.

For the homogeneous manifold ⇡ : G ! G/H we take the right action of H onG to be right multiplication so that, as we have just seen, ⇡(g · h) = ⇡(g). The localtriviality condition follows from the existence of local sections (see Theorem 1.1.5).Indeed, if s : U ! ⇡�1(U) is such a section, then

⇡�1(U) =[

[g]2U

⇢s([g]) · h : h 2 H

�

and we can define

(s([g]) · h) =�⇡(s([g]) · h), h

�= ( [g], h ).

One shows that is a di↵eomorphism of ⇡�1(U) onto U ⇥ H so that ⇡ : G ! G/Hhas the structure of a smooth, principal H-bundle.

There is a standard procedure (described in detail in Section 6.7 of [Nab2]) forconstructing from a smooth, principal H-bundle ⇡ : P ! X and a manifold M onwhich H acts smoothly on the left a smooth “fiber bundle” in which each H-fiber isreplaced by a copy of M. In particular, if M is a finite-dimensional vector space Vand the left action of H on V is given by a smooth representation of H on V, then theresult is a smooth “vector bundle” over X. We will need an analogous constructionin which V is replaced by an infinite-dimensional, complex, separable Hilbert spaceH on which some strongly continuous, unitary representation � : H ! U(H) ofH acts. The procedure is the same as in the finite-dimensional case, but becausethe representation is only assumed continuous the end result, called a C0-Hilbertbundle, will live in the topological category. We will sketch the construction.

Let ⇡ : P! X be a smooth, principal H-bundle, H a complex, separable Hilbertspace, and � : H ! U(H) a strongly continuous, unitary representation of H on H.Write �(h)(v) = h · v for all h 2 H and all v 2 H. On the product space P⇥H definea continuous right action of H by

(p, v) · h = (p · h, h�1 · v). (1.4)

This action defines an equivalence relation on P⇥H. Specifically, (p1, v1) ⇠ (p2, v2)if and only if there exists an h 2 H such that (p2, v2) = (p1, v1) · h. The equivalenceclass of (p, v) is denoted [p, v] and the set of all such is denoted P⇥�H. The naturalprojection of P ⇥H onto P ⇥� H is denoted q� : P ⇥H ! P ⇥� H and defined byq�(p, v) = [p, v]. We provide P ⇥� H with the quotient topology determined by q�and call it the orbit space of the action (1.4). Next define

⇡� : P ⇥� H ! X

1.4 Induced Representations 31

by ⇡�([p, v]) = ⇡(p). This is well-defined because ⇡(p ·h) = ⇡(p), surjective because⇡ is surjective, and continuous because its composition with q� is continuous. Forany x 2 X the fiber of ⇡� above x is just ⇡�1

� (x) = {[p, v] : v 2 H}, where p isany fixed point in ⇡�1(x). Each of these fibers can be supplied with the structure ofa complex, separable Hilbert space isomorphic to H by defining [p, v1] + [p, v2] =[p, v1 + v2], a[p, v] = [p, av] and h [p, v1], [p, v2] i = hv1, v2iH. All of this is easyto check. It takes just a bit more work to show that ⇡� : P ⇥� H ! X satisfies a(topological) local triviality condition, that is, that for each x0 2 X there is an openneighborhood U of x0 in X and a homeomorphism � : ⇡�1

� (U)! U ⇥H of ⇡�1� (U)

onto U⇥H of the form �([p, v]) = ( ⇡�([p, v]), �([p, v]) ) = ( ⇡(p), �([p, v]) ), wherethe restriction �|⇡�1

� (x) of � to each fiber of ⇡� is a linear homeomorphism. To seewhere � comes from begin with a local trivialization : ⇡�1(U) ! U ⇥ H of theprincipal bundle ⇡ : P ! X. Let s : U ! ⇡�1(U) be the natural associated sectiondefined by s(x) = �1(x, e). Now define ��1 : U ⇥ H ! ⇡�1

� (U) by ��1(x, v) =[s(x), v]. The � we are looking for is the inverse of this.

With the structure we have just described ⇡� : P ⇥� H ! X is an example of aC0-Hilbert bundle over X. A (global) section of ⇡� : P ⇥� H ! X is a continuousmap s : X ! P ⇥� H with the property that ⇡� � s = idX . One can show (seeSection 6.8 of [Nab2] and Remark 1.4.1 below) that these sections are in one-to-onecorrespondence with continuous maps f : P ! H that are equivariant with respectto the given actions of H on P and H, that is, that satisfy

f (p · h) = h�1 · f (p) = �(h)�1( f (p)).

For the time being we will identify continuous sections of ⇡� : P ⇥� H ! X withthese continuous, equivariant, H-valued maps on P.

Remark 1.4.1. In Section 2.8 we will need to undo this identification and convertthe equivariant maps to sections so we will note for the record that the section s f :X ! P ⇥� H corresponding to the equivariant map f : P! H is given by

s f (x) = [p, f (p)],

where p is any point in ⇡�1(x).

With this review behind us we can return to the problem of describing inducedrepresentations. Thus, we let G be a Lie group, H a closed subgroup of G, and� : H ! U(H) a strongly continuous, irreducible, unitary representation of H onthe complex, separable Hilbert space H. We would like to construct from this dataa strongly continuous, irreducible, unitary representation of G on some complex,separable Hilbert space H�. We now know that ⇡ : G ! G/H is a principal H-bundle over G/H in which the right action of H on G is just right multiplication byelements of H. This, together with � : H ! U(H), then gives rise to an associatedC0-Hilbert bundle ⇡� : G ⇥� H ! G/H. The Hilbert space H� will be the spaceof L2-sections of ⇡� : G ⇥� H ! G/H. Naturally, this requires that we be able tointegrate over G/H. Now G, being a locally compact group, has a left-invariant Haar


measure on the Borel sets of G that is unique up to a normalizing factor. However,this measure generally does not give rise to a G-invariant measure on G/H and thisis what we need. It does give rise to a quasi-invariant measure on G/H, meaningthat the G-action on G/H preserves sets of measure zero, and it turns out that thisis enough. Nevertheless, in all of the examples of interest to us G/H will admita natural G-invariant measure so for the remainder of our sketch we will simplyassume that G/H admits a measure µ that is invariant under the G-action on G/H.

Now the construction of H� proceeds as follows. Begin with the set CE0 (G,H)

of continuous maps f : G ! H satisfying

1. f (gh) = �(h)�1( f (g)) 8g 2 G 8h 2 H, and2.

�[g] 2 G/H : g 2 supp( f )

is compact in G/H.

Notice that, since � is unitary, k f (gh)kH = k f (g)kH for all g 2 G and all h 2 Hso k f (g)kH depends only on [g]. We can therefore define a real-valued function onG/H that we will denote k f ([g])k by

k f ([g])k = k f (g)kH,

where g is any element of [g]. Integrating with respect to the G-invariant measure µon G/H we define

k f k =✓ Z

G/Hk f ([g])k2dµ([g])

◆ 12.

This is a norm on CE0 (G,H) and arises from the inner product

h f1, f2i =Z

G/Hh f1([g]), f2([g])idµ([g]),

where h f1([g]), f2([g])i = h f1(g), f2(g)iH; this also depends only on [g]. Since theelements of CE

0 (G,H) are continuous, it is not complete with respect to this norm sowe take H� to be its Hilbert space completion. The elements of this completion canbe identified with Borel measurable functions f : G ! H, modulo equality almosteverywhere with respect to Haar measure on G, that satisfy

1. f (gh) = �(h)�1( f (g)) for all h 2 H and almost all g 2 G, and

2.R

G/H k f ([g])k2dµ([g]) < 1.

We can now fulfill our stated objective. Thus, we let G be a Lie group, H aclosed subgroup of G, and � : H ! U(H) a strongly continuous, irreducible,unitary representation of H on the complex, separable Hilbert space H. Then therepresentation of G induced by H and � is denoted

IndGH(�) : G ! U(H�)

and acts by left translation on the elements of H�, that is,

1.5 Representations of Semi-Direct Products 33

⇥ ⇥IndG

H(�)(g)⇤f⇤(g0) = f (g�1g0).

Exercise 1.4.1. Show that IndGH(�) is a unitary representation of G on H�.

We are most interested in the induced representation when G is given as a semi-direct product and it is to this case that we will turn in the next section.

1.5 Representations of Semi-Direct Products

We begin with some algebraic generalities on semi-direct products of groups. Let Nand H be groups and ✓ : H ! Aut(N) a homomorphism of H into the automorphismgroup of N. Then ✓ determines a left action of H on N which we will write as

h · n = ✓(h)(n)

for all h 2 H and all n 2 N.

Exercise 1.5.1. Verify that

h1 · (h2 · n) = (h1h2) · n

and

h · (n1n2) = (h · n1)(h · n2)

for all h1, h2, h 2 H and all n1, n2, n 2 N.

The semi-direct product of N and H determined by ✓ is the group

G = N o✓ H

whose underlying set is N ⇥ H = {(n, h) : n 2 N, h 2 H} and in which the groupoperations are defined by

(n1, h1)(n2, h2) = (n1(h1 · n2), h1h2),1G = (1N , 1H),

(n, h)�1 = (h�1 · n�1, h�1).

Exercise 1.5.2. Verify the group axioms and show that the maps

n 7! (n, 1H) : N ! N o✓ H


and

h 7! (1N , h) : H ! N o✓ H

are embeddings so that we can (and will) identify N and H with subgroups of No✓H.Notice also that, when ✓ is the trivial homomorphism that sends everything to theidentity, the semi-direct product reduces to the usual direct product N ⇥ H of thegroups N and H.

Remark 1.5.1. When the action of H on N is clear from the context we will oftenomit the subscript ✓ and write N o✓ H simply as N o H.

Now notice that

(1N , h)(n, 1H)(1N , h)�1 = (1N(h · n), h1H)(h�1 · 1�1N , h

�1)

= (h · n, h)(1N , h�1)

= ((h · n)1N , hh�1)= (h · n, 1H)= (✓(h)(n), 1H).

If we identify N and H with subgroups of G = N o✓ H in the manner described inthe previous exercise this simply says that the action of H on N is by conjugation inG, that is,

h · n = hnh�1.

Exercise 1.5.3. Identify N and H with subgroups of G = N o✓ H and show that

1. N is a normal subgroup of G,2. NH = G, and3. N \ H = {1G}.Show also that the last two conditions imply that every element g of G can be writtenuniquely as g = nh with n 2 N and h 2 H.

Turning matters around let us suppose that G is an arbitrary group and N and Hare subgroups of G satisfying the conditions

1. N is a normal subgroup of G,2. NH = G, and3. N \ H = {1G}.Then G is said to be the internal semi-direct product of N and H. One can thendefine an action of H on N by conjugation, that is, define ✓ : H ! Aut(N) by✓(h)(n) = hnh�1.


Exercise 1.5.4. Show that, in this case, the map � : N o✓ H ! G defined by

�(n, h) = nh

is an isomorphism. Consequently, if G is the internal semi-direct product of thesubgroups N and H, then it is also the (external) semi-direct product of the groupsN and H.

The examples of most interest to us will be described in Section 2.4, but in orderto see how semi-direct products arise in practice and to ease our way back into thecategory of Lie groups we will look at the following simpler example.

Example 1.5.1. (The Inhomogeneous Rotation Group) Fix an element R of SO(3)and an a in R3. Define a mapping (a,R) : R3 ! R3 by

x 2 R3 ! (a,R)(x) = Rx + a 2 R3,

where Rx is the matrix product of R and the (column) vector x. Thus, (a,R) rotatesby R and then translates by a so it is an isometry of R3. The composition of twosuch mappings is given by

x! R2x + a2 ! R1(R2x + a2) + a1 = (R1R2)x + (R1a2 + a1).

Since R1R2 2 SO(3) and R1a2 + a1 2 R3, this composition is just

(a1,R1) � (a2,R2) = (R1a2 + a1,R1R2)

so this set of mappings is closed under composition. Moreover, (0, id3⇥3) is clearlyan identity element and every (a,R) has an inverse given by

(a,R)�1 = (�R�1a,R�1)

so this collection of maps forms a group under composition. This group is the semi-direct product of R3 and SO(3) corresponding to the natural action of SO(3) on R3.We will denote it ISO(3) and refer to it as the inhomogeneous rotation group. Itselements are di↵eomorphisms of R3 onto itself and we can think of it as defining agroup action on R3.

(a,R) · x = Rx + a

Notice that the maps a ! (a, id3⇥3) and R ! (0,R) identify R3 and SO(3) withsubgroups of ISO(3) and that R3 is a normal subgroup since it is the kernel of theprojection (a,R) ! (0,R) and this is a homomorphism (the projection onto R3 isnot a homomorphism).

We would like to find an explicit matrix model for ISO(3). For this we identifyR3 with the subset of R4 consisting of (column) vectors of the form


0BBBBBBBBBBBB@

x1

x2

x3

1

1CCCCCCCCCCCCA=

x1

!

where x = (x1 x2 x3)T 2 R3. Now consider the set G of 4 ⇥ 4 matrices of the form

R a0 1

!=

0BBBBBBBBBBBB@

R11 R1

2 R13 a1

R21 R2

2 R23 a2

R31 R3

2 R33 a3

0 0 0 1

1CCCCCCCCCCCCA,

where R 2 SO(3) and a 2 R3. Notice that

R a0 1

! x1

!=

Rx + a

1

!

and R1 a10 1

! R2 a20 1

!=

R1R2 R1a2 + a1

0 1

!

so we can identify ISO(3) with G and its action on R3 with matrix multiplication.G is a matrix Lie group of dimension 6. Its Lie algebra can be identified with

the set of 4 ⇥ 4 real matrices that arise as velocity vectors to curves in G throughthe identity with matrix commutator as bracket. We find a basis for this Lie algebra(otherwise called a set of generators) by noting that if

↵a(t) =

id3⇥3 ta0 1

!,

then↵0a(0) =

0 a0 0

!

and if↵N(t) =

etN 00 1

!,

then↵0N(0) =

N 00 0

!.

(see Theorem A.2.2 for N). Taking a = (1, 0, 0), (0, 1, 0), (0, 0, 1) and n =(1, 0, 0), (0, 1, 0), (0, 0, 1) (again, see Theorem A.2.2 for n) we obtain a set of sixgenerators for the Lie algebra iso(3) of ISO(3) that we will write as follows.

N1 =

0BBBBBBBBBBBB@

0 0 0 00 0 �1 00 1 0 00 0 0 0

1CCCCCCCCCCCCA,N2 =

0BBBBBBBBBBBB@

0 0 1 00 0 0 0�1 0 0 00 0 0 0

1CCCCCCCCCCCCA,N3 =

0BBBBBBBBBBBB@

0 �1 0 01 0 0 00 0 0 00 0 0 0

1CCCCCCCCCCCCA


P1 =

0BBBBBBBBBBBB@

0 0 0 10 0 0 00 0 0 00 0 0 0

1CCCCCCCCCCCCA, P2 =

0BBBBBBBBBBBB@

0 0 0 00 0 0 10 0 0 00 0 0 0

1CCCCCCCCCCCCA, P3 =

0BBBBBBBBBBBB@

0 0 0 00 0 0 00 0 0 10 0 0 0

1CCCCCCCCCCCCA

N1,N2, and N3 are called the generators of rotations, while P1, P2, and P3 are calledthe generators of translations.

We will write [A, B] = AB � BA for the matrix commutator. Using ✏i jk for theLevi-Civita symbol (1 if i j k is an even permutation of 1 2 3, �1 if i j k is an odd per-mutation of 1 2 3 and 0 otherwise) we record the following commutation relationsfor these generators, all of which can be verified by simply computing the matrixproducts.

[Ni,Nj] = ✏i jkNk, i, j = 1, 2, 3[Pi, Pj] = 0, i, j = 1, 2, 3[Ni, Pj] = ✏i jkPk, i, j = 1, 2, 3

The following result on semi-direct products of Lie groups is proved in ChapterIV, Section XV, of [Chev].

Theorem 1.5.1. Let N and H be Lie groups. Then, with the compact-open topology,Aut(N) is also a Lie group and so any continuous group homomorphism ✓ : H !Aut(N) is smooth. The corresponding semi-direct product No✓ H is also a Lie group.

Our final objective in this section is to apply the construction of induced rep-resentations (Section 1.4) to certain semi-direct products of Lie groups in order todescribe the so-called Mackey machine for manufacturing all of the irreducible, uni-tary representations of such groups. Our primary goal is to apply this machine to thePoincare group and its universal double cover in Section 2.8.

We will consider two Lie groups N and H, where N is assumed Abelian, anda continuous (and therefore smooth) homomorphism ✓ : H ! Aut(N) of H intoAut(N). As usual, we will write ✓(h)(n) = h · n. Then

G = N o✓ H

is also a Lie group. It will be convenient to identify both N and H with closedsubgroups of G so that N is a normal subgroup, G = NH, N \ H = {e}, and theaction of H on N is by conjugation

h · n = hnh�1.

An important role in the construction will be played by the irreducible, unitaryrepresentations of N. Since N is Abelian these are all 1-dimensional (Corollary


1.2.2) and are described by the characters of N. We will pause for a moment toprovide some background information.

Remark 1.5.2. In our present circumstances N is an Abelian Lie group, but the dis-cussion that follows requires only that N be a locally compact, Hausdor↵, Abeliantopological group. A character of N is a continuous homomorphism ⇠ : N ! S 1

from N to the group of complex numbers of modulus one. Each of these determines aunitary representation U⇠ of N on C defined by U⇠(n)(z) = ⇠(n)z. The set of all char-acters of N is denoted N. Under pointwise multiplication ((⇠1⇠2)(n) = ⇠1(n)⇠2(n)) Nis an Abelian group called the character group, or dual group of N. Regard N as asubset of the space C0(N, S 1) of continuous maps from N to S 1 with the compact-open topology and provide it with the subspace topology. N thereby becomes asecond countable, locally compact, Hausdor↵, Abelian topological group. The lo-cal compactness is not at all obvious (see page 89 of [Fol2]). If N1, . . . ,Nk are lo-cally compact, Abelian groups, then so is N1 ⇥ · · · ⇥ Nk and the character group ofN1 ⇥ · · ·Nk is isomorphic, as a topological group, to N1 ⇥ · · · ⇥ Nk (see Proposition4.6 of [Fol2]). An isomorphism from N1⇥ · · ·⇥ Nk to the dual group of N1⇥ · · ·⇥Nkis given by (⇠1, . . . , ⇠k) 7! ⇠, where

⇠(n1, . . . , nk) = ⇠1(n1) · · · ⇠k(nk).

Example 1.5.2. We will first find all of the characters of the additive group R andshow that R is isomorphic to R. Thus, we let ⇠ : R ! S 1 be a continuous grouphomomorphism so that ⇠(x1 + x2) = ⇠(x1)⇠(x2) and ⇠(0) = 1. Then there exists an✏ > 0 such that ⇠([�✏, ✏]) is contained in Re(z) > 0. Since p 2 [� ⇡

2✏ ,⇡2✏ ] ) p✏ 2

[� ⇡2 , ⇡2 ] there is a unique p 2 [� ⇡2✏ ,

⇡2✏ ] such that

⇠(✏) = eip✏ .

Now, ⇠(✏) = ⇠( ✏2 +✏2 ) = ⇠( ✏2 )2 and therefore

⇠✓ ✏2

◆= eip(✏/2)

because �eip(✏/2) does not have positive real part. Iterating gives

⇠✓ ✏2n

◆= eip(✏/2n), n = 0, 1, 2, . . .

Thus, for any k 2 Z,

⇠✓ k2n ✏

◆= eip(k/2n)✏ .

Since the set of all k2n ✏, k 2 Z, n = 0, 1, 2, . . . is dense in R and ⇠ is continuous,


⇠(x) = eipx 8x 2 R.

Exercise 1.5.5. Show that, for each ⇠ 2 R, there exists a unique p in R such that⇠(x) = eipx for every x 2 R. Hint: Repeat the argument above for any ✏0 with0 < ✏0 < ✏.

Consequently, the map � : R ! R defined by �(p) = eipx is a group isomor-phism. All that remains is to show that it is also a homeomorphism and thereforean isomorphism of topological groups. To show that � is continuous it is enough toprove continuity at the identity p = 0 in R. A basic open neighborhood of �(0) = 1in the compact-open topology on R can be described as follows. Let r be a posi-tive real number and n a positive integer. Define Ur = {z 2 C : |1 � z| < r} andKn = [�n, n] ✓ R. Then

U(Kn,Ur) = {⇠ 2 R : ⇠(Kn) ✓ Ur} = {eipx : |1 � eipx| < r 8 |x| n}

is a basic open neighborhood of 1 2 R. Since |1 � eipx|2 = 4 sin2� px2�,

��1(U(Kn,Ur)) =⇢

p 2 R : 4 sin2✓ px

2

◆< r2 8 |x| n

�

so p is in ��1(U(Kn,Ur)) if and only if |p| < 2n arcsin

� r2�

and this is an open neigh-borhood of 0 2 R. This proves the continuity of �.

Exercise 1.5.6. Show that ��1 : R! R is continuous.

We conclude that, as topological groups,

R � R.

via the map

p 2 R 7! eipx 2 R.

Applying the result on products quoted above we find that

cRk � R⇥ k· · · ⇥ R = Rk � Rk

and that any element of cRk can be written in the form

⇠(x1, . . . , xk) = ei(p1 x1+···+pk xk),

where (p1, . . . , pk) 2 Rk is unique.

Notice that if G = N o✓ H, then the action of H on N induces an action of H onN as follows. For any h 2 H and any ⇠ 2 N we define h · ⇠ 2 N by


(h · ⇠)(n) = ⇠(h�1 · n)8n 2 N.

Now fix some ⇠0 2 N and consider the orbit

O⇠0 = H · ⇠0 = {h · ⇠0 : h 2 H}

of ⇠0 under the H-action on N. Also let

H⇠0 = {h 2 H : h · ⇠0 = ⇠0}

be the isotropy subgroup of ⇠0 with respect to this H-action. This is a closed sub-group of H and therefore also a Lie group. Physicists refer to H⇠0 as the little groupat ⇠0. Notice that if ⇠00 is another point in O⇠0 with, say, ⇠00 = h0 ·⇠0, then the orbit of ⇠00coincides with the orbit of ⇠0 and the isotropy subgroups H⇠00

and H⇠0 are conjugate(H⇠00

= h0H⇠0 h�10 ) and therefore isomorphic as Lie groups. In particular, any unitary

representation � of H⇠0 is unitarily equivalent to the unitary representation �0 of H⇠00

defined by �0(h0hh�10 ) = �(h0)�(h)�(h0)�1 for h 2 H⇠0 . On a fixed orbit in N, the

little groups all have the same unitary representations, up to unitary equivalence.

Remark 1.5.3. There is a technical assumption we will need in order to get the fullforce of Mackey’s theorem on irreducible, unitary representations of G = N o✓ H.We will say that the semi-direct product G is regular if, for each ⇠0 2 N, the orbit O⇠0

is locally closed in N, meaning that for any ⇠ 2 O⇠0 there is an open neighborhoodV of ⇠ in N such that O⇠0 \ V is closed in V . Certainly this is the case if all of theorbits are closed in N, as they will be for the examples of most interest to us.

Now, suppose we are given a strongly continuous, irreducible, unitary represen-tation� of the little group H⇠0 on some separable, complex Hilbert space H. Assum-ing that H/H⇠0 admits an H-invariant measure (as it will in our examples) we canfollow the procedure described in Section 1.4 to produce an induced representation

IndHH⇠0

(�) : H ! U(H�)

of H on the Hilbert space H� of L2-sections of the associated C0-Hilbert bundle.For any n 2 N we let U(n) be the unitary multiplication operator on H� defined by

[U(n) f ](h) = [(h · ⇠0)(n)] f (h)

for all f 2 H� and h 2 H. Finally, recalling that any element g of G = N o� H canbe written uniquely as g = nh, where n 2 N and h 2 H we can define LO⇠0 ,�

on G by

LO⇠0 ,�(g) = LO⇠0 ,�

(nh) = U(n) � IndHH⇠0

(�)(h).

For each orbit O⇠0 of H in N and each strongly continuous, irreducible, unitary rep-resentation � of the little group H⇠0 on some separable, complex Hilbert space H,


Mackey proves that LO⇠0 ,�is a strongly continuous, irreducible, unitary represen-

tation of G = N o� H on H�. Indeed, much more is true. The following result ofMackey is quite deep and we will simply refer to Theorem 6.24 of [Vara] for thedetails.

Theorem 1.5.2. (Mackey’s Theorem) LO⇠0 ,�is a strongly continuous, irreducible,

unitary representation of G = N o� H on H�. If G is a regular semi-direct product,then every strongly continuous, irreducible, unitary representation of G is unitarilyequivalent to some LO⇠0 ,�

. Furthermore, LO⇠00,�0 is unitarily equivalent to LO⇠0 ,�

ifand only if O⇠00

= O⇠0 and �0 is unitarily equivalent to �.

Here then is the Mackey machine for computing all of the irreducible, unitaryrepresentations of the regular semi-direct product G = N o✓ H of two Lie groups Nand H when N is Abelian. Identify N and H with subgroups of G so that G = NHand the action of H on N is by conjugation.

1. Select an orbit O of the H-action on the character group N of N and an arbitrarypoint ⇠0 in O so that O = O⇠0 .

2. Select an H-invariant measure µ on H/H⇠0 , where H⇠0 is the isotropy subgroupof ⇠0 in H.

Note: In general, such a measure need not exist and it is not really necessary forthe operation of the Mackey machine, but for the examples of interest to us wewill explicitly construct them.

3. Select a strongly continuous, irreducible, unitary representation � : H⇠0 ! U(H)of the isotropy subgroup H⇠0 of ⇠0 on some complex, separable Hilbert space H.

4. Construct the C0-Hilbert bundle associated with H, H⇠0 , and µ as follows: ⇡ :H ! H/H⇠0 is an H⇠0 -principal bundle, where the right action of H⇠0 on H isright multiplication. This, together with the representation � : H⇠0 ! U(H)determines an associated C0-vector bundle

⇡� : H ⇥� H ! H/H⇠0

over H/H⇠0 with H-fibers.

5. The L2-sections of the vector bundle ⇡� : H ⇥� H ! H/H⇠0 with respect tothe measure µ form a Hilbert space H� that can be identified with the space ofBorel measurable functions f : H ! H, modulo equality almost everywherewith respect to Haar measure on H, satisfying


a. f (hh0) = �(h0)�1( f (h)) 8h0 2 H⇠0 and for almost every h 2 H, and

b.R

H/H⇠0k f ([h]) k2 dµ([h]) < 1, where

k f ([g])k = k f (g)kH,

and g is any element of [g].

6. The representation of H induced by H⇠0 and � is denoted

IndHH⇠0

(�) : H ! U(H�)

and acts by left translation on the elements of H�, that is,⇥ ⇥

IndHH⇠0

(�)(h)⇤f⇤(h0) = f (h�1h0).

7. For any n 2 N let U(n) be the unitary multiplication operator on H� defined by

[U(n) f ](h) = [(h · ⇠0)(n)] f (h) = ⇠0(h�1 · n) f (h)

for all f 2 H� and h 2 H.

8. Define LO,� on G as follows. Write g 2 G uniquely as g = nh, where n 2 N andh 2 H. Then

LO,�(g) = LO,�(nh) = U(n) � IndHH⇠0

(�)(h).

Then LO,� is a strongly continuous, irreducible, unitary representation of G. Upto unitary equivalence this representation is independent of the choice of ⇠0 2 O.LO0,�0 is unitarily equivalent to LO,� if and only if O0 = O and �0 is unitarily equiv-alent to �. Moreover, every strongly continuous, irreducible, unitary representationof G is unitarily equivalent to some LO,�.

Needless to say, the machine operates only when the isotropy groups H⇠0 aresu�ciently simple that one can find all of their representations. Fortunately, this isoften, although not always, the case. In particular, we will find in Section 2.8 thatMackey’s procedure yields an explicit description of all of the strongly continuous,irreducible, unitary representations of the universal cover of the Poincare group.These, in turn, give the objects of interest in relativistic quantum mechanics, that is,the projective representations of the Poincare group.

Chapter 2Minkowski Spacetime

2.1 Introduction

Quantum Field Theory (QFT) arose from attempts to reconcile quantum mechanicswith the special theory of relativity. In this chapter we will attempt to provide thebackground in relativity required to understand the challenges that such a reconcili-ation must confront and how the formalism of QFT proposes to deal with them. Wewill discuss only those aspects of special relativity that are directly relevant to thisobjective. Our primary source for this material and our primary reference for a morecomprehensive introduction to special relativity is [Nab4]. In particular, Section 2.2is essentially the Introduction to [Nab4].

2.2 Motivation

Minkowski spacetime is generally regarded as the appropriate arena within whichto formulate those laws of physics that do not refer specifically to gravitational phe-nomena. We would like to spend a moment here at the outset briefly examiningsome of the circumstances that give rise to this belief.

We shall adopt the point of view that the basic problem of science in general isthe description of “events” that occur in the physical universe and the analysis of re-lationships between these events. We use the term “event”, however, in the idealizedsense of a “point-event,” that is, a physical occurrence that has no spatial extensionand no duration in time. One might picture, for example, an instantaneous collisionor explosion, or an “instant” in the history of some material point particle or photon.In this way the existence of a point particle or photon can be represented by a con-tinuous sequence of events called its worldline. We begin then with an abstract setM whose elements we call events. We will provide M with a mathematical struc-ture that reflects certain simple facts of human experience as well as some rathernontrivial results of experimental physics.

43

44 2 Minkowski Spacetime

Events are “observed” and we will be particularly interested in a certain class ofobservers, called admissible, and the means they employ to describe events. Since itis in the nature of our perceptual apparatus that we identify events by their “locationin space and time,” we must specify the means by which an observer is to accomplishthis in order to be deemed admissible. We begin the process as follows.

Each admissible observer presides over a 3-dimensional, right-handed, Carte-sian spatial coordinate system based on an agreed unit of length and relative towhich photons propagate rectilinearly in any direction.

A few remarks are in order. First, the expression “presides over” is not to be takentoo literally. An observer is in no sense ubiquitous. Indeed, we generally picture theobserver as just another material particle residing at the origin of his spatial coordi-nate system; any information regarding events that occur at other locations must becommunicated to him by means we will consider shortly. Second, the restriction onthe propagation of photons is a real restriction. The term “straight line” has meaningonly relative to a given spatial coordinate system and if, in one such system, lightdoes indeed travel along straight lines, then it certainly will not in another systemwhich, say, rotates relative to the first. Notice, however, that this assumption doesnot preclude the possibility that two admissible coordinate systems are in relativemotion. We shall denote the spatial coordinate systems of observers O, O, . . . by⌃(x1, x2, x3), ⌃(x1, x2, x3), . . .

We take it as a fact of human experience that each observer has an innate, intuitivesense of “temporal order” that applies to events which he experiences directly, thatis, to events on his worldline. This sense, however, is not quantitative; there is noprecise, reliable sense of “equality” for “time intervals.” We remedy this situationby giving him a watch.

Each admissible observer is provided with an ideal, standard clock based on anagreed unit of time with which to provide a quantitative temporal order to the eventson his worldline.

Notice that thus far we have assumed only that an observer can assign a time toeach event on his worldline. In order for an observer to be able to assign times toarbitrary events we must specify a procedure for the placement and synchroniza-tion of clocks throughout his spatial coordinate system. One possibility is simply tomass produce clocks at the origin, synchronize them and then move them to vari-ous other points throughout the coordinate system. However, it has been found thatmoving clocks about has a most undesirable e↵ect upon them. Two identical andvery accurate atomic clocks are manufactured in New York and synchronized. Oneis placed aboard a passenger jet and flown around the world. Upon returning to NewYork it is found that the two clocks, although they still “tick” at the same rate, are nolonger synchronized. The traveling clock lags behind its stay-at-home twin. Strange,indeed, but it is a fact and we shall come to understand the reason for it shortly.

2.2 Motivation 45

Remark 2.2.1. I didn’t make this up. The experiment was first performed by J.C.Hafele and R.E. Keating in 1971 (see [HK]).

To avoid this di�culty we will ask our admissible observers to build their clocksat the origins of their coordinate systems, transport them to the desired locations,set them down and return to the master clock at the origin. We assume that eachobserver has stationed an assistant at the location of each transported clock. Nowour observer must “communicate” with each assistant, telling him the time at whichhis clock should be set in order that it be synchronized with the clock at the ori-gin. As a means of communication we choose a signal which seems, among all thepossible choices, to be least susceptible to annoying fluctuations in reliability, thatis, light signals. To persuade the reader that this is an appropriate choice we willrecord some of the experimentally documented properties of light signals. First,however, a little experiment. From his location at the origin O an observer O emitsa light signal at the instant his clock reads t0. The signal is reflected back to himat a point P and arrives again at O at the instant t1. Assuming there is no delay atP when the signal is bounced back, O will calculate the speed of the signal to bedistance(O, P)/ 1

2 (t1 � t0), where distance(O, P) is computed from the Cartesian co-ordinates of P in ⌃(x1, x2, x3). This technique for measuring the speed of light wecall the Fizeau procedure in honor of the gentleman who first carried it out withcare.

Remark 2.2.2. Notice that we must bounce the signal back to O since we do not yethave a clock at P that is synchronized with that at O.

For each admissible observer the speed of light in vacuo as determined by theFizeau procedure is independent of when the experiment is performed, the arrange-ment of the apparatus (that is, the choice of P), the frequency (energy) of the signaland, moreover, has the same numerical value c (approximately 3.0⇥ 108 meters persecond) for all such observers.

Here we have the conclusions of numerous experiments performed over theyears, most notably those first performed by Michelson-Morley and Kennedy-Thorndike (see Exercises 33 and 34 of [TW] for a discussion of these experiments).The results may seem odd. Why is a photon so unlike an electron (or a baseball)whose speed certainly will not have the same numerical value for two observers inrelative motion? Nevertheless, they are incontestable facts of nature and we mustdeal with them. We will exploit these rather remarkable properties of light signalsimmediately by asking all of our observers to multiply each of their time readingsby the constant c and thereby measure time in units of distance (light travel time).For example, one meter of time is the amount of time required by a light signal totravel one meter in vacuo. With these units, all speeds are dimensionless and c = 1.Such time readings for observers O, O, . . . will be denoted x0(= ct ), x0(= ct ), . . . .

Now we provide each of our admissible observers with a system of synchronizedclocks in the following way. At each point P of his spatial coordinate system, place


a clock identical to that at the origin. At some time x0 at O emit a spherical elec-tromagnetic wave (photons in all directions). As the wavefront encounters P set theclock placed there at time x0 + distance(O, P) and set it ticking, thus synchronizedwith the clock at the origin.

At this point each of our observers O, O, . . . has established a frame of referenceS(x↵) = S(x0, x1, x2, x3), S(x↵) = S(x0, x1, x2, x3), . . .. A useful intuitive visualiza-tion of such a reference frame is a lattice work of spatial coordinate lines with, ateach lattice point, a clock and an assistant whose task it is to record locations andtimes for events occurring in his immediate vicinity; the data can later be collectedfor analysis by the observer.

How are the S-coordinates of an event related to the S-coordinates? That is,what can be said about the mapping F : R4 ! R4 defined by F(x0, x1, x2, x3) =(x0, x1, x2, x3)? Certainly, it must be one-to-one and onto. Indeed, F�1 : R4 ! R4

must be the coordinate transformation from hatted to unhatted coordinates. To saymore we require a seemingly innocuous Causality Assumption.

Any two admissible observers agree on the temporal order of any two events onthe worldline of a photon, that is, if two such events have coordinates (x0, x1, x2, x3)and (x0

0, x10, x

20, x

30) in S and (x0, x1, x2, x3) and (x0

0, x10, x

20, x

30) in S, then �x0 = x0� x0

0and �x0 = x0 � x0

0 have the same sign.

Notice that we do not prejudge the issue by assuming that �x0 and �x0 are equal,but only that they have the same sign, that is, that O and O agree as to which ofthe events occurred first. Thus, F preserves order in the 0th-coordinate, at least forevents that lie on the worldline of a photon. How are two such events related? Sincephotons propagate rectilinearly with speed 1 in any admissible frame of reference,two events on the worldline of a photon must have coordinates in S that satisfy

(x0 � x00)2 � (x1 � x1

0)2 � (x2 � x20)2 � (x3 � x3

0)2 = 0 (2.1)

and coordinates in S that satisfy

(x0 � x00)2 � (x1 � x1

0)2 � (x2 � x20)2 � (x3 � x3

0)2 = 0. (2.2)

Consequently, the coordinate transformation map F : R4 ! R4 must carry the cone(2.1) onto the cone (2.2) and satisfy

x0 > x00 , x0 > x0

0 (2.3)

on the cones. Furthermore, F�1 : R4 ! R4 carries (2.2) onto (2.1) and satisfies(2.3). In 1964, Zeeman [Z] called such a mapping F a causal automorphism andproved the remarkable fact that any causal automorphism is a composition of thefollowing three basic types.

1. Translations: x↵ = x↵ + a↵,↵ = 0, 1, 2, 3, for some constants a↵,↵ = 0, 1, 2, 3.2. Positive Scalar Multiples: x↵ = kx↵,↵ = 0, 1, 2, 3, for some positive constant k.

2.2 Motivation 47

3. Linear transformations

x↵ = ⇤↵�x�, ↵ = 0, 1, 2, 3, (2.4)

where the matrix ⇤ = (⇤↵�)↵,�=0,1,2,3 satisfies ⇤T⌘⇤ = ⌘, where T means trans-pose, ⌘ = (⌘↵�)↵,�=0,1,2,3 is the matrix

0BBBBBBBBBBBB@

1 0 0 00 �1 0 00 0 �1 00 0 0 �1

1CCCCCCCCCCCCA,

and ⇤00 � 1.

Remark 2.2.3. Notice that it is not even assumed at the outset that F is continuous(much less a�ne). A proof of Zeeman’s Theorem is available either in [Z] or inSection 1.6 of [Nab4]. We should point out that this result was actually proved in1950 by Alexandrov [Alex], but the paper was in Russian and, sadly, was not widelyknown in the West.

Since two frames of reference related by a mapping of type (2) di↵er only bya rather trivial and unnecessary change of scale, we shall banish them from fur-ther consideration. In some circumstances one can adopt a similar attitude towardmappings of type (1). The constants a↵,↵ = 0, 1, 2, 3, can be regarded as the S-coordinates of S’s spacetime origin and we may request that all of our observers co-operate to the extent that they select a common event to act as origin, in which casea↵ = 0,↵ = 0, 1, 2, 3. This is particularly useful when one is interested in purely ge-ometrical questions. On the other hand, in classical mechanics translations in spaceand time are important symmetries with important conservation laws (momentumand energy) so, when the issue is dynamics, translations will play a fundamentalrole (see Sections A.2 and A.3).

The (linear) coordinate transformations of type (3), which we will soon chris-ten “orthochronous Lorentz transformations”, contain all of the novel kinematicfeatures of special relativity (“time dilation”, “length contraction”, etc.). We willfind that these are precisely the linear maps that leave invariant the quadratic form(x0)2 � (x1)2 � (x2)2 � (x3)2 (analogous to orthogonal transformations of R3, whichleave invariant the usual squared length x2+ y2+ z2) and preserve “time orientation”in the sense described in our Causality Assumption. We will see that any such ⇤has determinant ±1. Those with det ⇤ = 1 are called “proper” and the collectionof all proper, orthochronous Lorentz transformations is denoted L"+. We will showthat L"+ forms a group under matrix multiplication (that is, composition). The groupgenerated by L"+ and the translations ofR4 in (1) is called the Poincare group and isdenoted P"+.

The mathematical structure that appears to be emerging for M is that of a 4-dimensional real vector space with a distinguished quadratic form and a group of


transformations that preserve the quadratic form. Admissible observers coordinatizeM and these coordinates are related by elements of the group.

With this we conclude our attempt at motivation for the definitions to follow inSection 2.3. There is, however, one more item on the agenda of our introductoryremarks. It is the cornerstone upon which the special theory of relativity is built.

The Relativity Principle : All admissible frames of reference are completelyequivalent for the formulation of the laws of physics.

You will object that this is rather vague and we will not dispute the point. Onecould try to be more precise about what “completely equivalent” means and what“laws of physics” we have in mind, but this would, in some sense, be wrongheaded.It is most profitable to think of the Relativity Principle primarily as a heuristic prin-ciple asserting that there are no “distinguished” admissible observers, that is, thatnone can claim to have a privileged view of the universe. In particular, no such ob-server can claim to be “at rest” while the others are moving; they are all simplyin relative motion. Admissible observers can disagree about some rather startlingthings (for example, whether or not two given events are “simultaneous”) and theRelativity Principle will prohibit us from preferring the judgement of one to anyof the others. Although we will not dwell on the experimental evidence in favor ofthe Relativity Principle, it should be observed that its roots lie in such common-place observations as the fact that a passenger in a (smooth, quiet) airplane travelingat constant groundspeed in a straight line cannot “feel” his motion relative to theearth; no physical e↵ects are apparent in the plane that would serve to distinguish itfrom the (quasi-) admissible frame rigidly attached to the earth.

Our task then is to study the geometry and physics of these “admissible framesof reference.” Before embarking on such a study, however, it is only fair to concedethat, in fact, no such thing exists. As with any intellectual construct with which weattempt to model the physical universe, the notion of an admissible frame of refer-ence is an idealization, a rather fanciful generalization of circumstances which, tosome degree of accuracy, we encounter in the world. In particular, it has been foundthat the existence of gravitational fields imposes severe restrictions on the “extent”(both in space and in time) of an admissible frame (for more on this see Section4.2 of [Nab4]). Knowing this we intentionally avoid the di�culty by restricting ourattention to situations in which the e↵ects of gravity are “negligible.”

2.3 Geometrical Structure of M

Minkowski spacetime is a 4-dimensional real vector space M on which is defined anondegenerate, symmetric, bilinear form

h , iM : M ⇥M! R

of index 3. This means that there exists a basis {e0, e1, e2, e3} for M with

2.3 Geometrical Structure of M 49

he↵, e�iM = ⌘↵� =

8>>>>><>>>>>:

0, if ↵ , �1, if ↵ = � = 0�1, if ↵ = � = 1, 2, 3

h , iM is called the Lorentz inner product or Minkowski inner product on M and anysuch basis is said to be M-orthonormal, or simply orthonormal.

Remark 2.3.1. Unless some confusion is likely to arise we will tend to write thisand any other inner product that is likely to arise simply as h , i .

The quadratic form corresponding to h , i is the map Q : M! R defined by

Q(x) = hx, xi

for all x 2M. The points in M are called events.

Remark 2.3.2. As is customary in linear algebra we will have no qualms aboutblurring the distinction between a “point” in M and a “vector” in M since the contextinvariably makes clear which geometrical picture we have in mind. If one has moralobjections to this the appropriate course is to define Minkowski spacetime as ana�ne space containing the “points” and distinguish it from the corresponding vectorspace of “displacement vectors” determined by two “points” (a “tip” and a “tail”).

An x 2 M is said to be spacelike, timelike, or null if Q(x) is < 0, > 0, or = 0,respectively. We introduce a 4 ⇥ 4 matrix ⌘ defined by

⌘ = (⌘↵�)↵,�=0,1,2,3 =

0BBBBBBBBBBBB@

1 0 0 00 �1 0 00 0 �1 00 0 0 �1

1CCCCCCCCCCCCA.

The inverse of ⌘ is the same matrix, but we will denote its entries by ⌘↵�.

⌘�1 = (⌘↵�)↵,�=0,1,2,3 =

0BBBBBBBBBBBB@

1 0 0 00 �1 0 00 0 �1 00 0 0 �1

1CCCCCCCCCCCCA

Writing, with the summation convention, x = x↵e↵ we define x� = ⌘↵�x↵ for � =0, 1, 2, 3. Then x↵ = ⌘↵�x� and

hx, yi = hx↵e↵, y�e�i = ⌘↵�x↵y� = x↵y↵ = x�y�.

The basic geometrical structure of M is discussed in considerable detail in[Nab4]. We will now summarize some of its most important features and then turnto a more detailed discussion of those items that we will specifically call upon in the


sequel. Two vectors x and y in M are said to be M-orthogonal, or simply orthogonalif hx, yi = 0. The following results are Theorem 1.2.1 and Corollary 1.3.2 of [Nab4].

Theorem 2.3.1. Two nonzero null vectors x and y in M are orthogonal if and onlyif they are parallel, that is, if and only if there is a t , 0 in R such that y = tx.

Theorem 2.3.2. If a nonzero vector in M is orthogonal to a timelike vector, then itmust be spacelike.

Theorem 2.3.2 follows from our next result which provides more detailed infor-mation; it is Theorem 1.3.1 of [Nab4].

Theorem 2.3.3. Suppose x is timelike and y is either timelike or null and nonzero.Let {e↵}↵=0,1,2,3 be an orthonormal basis for M with x = x↵e↵ and y = y↵e↵. Theneither

1. x0y0 > 0, in which case hx, yi > 0, or2. x0y0 < 0, in which case hx, yi < 0.

With this we can define an equivalence relation ⇠ on the collection T of all time-like vectors in M as follows. For x, y 2 T,

x ⇠ y, hx, yi > 0.

In this case we say that x and y have the same time orientation.. The equivalencerelation has precisely two equivalence classes. We arbitrarily select one of these,denote it T+, and call its elements future directed. The other equivalence class isdenoted T� and its elements are called past directed. T+ and T� are cones in M, thatis, if x and y are in T+ (respectively, T�) and if r is a positive real number, then rxand x + y are in T+ (respectively, T�). One can extend this distinction to nonzeronull vectors n by noting that hx, ni has the same sign for all x 2 T+. Thus, we cansay that n is future directed if hx, ni > 0 for all x 2 T+ and past directed otherwise.

For each x0 in M we define the time cone CT (x0), future time cone C+T (x0), andpast time cone C�T (x0) at x0 by

CT (x0) = {x 2M : x � x0 2 T}C+T (x0) = {x 2M : x � x0 2 T+}C�T (x0) = {x 2M : x � x0 2 T�}.

Similarly, the null cone CN(x0), future null cone C+N(x0), and past null cone C�N(x0)at x0 are given by


CN(x0) = {x 2M : x � x0 is null}C+N(x0) = {x 2M : x � x0 is null and future directed}C�N(x0) = {x 2M : x � x0 is null and past directed}.

There is no notion of time orientation for spacelike vectors and the set of spacelikevectors is not a cone so, at each x0 2M, one simply defines

E(x0) = {x 2M : x � x0 is spacelike}

and calls it elsewhere.With the choice of T+ we have provided M with what is called a time orientation.

Independently of this we will also select some orientation for the vector space M,that is, some equivalence class of ordered bases for M. Henceforth we will consideronly orthonormal bases {e0, e1, e2, e3} for M that are in the chosen orientation classand for which e0 is timelike and future directed. Such a basis is called an admissiblebasis for M. The coordinates (x0, x1, x2, x3) of an event x in M relative to such abasis are identified with the spatial (x1, x2, x3) and time (x0) coordinates of the eventprovided by some admissible observer (see Section 2.2). The choice of such a basisidentifies the vector space M with the vector space R4 and transfers the Lorentzinner product to R4. To emphasize the signature of the Lorentz inner product wewill write this copy of R4 as R1,3.

M � R1,3

Although the inner product is di↵erent, the topology and di↵erentiable structure ofR1,3 are taken to be the usual ones of R4.

Remark 2.3.3. It is not unreasonable to argue that the Euclidean topology for R1,3

does not make a great deal of physical sense since the Euclidean inner product has noinvariant physical interpretation for all admissible observers. Alternative topologieshave been proposed and one of them is discussed in some detail in Appendix A of[Nab4]. Even if one acquiesces to the choice of the Euclidean topology, the di↵eren-tiable structure is not determined. Deep results of Michael Freedman on topological4-manifolds and Simon Donaldson on smooth 4-manifolds combine to show thatR4 admits (many) non-di↵eomorphic di↵erentiable structures. This cannot occurfor any Rn of dimension n , 4.

The linear subspace spanned by e0 is called the time axis of the correspondingadmissible observer and its orthogonal complement

(e0)? = {x 2M : he0, xi = 0} = Span {e1, e2, e3}

is the observer’s space. We orient (e0)? by decreeing that {e1, e2, e3} is an orientedbasis. We will feel free to view Span {e0} and Span {e1, e2, e3} either in M or inR1,3.

If {e↵}↵=0,1,2,3 and {e↵}↵=0,1,2,3 are two orthonormal bases for M, then there existsa unique linear transformation L : M!M such that Le↵ = e↵ and this satisfies


hLx, Lyi = hx, yi

for all x, y 2M. Such a linear transformation is called an M-orthogonal transforma-tion, or simply an orthogonal transformation. If x 2M and if we write x = x↵e↵ andx = x↵e↵, then (x↵) and (x↵) are interpreted as the spacetime coordinates of the eventx in the two frames of reference corresponding to {e↵}↵=0,1,2,3 and {e↵}↵=0,1,2,3. Weare interested in the coordinate transformation relating these two sets of coordinates.For this we write

e� = ⇤↵�e↵, � = 0, 1, 2, 3

for some real numbers ⇤↵�,↵, � = 0, 1, 2, 3. Then the orthogonality conditionshe�, e�i = ⌘�� can be written

⇤↵�⇤��⌘↵� = ⌘��, �, � = 0, 1, 2, 3, (2.5)

or, equivalently,

⇤↵�⇤��⌘

�� = ⌘↵�, ↵, � = 0, 1, 2, 3. (2.6)

We introduce the 4 ⇥ 4 matrix

⇤ = (⇤↵�) =

0BBBBBBBBBBBB@

⇤00 ⇤0

1 ⇤02 ⇤0

3⇤1

0 ⇤11 ⇤1

2 ⇤13

⇤20 ⇤2

1 ⇤22 ⇤2

3⇤3

0 ⇤31 ⇤3

2 ⇤33

1CCCCCCCCCCCCA.

Then the orthogonality conditions (2.5) or, equivalently, (2.6) can be written

⇤T⌘⇤ = ⌘, (2.7)

where “T” means “transpose”. The coordinate transformation from unhatted to hat-ted coordinates is just the matrix product

0BBBBBBBBBBBB@

x0

x1

x2

x3

1CCCCCCCCCCCCA=

0BBBBBBBBBBBB@

⇤00 ⇤0

1 ⇤02 ⇤0

3⇤1

0 ⇤11 ⇤1

2 ⇤13

⇤20 ⇤2

1 ⇤22 ⇤2

3⇤3

0 ⇤31 ⇤3

2 ⇤33

1CCCCCCCCCCCCA

0BBBBBBBBBBBB@

x0

x1

x2

x3

1CCCCCCCCCCCCA

or, more concisely,

x↵ = ⇤↵�x�, ↵ = 0, 1, 2, 3. (2.8)

Any real, 4 ⇥ 4 matrix ⇤ satisfying (2.7) is called a (general) Lorentz transforma-tion. Any such ⇤ satisfies det ⇤ = ±1 and either ⇤0

0 � 1 or ⇤00 �1. To ensure

that the basis {e↵}↵=0,1,2,3 has the proper orientation we will consider only Lorentztransformations ⇤ that are proper, that is, satisfy


det ⇤ = 1.

To ensure that e0 is future directed we assume also that ⇤ is orthochronous, that is,satisfies

⇤00 � 1.

Remark 2.3.4. Orthochronous Lorentz transformations actually preserve the timeorientation (future directed or past directed) of all timelike and nonzero null vectors(see Theorem 1.3.3 of [Nab4]).

The set of all proper, orthochronous Lorentz transformations is denoted L"+.

Exercise 2.3.1. Show that L"+ is a group under matrix multiplication. Hint: Showthat any (general) Lorentz transformation ⇤ has an inverse given by

⇤�1 = ⌘⇤T⌘.

We will denote the entries of the matrix ⇤�1 by ⇤↵�, where ↵ labels the columnand � labels the row. Thus,

⇤�1 =

0BBBBBBBBBBBBB@

⇤00 ⇤1

0 ⇤20 ⇤3

0

⇤01 ⇤1

1 ⇤21 ⇤3

1

⇤02 ⇤1

2 ⇤22 ⇤3

2

⇤03 ⇤1

3 ⇤23 ⇤3

3

1CCCCCCCCCCCCCA=

0BBBBBBBBBBBB@

⇤00 �⇤1

0 �⇤20 �⇤3

0�⇤0

1 ⇤11 ⇤2

1 ⇤31

�⇤02 ⇤1

2 ⇤22 ⇤3

2�⇤0

3 ⇤13 ⇤2

3 ⇤33

1CCCCCCCCCCCCA

Exercise 2.3.2. Prove each of the following.

1. ⇤↵� = ⌘↵�⌘��⇤��, ↵, � = 0, 1, 2, 3

2. ⇤↵� = ⌘↵�⌘��⇤��, ↵, � = 0, 1, 2, 3

3. ⇤↵�⇤��⌘↵� = ⌘��, �, � = 0, 1, 2, 3

4. ⇤↵�⇤��⌘�� = ⌘↵�, ↵, � = 0, 1, 2, 3

Exercise 2.3.3. Show that, if ⇤ 2 L"+, then (⇤�1)T 2 L"+.

L"+ has an important subgroup R indexR consisting of those R = (R↵�)↵,�=0,1,2,3

of the form

R =

0BBBBBBBBBBBB@

1 0 0 00 R1

1 R12 R1

30 R2

1 R22 R2

30 R3

1 R32 R3

3

1CCCCCCCCCCCCA,


where (Rab)a,b=1,2,3 is an element of the rotation group SO(3). The coordinate trans-

formation associated with R corresponds physically to a rotation of the spatial coor-dinate axes within a given frame of reference. R is called the rotation subgroup ofL"+. The following is Lemma 1.3.4 of [Nab4].

Theorem 2.3.4. Let ⇤ = (⇤↵�)↵,�=0,1,2,3 be a proper, orthochronous Lorentz trans-formation. Then the following are equivalent.

1. ⇤ is a rotation in L"+.2. ⇤0

1 = ⇤02 = ⇤0

3 = 03. ⇤1

0 = ⇤20 = ⇤3

0 = 04. ⇤0

0 = 1

Exercise 2.3.4. Let µ and ⌫ be real numbers with ⌫ > 0 and ⌫2 � µ2 = 1. Show that

⇤µ,⌫ =

0BBBBBBBBBBBB@

⌫ µ 0 0µ ⌫ 0 00 0 1 00 0 0 1

1CCCCCCCCCCCCA

is in L"+.

Taking ⌫ = cosh ✓ and µ = sinh ✓ for some real number ✓ > 0 in Exercse 2.3.4one obtains what is called a boost in the x1-direction.

⇤1(✓) =

0BBBBBBBBBBBB@

cosh ✓ sinh ✓ 0 0sinh ✓ cosh ✓ 0 0

0 0 1 00 0 0 1

1CCCCCCCCCCCCA.

The physical interpretation of the corresponding coordinate transformation is dis-cussed in some detail on pages 21-27 of [Nab4]. It describes the relationship be-tween the spacetime coordinates in two admissible frames of reference S and Sfor which the spatial coordinate axes initially coincide and remain parallel, but forwhich those of S move in the positive direction along the common x1, x1-axis withspeed � = tanh ✓. One can define boosts in the x2- and x3-directions in an entirelyanalogous way. Their matrices are

⇤2(✓) =

0BBBBBBBBBBBB@

cosh ✓ 0 sinh ✓ 00 1 0 0

sinh ✓ 0 cosh ✓ 00 0 0 1

1CCCCCCCCCCCCA, ⇤3(✓) =

0BBBBBBBBBBBB@

cosh ✓ 0 0 sinh ✓0 1 0 00 0 1 0

sinh ✓ 0 0 cosh ✓

1CCCCCCCCCCCCA.


Remark 2.3.5. These boosts along the spatial coordinate axes of a given frame ofreference are generally called special Lorentz transformations and we have writtenthem in what is called hyperbolic form. They are more traditionally written in termsof the relative velocity � = tanh ✓ of the two frames and � = (1 � �2)�1/2 in whichcase cosh ✓ = � and sinh ✓ = ��.

Boosts in the x1-direction and rotations su�ce to describe all of the elements ofthe proper, orthochronous Lorentz group. The following is Theorem 1.3.5 of [Nab4].

Theorem 2.3.5. Let ⇤ be a proper, orthochronous Lorentz transformation. Thenthere exists a real number ✓ and two rotations R1 and R2 in R such that

⇤ = R1⇤1(✓)R2.

The Theorem suggests that the boosts ⇤1(✓) contain a great deal of the kinematicinformation contained in L"+ and this is, indeed, the case. All of the well-knownkinematic e↵ects of special relativity such as the relativity of simultaneity, time di-lation, length contraction, the relativistic addition of velocities formula, and thequite inappropriately named twin paradox are easily derived from the properties ofthe coordinate transformation corresponding to ⇤1(✓). This, however, is not reallyour business here so, for most of this, we will simply refer to the rather detaileddiscussions in [Nab4] (particularly pages 29-42). We will, however, need some ad-ditional information about timelike vectors and curves that is closely related thesephenomena and we will conclude this section with a brief synopsis of what we re-quire.

If v 2 M is a timelike vector we define its duration ⌧(v) by ⌧(v) =p

Q(v) =phv, vi. If v = x � x0 is the displacement vector between two events x and x0 in M,then it is always possible to find an admissible basis in which x0 and x occur at thesame spatial location, one after the other, and ⌧(x � x0) is interpreted physically asthe time separation of x and x0 in any such frame (see pages 42-43 of [Nab4]). Inthis case ⌧(x � x0) is called the proper time separation of x0 and x.

The signature of the Lorentz inner product reverses some of the familiar inequal-ities from Euclidean geometry. The following results are Theorems 1.4.1 and Theo-rem 1.4.2 of [Nab4].

Theorem 2.3.6. (Reversed Schwartz Inequality) If v and w are timelike vectors inM, then

hv,wi2 � hv, vihw,wi

and equality holds if and only if v and w are parallel.


Theorem 2.3.7. (Reversed Triangle Inequality) If v and w are timelike vectors withthe same time orientation (that is, hv,wi > 0), then v + w is timelike and

⌧(v + w) � ⌧(v) + ⌧(w).

Equality holds if and only if v and w are parallel.

Theorem 2.3.7 extends to any finite sum of timelike vectors all of which have thesame time orientation (Corollary 1.4.4 of [Nab4]).

Now suppose I is an interval in R and ↵ : I ! M is a smooth curve in M. Then↵ is said to be spacelike, timelike, or null, respectively, if its velocity vector ↵0(t),identified with a vector in M, is spacelike, timelike or null, respectively, for eacht 2 I.

Remark 2.3.6. Note that if I has endpoints, then “smooth” means that ↵ extends toan open interval on which it is C1.

A smooth timelike or null curve ↵ : I ! M in M is future directed (respectively,past directed) if ↵0(t) is future directed (respectively, past directed) for every t 2 I. Afuture directed timelike curve is also called a timelike worldline, or the worldline ofa material particle and is interpreted physically as the set of all events in the historyof some material particle.

If ↵ : [a, b] ! M is a timelike worldline joining ↵(a) and ↵(b), then we definethe proper time length L(↵) of ↵ by

L(↵) =Z b

ah↵0(t),↵0(t)i1/2 dt =

Z b

a⌧(↵0(t)) dt.

The physical interpretation of L(↵) is a bit more subtle than one might expect anddepends on additional physical input called the “Clock Hypothesis” (see pges 48-50 of [Nab4]). The generally accepted interpretation is that L(↵) is the time lapsebetween the events ↵(a) and ↵(b) as measured by an ideal standard clock carriedalong by the particle whose worldline is ↵. This is always less that or equal to theproper time separation of ↵(a) and ↵(b); the following result combines Theorem1.4.6 and Theorem 1.4.8 of [Nab4].

Theorem 2.3.8. (Twin Paradox) Let ↵ : [a, b] ! M be a timelike worldline from↵(a) to ↵(b). Then the displacement vector ↵(b)�↵(a) is timelike and future directedand

L(↵) ⌧(↵(b) � ↵(a)).

Equality holds if and only if ↵ is a parametrization of the straight line joining ↵(a)and ↵(b).

2.4 Lorentz and Poincare Groups 57

Exercise 2.3.5. Do a little outside reading and decide for yourself why we choose tocall Theorem 2.3.8 the Twin Paradox. Also decide for yourself if there is anythingparadoxical about it.

We now define, for any timelike worldline ↵ : [a, b] ! M, what is for mostpurposes its most convenient parametrization. The proper time function ⌧ = ⌧(t) on[a, b] is defined by

⌧ = ⌧(t) =Z t

a⌧(↵0(u)) du =

Z t

ah↵0(u),↵0(u)i1/2 du

for every t in [a, b]. Since ↵ is timelike, d⌧dt is smooth and positive so the inverse of

⌧ = ⌧(t) exists and is smooth with a positive derivative. We can therefore parametrize↵ by ⌧ and we will abuse the notation a bit and write this parametrization simply↵(⌧). Physically, we are parametrizing the timelike worldline by time readings actu-ally recorded along the worldline, assuming the time is set to zero at ↵(a). Relativeto any admissible basis we write

↵(⌧) = xµ(⌧)eµ.

The velocity vector

↵0(⌧) =dxµ

d⌧eµ

is called the 4-velocity of the timelike worldline and

↵00(⌧) =d2xµ

d⌧2 eµ

is its 4-acceleration.


1. h↵0(⌧),↵0(⌧)i = 1 for all ⌧ in [0, L(↵)].

2. h↵0(⌧),↵00(⌧)i = 0 for all ⌧ in [0, L(↵)].

Thus, the 4-velocity is a unit timelike vector and the 4-acceleration is spacelike ateach point.

2.4 Lorentz and Poincare Groups

In this section we will look a bit more closely at the groups that will be of particularinterest to us. The proper, orthochronous Lorentz group L"+ was introduced in the


previous section and consists of all 4 ⇥ 4 real matrices ⇤ = (⇤↵�)↵,�=0,1,2,3 satisfying⇤T⌘⇤ = ⌘, ⇤0

0 � 1 and det⇤ = 1. Physically, these are the coordinate transforma-tion matrices between two admissible frames of reference that agree on a commonevent as the spacetime origin. L"+ inherits a topology as a subspace of the real gen-eral linear group GL(4,R), which is an open subset of R42 . With this topology L"+is a Hausdor↵ topological group. Being a closed subgroup of GL(4,R), L"+ is, infact, a smooth submanifold and a Lie group (Theorem 1.1.2). In Section 2.6 we willshow that L"+ is di↵eomorphic to R3 ⇥ SO(3).

L"+ � R3 ⇥ SO(3)

In particular, L"+ is 6-dimensional, connected, and has fundamental group Z2 (forthe last statement see Appendix B of [Nab4]).

Admissible observers that do not share a common spacetime origin will assigncoordinates that di↵er by a translation and a Lorentz transformation. Specifically, ifS and S are two such frames of reference assigning coordinates x and x, respectively,then there exists an a 2 R1,3 and a ⇤ 2 L"+ such that x = a + ⇤x. The compositionof two such transformations is given by

x 7! a2 + ⇤2x 7! a1 + ⇤1(a2 + ⇤2x) = (a1 + ⇤1a2) + (⇤1⇤2)x

and this is a transformation of the same type with a = a1 + ⇤1a2 and ⇤ = ⇤1⇤2.Somewhat more formally we define, for each (a,⇤) 2 R1,3 ⇥ L"+ an a�ne mapping(a,⇤) : R1,3 ! R1,3 by

(a,⇤)(x) = a + ⇤x

for every x 2 R1,3. Then

(a1,⇤1) � (a2,⇤2) = (a1 + ⇤1a2,⇤1⇤2).

This we recognize (Section 1.5) as the multiplication for the semi-direct product ofR1,3 (regarded as an additive translation group) and L"+ determined by the naturalaction of L"+ on R1,3. We will suppress this natural action from the notation andwrite the semi-direct product simply as R1,3

o L"+. It is called the Poincare groupand is denoted

P"+ = R1,3o L"+.

P"+ has the topology and manifold structure of R1,3 ⇥ L"+ and is a 10-dimensional,connected Lie group with fundamental group ⇡1(P"+) � Z2. For the record we recallthat the identity element in P"+ is (0, id4⇥4) and that the inverse of any (a,⇤) in P"+ isgiven by

(a,⇤)�1 = (⇤�1(�a),⇤�1) = (�⇤�1a,⇤�1).


It will be convenient to have an explicit matrix model of P"+ and this is easilydone. Define a mapping � : P"+ ! GL(5,R) by

�(a,⇤) =

0BBBBBBBBBBBBBBBBB@

1 0 0 0 0a0 ⇤0

0 ⇤01 ⇤0

2 ⇤03

a1 ⇤10 ⇤1

1 ⇤12 ⇤1

3a2 ⇤2

0 ⇤21 ⇤2

2 ⇤23

a3 ⇤30 ⇤3

1 ⇤32 ⇤3

3

1CCCCCCCCCCCCCCCCCA=

1 0a ⇤

!

Now identify R1,3 with the subspace of R5 consisting of all

X =

0BBBBBBBBBBBBBBBBB@

1x0

x1

x2

x3


1x

!.

Then�(a,⇤)X =

1 0a ⇤

! 1x

!=

1

a + ⇤x

!

Exercise 2.4.1. Show that � is a Lie group isomorphism of P"+ onto its image inGL(5,R).

General topological considerations imply that, because the fundamental groupsof L"+ and P"+ are Z2, each of these groups has a “universal double covering group”.However, we will need explicit constructions of these so we will build them andexplain as we go along what “universal double covering group” means (also seepage 13). The construction depends on a rather remarkable reformulation of bothMinkowski spacetime and the Lorentz group which we now describe.

We begin by considering the real vector space H2 of 2 ⇥ 2, complex matrices Xthat are Hermitian (X

T= X). This is precisely the set of matrices that can be written

in the form

X =

x0 + x3 x1 � ix2

x1 + ix2 x0 � x3

!= x0�0 + x1�1 + x2�2 + x3�3 = x↵�↵,

where x0, x1, x2, and x3 are real numbers, �0 is the 2 ⇥ 2 identity matrix and �1,�2,and �3 are the Pauli spin matrices (see Exercise 1.2.6). Notice that

det X = det

x0 + x3 x1 � ix2

x1 + ix2 x0 � x3

!= (x0)2 � (x1)2 � (x2)2 � (x3)2.

Define an inner product on H2 by polarization of this quadratic form, that is,


hX,YiH2 =14

[det(X + Y) � det(X � Y)] = x0y0 � x1y1 � x2y2 � x3y3.

Finally, notice that the map x = (x0, x1, x2, x3) 2 R1,3 7! X == x0�0+ x1�1+ x2�2+x3�3 2 H2, which is clearly linear, has an inverse given by

x↵ =12

Trace (�↵X), ↵ = 0, 1, 2, 3,

We conclude that this map is an isomorphism ofR1,3 onto H2 that sends the Lorentzinner product to the H2-inner product so that we are free to fully identify R1,3 andH2.

Next we consider the Lie group SL(2,C) of 2 ⇥ 2 complex matrices with deter-minant one. For every A 2 SL(2,C) we define a mapping MA : H2 ! H2 by

MA(X) = AXAT

for every X 2 H2.

Exercise 2.4.2. Show that MA(X) is in H2 and det MA(X) = det X for every A 2SL(2,C) and every X 2 H2.

But then MA(X) can be uniquely written in the form

MA(X) =

x0 + x3 x1 � ix2

x1 + ix2 x0 � x3

!= x0�0 + x1�1 + x2�2 + x3�3 = x↵�↵,

where x0, x1, x2, and x3 are real numbers. Thus,

(x0)2 � (x1)2 � (x2)2 � (x3)2 = (x0)2 � (x1)2 � (x2)2 � (x3)2.

Consequently, the mapping x = (x↵)↵=0,1,2,3 7! x = (x↵)↵=0,1,2,3 defined by

x0 + x3 x1 � ix2

x1 + ix2 x0 � x3

!= A

x0 + x3 x1 � ix2

x1 + ix2 x0 � x3

!A

T,

which is clearly linear for each fixed A 2 SL(2,C), preserves the quadratic form⌘↵�x↵x� and therefore, by polarization, preserves the Lorentz inner product ⌘↵�x↵y�.The mapping is therefore a general Lorentz transformation which we will denote⇤A. Notice that ⇤�A = ⇤A for every A 2 SL(2,C). One can write out the matrix ⇤Aexplicitly in terms of the entries in A and we will do so in a moment, but for manypurposes it is more useful to note that

(⇤A)↵� =12

Trace��↵A��A

T �, ↵, � = 0, 1, 2, 3.

To see this we compute


12

Trace��↵A��A

T �x� =

12

Trace��↵A(x��)A

T �

=12

Trace��↵AXA

T �

=12

Trace��↵MA(X)

�

= x↵.

Notice that (⇤A)00 =

12 Trace

�AA

T �which is one-half the sum of the squared moduli

of the entries in A. This is positive so ⇤A is orthochronous. To see that ⇤A is properwe proceed as follows. In Exercise 2.6.3 you will show that SL(2,C) is di↵eomor-phic to R3 ⇥ SU(2).

SL(2,C) � R3 ⇥ SU(2)

Since SU(2) is homeomorphic to the 3-sphere S 3 (Theorem 1.1.4 of [Nab2]) andS 3 is connected and simply connected (pages 118-119 of [Nab2]), it follows thatSL(2,C) is connected and simply connected (see Theorem 2.4.10 of [Nab2]). Now,being Lorentz transformations, every ⇤A has det⇤A = ±1. But det⇤A is a continuousfunction of the entries in A so it must be constant on the connected space SL(2,C).When A is the 2 ⇥ 2 identity matrix, ⇤A is the 4 ⇥ 4 identity matrix and this hasdeterminant 1. Consequently, det⇤A = 1 for all A 2 SL(2,C). We conclude that⇤A 2 L"+ for every A 2 SL(2,C). Thus, we have a mapping

: SL(2,C)! L"+(A) = ⇤A

of SL(2,C) to L"+.Next we would like to show that is a group homomorphism, that is,

(BA) = (B)(A)

for all A, B 2 SL(2,C). What we want to show then is that ⇤BA = ⇤B⇤A and for thisit will be enough to show that, for any x = (x�)�=0,1,2,3 2 R1,3,

(⇤BA)��x� = (⇤B)�↵(⇤A)↵�x�.

First note that, as we showed above,

x↵ = (⇤A)↵�x� =12

Trace��↵AXA

T �=

12

Trace��↵MA(X)

�

and similarly

(⇤B)�↵ x↵ =12

Trace��MB(X)

�=

12

Trace��(MB � MA)(X)

�.


Finally,

(⇤BA)��x� =12

Trace��(BA)X(BA)T �

=12

Trace��B(AXA

T)B

T �

=12

Trace��(MB � MA)(X)

�

= (⇤B)�↵ x↵

= (⇤B)�↵(⇤A)↵�x�

as required.Next we will show that the map is surjective and precisely two-to-one. For this

it will be convenient to have in hand an explicit representation of ⇤A in terms of theentries

A =

a bc d

!

of A. Arriving at this is routine, albeit tedious. One can either write out thematrix product AXA

Tand read o↵ the coe�cients of the x↵ or use (⇤A)↵� =

12 Trace

��↵A��A

T �. In either case the result is

12

0BBBBBBBBBBBBBB@

|a|2 + |b|2 + |c|2 + |d|2 ac + ac + bd + bd i(ac + bd � ac � bd) |a|2 + |b|2 � |c|2 � |d|2ab + ab + cd + cd ad + ad + bc + bc i(ad � bc � ad + bc) ab + ab � cd � cd

i(ab � ab + cd � cd) i(ad + bc � ad � bc) ad + ad � bc � bc i(ab + cd � ab � cd)|a|2 � |b|2 + |c|2 � |d|2 ac + ac � bd � bd i(ac + bd � ac � bd) |a|2 � |b|2 � |c|2 + |d|2

1CCCCCCCCCCCCCCA

We have already seen that (�A) = (A) and would now like to show that isprecisely two-to-one. Since is a group homomorphism we need only show that thekernel of is {±I}, where I is the 2 ⇥ 2 identity matrix.

Exercise 2.4.3. Equate the explicit matrix representation for ⇤A to the 4⇥ 4 identitymatrix and show that A = ±I.

Exercise 2.4.4. LetA1(✓) =

cosh (✓/2) sinh (✓/2)sinh (✓/2) cosh (✓/2)

!

Then A1(✓) is in SL(2,C). Show that (A1(✓)) is the element of L"+ representing aboost in the x1-direction.

(A1(✓)) = ⇤1(✓)

If one has still not wearied of this laborious arithmetic one can check that, forany t 2 [0, ⇡] and any unit vector n = (n1, n2, n3) in R3, the matrix exponential


e�i(t/2)(n1�1+n2�2+n3�3) (2.9)

is in SU(2) ✓ SL(2,C) and its image under is the rotation in L"+ given by

R =

0BBBBBBBBBBBB@

1 0 0 000 R(n, t)0

1CCCCCCCCCCCCA,

where R(n, t) is the rotation in SO(3) described in Theorem A.2.2. Since every ro-tation in SO(3) is of this form we can conclude from this that maps the SU(2)subgroup of SL(2,C) onto the rotation subgroup R of L"+. Then we appeal to The-orem 2.3.5 which asserts that any ⇤ 2 L"+ can be written as ⇤ = R1⇤1(✓)R2, whereR1 and R2 are rotations in L"+ and to the fact that is a homomorphism to concludethat maps SL(2,C) onto L"+.

Exercise 2.4.5. Prove each of the following directly from the explicit representationfor ⇤A in terms of the entries for A.

( et (� i2�1) ) =

cos (t/2) �i sin (t/2)�i sin (t/2) cos (t/2)

!=

0BBBBBBBBBBBB@

1 0 0 00 1 0 00 0 cos t sin t0 0 �sin t cos t

1CCCCCCCCCCCCA

( et (� i2�2) ) =

cos (t/2) �sin(t/2)sin (t/2) cos (t/2)

!=

0BBBBBBBBBBBB@

1 0 0 00 cos t �sin t 00 0 1 00 sin t cos t 0

1CCCCCCCCCCCCA

( et (� i2�3) ) =

e�it/2 0

0 eit/2

!=

0BBBBBBBBBBBB@

1 0 0 00 cos t sin t 00 �sin t cos t 00 0 0 1

1CCCCCCCCCCCCA

There is one final consequence we would like to draw from the fact that :SL(2,C) ! L"+ is a surjective homomorphism with discrete kernel Z2 = {±I}.From the explicit matrix expression for (A) it is clear that is continuous. ByCorollary 1.1.3, it is smooth. Since SL(2,C) and L"+ are both connected we canapply Theorem 1.1.7 to conclude that is a smooth covering map. Since SL(2,C)is simply connected, it is, in fact, the universal covering group of L"+. Because istwo-to-one, SL(2,C) is called the universal double cover of L"+. SU(2) is also simplyconnected so it is the universal double cover of the rotation group R � SO(3). Themap itself is referred to either as the covering map or, on occasion, the spinor map.

All of this extends at once to the Poincare group which, we recall, is the semi-direct product


P"+ = R1,3o L"+.

determined by the natural action of L"+ onR1,3. Now define inhomogeneous SL(2,C),denoted ISL(2,C), to be the semi-direct product

ISL(2,C) = R1,3o SL(2,C),

where the action of SL(2,C) on R1,3 is determined by the covering map :SL(2,C)! L"+ as follows.

A · a = (A)a = ⇤Aa

for all A 2 SL(2,C) and all a 2 R1,3.

Exercise 2.4.6. Show that the map idR1,3 ⇥ : ISL(2,C)! P"+ defined by

(idR1,3 ⇥ )(a, A) = (a, (A))

for all (a, A) 2 ISL(2,C) is a smooth homomorphism of the semi-direct productswith kernel equal to ±idISL(2,C). Conclude that ISL(2,C) is the universal doublecover of P"+.

Exercise 2.4.7. Show that the kernel of idR1,3 ⇥ : ISL(2,C) ! P"+ is precisely thecenter of ISL(2,C).

2.5 Poincare Algebra

2.5.1 Introduction

In Sections A.2 and A.3 we noted the intimate connection between the “infinitesimalsymmetries” of a classical mechanical system and its conservation laws. To exploitsimilar ideas in the relativistic context we will need to understand the structure of theLie algebras of the Lorentz and Poincare groups. The proper, orthochronous Lorentzgroup L"+ is given as a matrix group; it is the connected component containing theidentity in the semi-orthogonal group O(1, 3) (see Example 1.1.1 (6) and (7)) sothese two have isomorphic Lie algebras. Once this Lie algebra is determined as aset of matrices by computing velocity vectors to smooth curves through the identitythe Lie bracket is just the matrix commutator. The Poincare group P"+ is given as asemi-direct product, but we have described an explicit matrix model of it in Section2.4 so we can follow the same procedure for it (a simpler example of this procedureis described in Example 1.1.2). Furthermore, the universal double cover of L"+ is

2.5 Poincare Algebra 65

SL(2,C) so these have isomorphic Lie algebras and, similarly, P"+ and ISL(2,C)have isomorphic Lie algebras.

2.5.2 Lie Algebra of L"+

Exercise 2.5.1. The semi-orthogonal group O(1, 3) is the set of 4⇥4 real matrices Asatisfying AT⌘A = ⌘ (Example 1.1.1 (6)). Follow the same procedure as in Example1.1.2 to show that the Lie algebra of O(1, 3), and therefore of L"+, is given by

o(1, 3) =�X 2 gl(4,R) : XT = �⌘X⌘

and that these are precisely the real matrices of the form

X =

0BBBBBBBBBBBB@

0 X01 X0

2 X03

X01 0 �X1

2 X13

X02 X1

2 0 �X23

X03 �X1

3 X23 0

1CCCCCCCCCCCCA.

From the explicit description of the elements of the Lie algebra o(1, 3) in thisExercise we can read o↵ a basis for o(1, 3).

M1 =

0BBBBBBBBBBBB@

0 0 0 00 0 0 00 0 0 �10 0 1 0

1CCCCCCCCCCCCA,M2 =

0BBBBBBBBBBBB@

0 0 0 00 0 0 10 0 0 00 �1 0 0

1CCCCCCCCCCCCA,M3 =

0BBBBBBBBBBBB@

0 0 0 00 0 �1 00 1 0 00 0 0 0

1CCCCCCCCCCCCA

N1 =

0BBBBBBBBBBBB@

0 1 0 01 0 0 00 0 0 00 0 0 0

1CCCCCCCCCCCCA,N2 =

0BBBBBBBBBBBB@

0 0 1 00 0 0 01 0 0 00 0 0 0

1CCCCCCCCCCCCA,N3 =

0BBBBBBBBBBBB@

0 0 0 10 0 0 00 0 0 01 0 0 0

1CCCCCCCCCCCCA

Notice that the Mj, j = 1, 2, 3, are skew-symmetric whereas the Nj, j = 1, 2, 3, aresymmetric.

Computing the matrix commutators one finds that these basis elements satisfythe following commutation relations (✏ jkl is the Levi-Civita symbol).

[Mj,Mk] = ✏ jklMl, j, k = 1, 2, 3 (2.10)

[Nj,Nk] = �✏ jklMl, j, k = 1, 2, 3 (2.11)

[Mj,Nk] = ✏ jklNl, j, k = 1, 2, 3 (2.12)


Comparing the first of these with Exercise 1.1.2 we find that {M1,M2,M3} generatea Lie algebra isomorphic to the Lie algebra so(3) of the rotation group. In particular,since the exponential map on so(3) is surjective (Theorem A.2.2) any rotation in L"+can be written in the form

e✓j Mj = e✓

1 M1+✓2 M2+✓3 M3

for some real numbers ✓ j, j = 1, 2, 3. M1,M2, and M3 are called the generators ofrotations in L"+.


e✓1 M1 =

0BBBBBBBBBBBB@

1 0 0 00 1 0 00 0 cos ✓1 �sin ✓10 0 sin ✓1 cos ✓1

1CCCCCCCCCCCCA

and then compute e✓2 M2 and e✓3 M3 .

Remark 2.5.1. Except for the 0th row of zeros and 0th column of zeros, M1,M2,and M3 are just the basis vectors X1, X2, and X3 for so(3) introduced in Exercise1.1.2. These are, of course, technically di↵erent objects, but it is often convenientto simply identify Mj and Xj for j = 1, 2, 3 so that e✓ j M j is either a rotation in L"+or the same rotation in SO(3) depending on the context. We will even take this onestep further and notice that these, in turn, can, by Exercise 1.2.6 (10), be identifiedwith the basis vectors � i

2� j, j = 1, 2, 3, of su(2) in which case e✓ j M j is an elementof SU(2) which corresponds to a rotation via the double covering.

M1,M2,M3 ! X1, X2, X3 ! � i2�1,�

i2�2,�

i2�3


e⇠1N1 =

0BBBBBBBBBBBB@

cosh ⇠1 sinh ⇠1 0 0sinh ⇠1 cosh ⇠1 0 0

0 0 1 00 0 0 1

1CCCCCCCCCCCCA

and then compute e⇠2N2 and e⇠3N3 .

Motivated by Exercise 2.5.3, N1,N2 and N3 are called the generators of boostsin L"+. Notice, however, that N1,N2 and N3 do not close under commutator since[Nj,Nk] = �✏ jklMl and therefore, unlike M1,M2, and M3, they do not span a Liealgebra. This is because the composition of two boosts in non-collinear directions


is not a boost, but is the composition of a boost and a rotation (called a Wignerrotation); there is an elementary discussion of this in [O’DV].

An arbitrary element of the Lie algebra of L"+ is of the form ✓ jMj + ⇠ jN j for realnumbers ✓ j, ⇠ j, j = 1, 2, 3. Consequently, every

e✓j Mj+⇠ jN j (2.13)

is in L"+. In fact, it is true, conversely, that every element of L"+ can be written inthe form (2.13). Stated otherwise, the exponential map on the Lie algebra of L"+is surjective. This is not obvious, however, and we will simply refer to the proofavailable at http://www.cis.upenn.edu/ cis610/cis61005sl8.pdf.

We now have an explicit description of the Lie algebra of L"+, but for variousreasons that we will mention as we proceed, it is useful to obtain a few alternatedescriptions of the generators of the Lorentz transformations. We begin by consol-idating all of the matrices Mj,Nj, j = 1, 2, 3, into a single 4 ⇥ 4 skew-symmetricmatrix Lµ⌫, µ, ⌫ = 0, 1, 2, 3, defined by

(Lµ⌫)µ,⌫=0,1,2,3 =

0BBBBBBBBBBBB@

0 �N1 �N2 �N3N1 0 �M3 M2N2 M3 0 �M1N3 �M2 M1 0

1CCCCCCCCCCCCA.

The entries of (Lµ⌫)µ,⌫=0,1,2,3 are themselves 4 ⇥ 4-matrices. Specifically,

Nj = Lj0 = �L0 j, j = 1, 2, 3

and

Mj = Lkl = �Llk, j = 1, 2, 3,

where jkl is an even permutation of 123. For fixed µ and ⌫ the entries in the matrixLµ⌫ will be designated Lµ⌫↵�. Thus, for example, L23

↵� are just the entries of M1 so

L23↵� =

8>>>>><>>>>>:

�1, if ↵ = 2, � = 31, if ↵ = 3, � = 20, otherwise.

Notice that this can be written

L23↵� = ⌘3��

↵2 � ⌘2��

↵3 .

Exercise 2.5.4. Check a few more cases to persuade yourself that, for all µ, ⌫ =0, 1, 2, 3 and all ↵, � = 0, 1, 2, 3,

Lµ⌫↵� = ⌘⌫��↵µ � ⌘µ��↵⌫ .


Then lower the index ↵, that is, define Lµ⌫↵� = ⌘↵�Lµ⌫��, and show that

Lµ⌫↵� = ⌘µ↵⌘⌫� � ⌘µ�⌘⌫↵

for all µ, ⌫,↵, � = 0, 1, 2, 3.

We can now write all of the commutation relations (2.10), (2.11), and (2.12) as asingle relation. For each fixed µ, ⌫,↵, � = 0, 1, 2, 3 we have

[Lµ⌫, L↵�] = �⌘µ↵L⌫� + ⌘µ�L⌫↵ + ⌘⌫↵Lµ� � ⌘⌫�Lµ↵. (2.14)

For example, if µ = 0, ⌫ = 1, ↵ = 0, and � = 2,

[L01, L02] = [�N1,�N2] = [N1,N2] = �✏123M3 = �M3

and

�⌘00L12 + ⌘02L10 + ⌘10L02 � ⌘12L00 = �⌘00L12 = �L12 = �M3

so (2.14) reduces to (2.11).

Exercise 2.5.5. Write out as many more of these as it takes to convince you that(2.14) contains all of the commutation relations (2.10), (2.11), and (2.12).

Exercise 2.5.6. Show that, if X = (Xµ⌫)µ,⌫=0,1,2,3 is in the Lie algebra of L"+ andXµ⌫ = ⌘�⌫Xµ�, then the image ⇤ of X in L"+ under the exponential map can bewritten

⇤ = e12 Xµ⌫Lµ⌫ .

There are advantages to viewing what we have just done from the complexifiedperspective (see Remark 1.1.2). For this we note that the complexification of the Liealgebra of L"+ contains, in particular, the matrices

J j = iMj, j = 1, 2, 3,

and

Kj = iN j, j = 1, 2, 3.

In terms of these the commutation relations (2.10), (2.11), and (2.12) take the form

[J j, Jk] = i✏ jkl Jl, j, k = 1, 2, 3 (2.15)

[Kj,Kk] = �i✏ jkl Jl, j, k = 1, 2, 3 (2.16)


[J j,Kk] = i✏ jklKl, j, k = 1, 2, 3. (2.17)

Note: Unless some confusion is likely to arise we will adopt the usual custom of sup-pressing the subscript C in the notation [ , ]C for the bracket of the complexification(Remark 1.1.2).

Notice that J1, J2, J3,K1,K2, and K3 can still be regarded as generators of theLorentz transformations by simply including a factor of �i in the argument of theexponential map, that is, every element of L"+ can be written in the form

e�i(✓ j J j+⇠ jK j). (2.18)

Next we consolidate J1, J2, J3,K1,K2, and K3 into a single skew-symmetric ma-trix of matrices given by

(Mµ⌫)µ,⌫=0,1,2,3 =

0BBBBBBBBBBBB@

0 �K1 �K2 �K3K1 0 �J3 J2K2 J3 0 �J1K3 �J2 J1 0

1CCCCCCCCCCCCA

and note that (2.14) becomes

[Mµ⌫,M↵�] = �i�⌘µ↵M⌫� � ⌘µ�M⌫↵ � ⌘⌫↵Mµ� + ⌘⌫�Mµ↵

�. (2.19)

If X = (Xµ⌫)µ,⌫=0,1,2,3 is in the Lie algebra of L"+ and Xµ⌫ = ⌘�⌫Xµ�, then the image⇤ of X in L"+ under the exponential map can be written

⇤ = e�i2 Xµ⌫Mµ⌫ .

Another useful set of complex generators for L"+ is obtained in the following way.Define

S j =12

(J j + iK j), j = 1, 2, 3

and

T j =12

(J j � iK j), j = 1, 2, 3.

Exercise 2.5.7. Show that, in terms of these, the commutation relations (2.15),(2.16), and (2.17) decouple as follows.

[S j, S k] = i✏ jklS l, j, k = 1, 2, 3 (2.20)


[T j,Tk] = i✏ jklTl, j, k = 1, 2, 3 (2.21)

[S j,Tk] = 0, j, k = 1, 2, 3 (2.22)

2.5.3 Lie Algebra of P"+

We now turn to the Poincare algebra. The Poincare group P"+ is given as a semi-direct product R1,3

o L"+ and one could appeal to general results on Lie algebrasof semi-direct products (see pages 301-306 of [Nab5] for a brief description andan example or Section I.4 of [Knapp] for the details). We prefer to follow a morepedestrian route using the explicit matrix model of P"+ constructed in Section 2.4.Recall that P"+ is isomorphic to the closed subgroup of GL(5,R) consisting of all

0BBBBBBBBBBBBBBBBB@

1 0 0 0 0a0 ⇤0

0 ⇤01 ⇤0

2 ⇤03

a1 ⇤10 ⇤1

1 ⇤12 ⇤1

3a2 ⇤2

0 ⇤21 ⇤2

2 ⇤23

a3 ⇤30 ⇤3

1 ⇤32 ⇤3

3


1 0a ⇤

!

where a 2 R1,3 and ⇤ 2 L"+. The Lie algebra is therefore a collection of real, 5 ⇥ 5matrices with the bracket given by matrix commutator and we would like to findgenerators and commutation relations for it.. Taking a = 0 one obtains a closedsubgroup of P"+ isomorphic to L"+ and we will abuse the notation a bit by continuingto denote this L"+. Similarly, taking ⇤ = id4⇥4 gives a closed subgroup isomorphic toR1,3 which we also denote R1,3. Since the underlying manifold of P"+ is the productR1,3 ⇥ L"+, the tangent space at the identity

1 00 id4⇥4

!

is just the vector space direct sum of the tangent spaces to R1,3 and L"+ at the iden-tity. We have already determined generators and commutation relations for the Liealgebra of the Lorentz group L"+. Thought of as living in the Lie algebra of P"+ theseare

0 00 Mj

!

and 0 00 Nj

!


for j = 1, 2, 3 and we will persist in our abuse of the notation by writing thesesimply as Mj and Nj, respectively. They satisfy the commutation relations (2.10),(2.11), and (2.12). The matrices

O0 =

0BBBBBBBBBBBBBBBBB@

0 0 0 0 01 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

1CCCCCCCCCCCCCCCCCA,O1 =

0BBBBBBBBBBBBBBBBB@

0 0 0 0 00 0 0 0 01 0 0 0 00 0 0 0 00 0 0 0 0

1CCCCCCCCCCCCCCCCCA

O2 =

0BBBBBBBBBBBBBBBBB@

0 0 0 0 00 0 0 0 00 0 0 0 01 0 0 0 00 0 0 0 0

1CCCCCCCCCCCCCCCCCA,O3 =

0BBBBBBBBBBBBBBBBB@

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 01 0 0 0 0

1CCCCCCCCCCCCCCCCCA

are generators for the Lie algebra of the translation subgroup R1,3 ✓ P"+ and theysatisfy the commutation relations

[Oµ,O⌫] = 0, µ, ⌫ = 0, 1, 2, 3.

Inserting a factor of i into each of the generators Mj and Nj, j = 1, 2, 3, we obtainthe complex generators of rotations and boosts

J j =

0 00 iMj

!, j = 1, 2, 3

andKj =

0 00 iN j

!, j = 1, 2, 3

and thereby the matrix (Mµ⌫)µ,⌫=0,1,2,3 satisfying the commutation relations (2.19).For the Oµ, µ = 0, 1, 2, 3, we introduce a factor of �i and define

Pµ = �iOµ, µ = 0, 1, 2, 3. (2.23)

These satisfy

[Pµ, P⌫] = 0, µ, ⌫ = 0, 1, 2, 3.

All that remains is to compute the brackets [Mµ⌫, P↵] and we claim that these aregiven by

[Mµ⌫, P↵] = i (⌘⌫↵Pµ � ⌘µ↵P⌫), µ, ⌫,↵ = 0, 1, 2, 3.

Exercise 2.5.8. Check this for µ = 1, ⌫ = 0,↵ = 0 and as many other cases as yourconscience requires.


We are now in a position to summarize the structure of the Poincare algebra.This is a complex Lie algebra p of dimension 10 (the complexification of the Liealgebra of the Poincare group P"+). It has complex generators Pµ, µ = 0, 1, 2, 3,and Mµ⌫ = �M⌫µ, with µ, ⌫ = 0, 1, 2, 3, and µ , ⌫ determined by the followingcommutation relations.

[Pµ, P⌫] = 0 (2.24)

[Mµ⌫,M↵�] = �i�⌘µ↵M⌫� � ⌘µ�M⌫↵ � ⌘⌫↵Mµ� + ⌘⌫�Mµ↵

�(2.25)

[Mµ⌫, P↵] = i (⌘⌫↵Pµ � ⌘µ↵P⌫) (2.26)

The complex generators J j and Kj of rotations and boosts, respectively, are givenby

J j = ✏ jklMkl, j = 1, 2, 3,

and

Kj = Mj0, j = 1, 2, 3,

so the commutation relations (2.24), (2.25), and (2.26) can be written in terms ofthese by making specific choices for the subscripts in Mµ⌫. For example,

[J1, J2] = [M23,M31] = �i(�⌘33M21) = �i(�M12) = iJ3.

Continuing in this way one obtains the following set of commutation relations forthe Poincare algebra p.

[J j, Jk] = i✏ jkl Jl, j, k = 1, 2, 3 (2.27)

[J j,Kk] = i✏ jklKl, j, k = 1, 2, 3 (2.28)

[Kj,Kk] = �i✏ jkl Jl, j, k = 1, 2, 3 (2.29)

[J j, Pk] = i✏ jklPl, j, k = 1, 2, 3 (2.30)

[Kj, Pk] = iP0� jk, j, k = 1, 2, 3 (2.31)

[J j, P0] = [Pj, P0] = [P0, P0] = 0, j = 1, 2, 3 (2.32)


[Kj, P0] = iP j, j = 1, 2, 3 (2.33)

In physics it is generally the commutation relations of p that play the most promi-nent role rather than any particular realization of them as operators. We have alreadyseen one concrete representation of these relations as 5⇥5 matrices, that is, as oper-ators onR5. We will conclude this section with another realization of p, this time asoperators on an infinite-dimensional vector space. We will find that these two man-ifestations of the Poincare algebra will suggest a link between the abstract bracketstructure of p and the physics of relativistic systems, both classical and quantum.

Before getting started, however, we should point out that in much of what fol-lows we will often treat Pµ and Mµ⌫ as if they were the components of objects thatlived in R1,3 rather than as elements of the Lie algebra p. We will, for example,“raise indices” with ⌘ to define Pµ = ⌘µ↵P↵ and Mµ⌫ = ⌘µ↵⌘⌫�M↵� and, in the nextsection, form products such as PµPµ. The motivation for this is as follows. We sawin Example 1.1.5 that the Poincare group P"+, and therefore its subgroup L"+, acts onP"+ on the right by conjugation and that this induces a right action of L"+ on the Liealgebra p. In particular, ⇤ 2 L"+ ✓ P"+ acts on each matrix Pµ by ⇤�1Pµ⇤. We willask you to show now that this has the same e↵ect as transforming the Pµ as if theywere the components of a 4-vector in R1,3.

Exercise 2.5.9. Compute the indicated matrix products and show that

⇤�1Pµ⇤ = ⇤µ⌫P⌫, µ = 0, 1, 2, 3. (2.34)

On the other hand, one thinks of (2.35) below as saying that the generators Mµ⌫transform as a second rank 4-tensor under L"+.


⇤�1Mµ⌫⇤ = ⇤µ↵⇤⌫�M↵�, µ, ⌫ = 0, 1, 2, 3. (2.35)

Remark 2.5.2. One often sees similar terminology used in the physics literature, butin a di↵erent context so we should explain. First recall (Section 1.1) that the actionof a matrix Lie group G on its Lie algebra g by conjugation is called the adjointaction of G and is denoted Ad : G ! Aut(g). The derivative of Ad at the identitye 2 G determines an action ad = Ad⇤e : g ! Der(g) of g on g. The value of adat X 2 g is denoted adX : g ! g and is given by adXY = [X,Y] for every Y 2 g.This is called the adjoint action of g. Intuitively, g (the tangent space at e 2 G) isthought of as an “infinitesimal” version of G. An element of so(3), for example,is an “infinitesimal rotation”. From this point of view the adjoint action of g is aninfinitesimal version of the adjoint action of G.

Consider, for example, the adjoint action of L"+ ✓ P"+ on p. Exercise 2.5.9 assertsthat under the action Ad⇤�1 of the Lorentz group on p, the generators Pµ transform


like a 4-vector so that, even though it lives in p, one can treat P = (P0, P1, P2, P3) asif it were a vector inR1,3. Now think of the elements of the Lie algebra o(1, 3) ✓ p ofL"+ as “infinitesimal Lorentz transformations”. The adjoint action of o(1, 3), given bybracket, is then thought of as the action of infinitesimal Lorentz transformations onp and one can ask how the elements of p transform under this action. For example,let’s consider the generators K1,K2,K3 in p and ask how these transform underinfinitesimal rotations. The generators of the rotations are J1, J2, and J3 so we areinterested in the action of J j on Kk, that is,

adJ j Kk = [J j,Kk].

But, according to (2.28),

adJ j Kk = i✏ jklKl

so (K1,K2,K3) transforms like a vector under infinitesimal rotations. Physicists areinclined to omit the word “infinitesimal” and to write

K = (K1,K2,K3)

as a reminder that K behaves in some ways like a vector in R3.

Exercise 2.5.11. Check that the commutation relations (2.24), (2.25), and (2.26) arethe same with all of the indices raised, that is,

[Pµ, P⌫] = 0 (2.36)

[Mµ⌫,M↵�] = �i�⌘µ↵M⌫� � ⌘µ�M⌫↵ � ⌘⌫↵Mµ� + ⌘⌫�Mµ↵

�(2.37)

[Mµ⌫, P↵] = i (⌘⌫↵Pµ � ⌘µ↵P⌫) (2.38)

Example 2.5.1. Now we move on to our second realization of p. We will consideradmissible coordinates x0, x1, x2 and x3 on R1,3 and will let C1(R1,3;C) be thevector space of smooth, complex-valued functions on R1,3. We will write @µ for @

@xµand will raise the indices with ⌘ to obtain @µ = ⌘µ↵@↵ so that @0 = @0 and @i =�@i, i = 1, 2, 3. Now define operators Pµ and Mµ⌫, µ, ⌫ = 0, 1, 2, 3, on C1(R1,3;C)as follows.

Pµ = i @µ, µ = 0, 1, 2, 3,

and

Mµ⌫ = xµP⌫ � x⌫Pµ, µ, ⌫ = 0, 1, 2, 3.


Thus,

P0 = i@0 and Pj = �i@ j, j = 1, 2, 3

M jk = i(xk@ j � x j@k), j, k = 1, 2, 3 and M0 j = �M j0 = �i(x0@ j + x j@0), j = 1, 2, 3.

We claim that, when the bracket is taken to be the operator commutator, these oper-ators satisfy the commutation relations defining the Poincare algebra, that is, (2.36),(2.37), and (2.38). The first of these is clear from the equality of mixed secondorder partial derivatives. We will check just two of the remaining cases to see howthings go and then leave the rest to you. Specifically, we will first verify (2.37) whenµ = 0, ⌫ = 1,↵ = 0, and � = 2. Notice that, in this case, the right-hand side of (2.37)evaluated at ' 2 C1(R1,3;C) reduces to

�i (⌘00M12') = x2@1' � x1@2'

since all of the remaining ⌘ factors are zero. To see that the left-hand side of (2.37)is the same we just compute as follows.

[M01,M02]' = [�i (x0@1 + x1@0)][�i (x0@2 + x2@0)]'

� [�i (x0@2 + x2@0)][�i (x0@1 + x1@0)]'

= �x0x0@1@2' � x0x2@1@0' � x1@0(x0@2') � x1x2@0@0'

+ x0x0@2@1' + x0x1@2@0' + x2@0(x0@1') + x2x1@0@0'

= x2@1' � x1@2'

as required.Next we will check (2.38) when µ = 1, ⌫ = 2, and ↵ = 1. Evaluated at ', the

right-hand side contains only one nonzero term, namely,

i (�⌘11P2') = i(�i@2') = @2'.

For the left-hand side we compute

[M12, P1]' = M12(P1') � P1(M12')

= i(x2@1 � x1@2)(�i@1') + i@1(i(x2@1' � x1@2'))

= x2@1@1' � x1@2@1' � x2@1@1' + @1(x1@2')= @2'. (2.39)

Exercise 2.5.12. Complete the verification of (2.37) and (2.38).

In order to search for some underlying connection with physics in this realizationwe will compare it with what we know about classical and quantum mechanics


from Appendix A. In the quantum arena, however, one should keep in mind thatC1(R1,3;C) consists of smooth objects and therefore is not a Hilbert space so a moreprecise analogy will require self-adjoint extensions of the operators to somethingthat is a Hilbert space. At the moment we are looking only for formal similaritiesthat we can use as motivation later on. To facilitate the comparisons we will alsoadopt units in which ~ = 1.

We begin by restricting our attention to some spatial cross section of R1,3 corre-sponding to a fixed time x0 in a fixed admissible frame of reference. This is a copyof R3. The operators

Pj = �i@

@x j , j = 1, 2, 3,

on C1(R3;C) are then just the restrictions of the quantum mechanical momentumoperators (Example A.4.1 and specifically (A.25)) on L2(R3) in the given frame ofreference. This would seem to suggest that the generators P1, P2 and P3 have some-thing to do with linear momentum. The suggestion is strengthened when we recallthat these are the generators of the spatial translations in the R1,3 subgroup of P"+and that, in classical mechanics, spatial translation symmetry implies conservationof linear momentum (Example A.2.1).

In the relativistic context, however, the three spatial components of non-relativisticmomentum have no physical significance. Rather, they appear as the non-relativisticapproximations to the spatial components of the momentum 4-vector whose timecomponent is the total relativistic energy (see Remark 2.6.1). In classical mechan-ics conservation of total energy follows from time-translation symmetry (Exam-ple A.2.1) and P0 is the generator of time translations in R1,3. Moreover, in non-relativistic quantum mechanics the total energy is given by the classical Hamiltonianoperator H on L2(R3) and this, according to the Schrodinger equation (see (A.16)),is related to the time evolution of the wave function by

iddt

( (t)) = H( (t)),

where the time evolution (t) of the wave function is regarded as a curve in L2(R3)and the t-derivative is the tangent vector to this curve in L2(R3). One can regard thewave function as defined on R1,3 and, under certain circumstances, one can iden-tify the L2(R3)-derivative d

dt ( (t)) with the time partial derivative of in the givenframe of reference (see Remark 6.2.14 of [Nab5] for more on this). The Schrodingerequation is then written

i@

@x0 = H

and this essentially identifies the operator

P0 = i@

@x0


with the total energy operator H. Now, the Schrodinger equation is not relativis-tically invariant so it has no real status on R1,3. However, the Relativity Principleasserts that all admissible frames are physically equivalent and this suggests that theSchrodinger equation should describe the non-relativistic limit of the time evolutionin any admissible frame of reference. This, in turn, suggests that P0 is the appro-priate relativistic analogue of the 0th-component of the 4-momentum in quantumtheory.

The preceding arguments were informal and, perhaps, not entirely persuasive,but we are simply trying to motivate “associating” the relativistic 4-momentum tothe abstract generators (P0, P1, P2, P3) of the Poincare algebra p. Precisely how thisrather vague “association” is made use of in practice will be addressed at the end ofthis section. In any case, we will proceed now to attempt something similar for theremaining generators Mµ⌫, µ, ⌫ = 0, 1, 2, 3. Only six of these are independent so wewill look just at the following generators.

M23 = i(x3@2 � x2@3), M31 = i(x1@3 � x3@1), M12 = i(x2@1 � x1@2)

and

M01 = �i(x0@1 + x1@0), M02 = �i(x0@2 + x2@0), M03 = �i(x0@3 + x3@0).

The appropriate interpretations for M23,M31, and M12 seem clear since these arejust the operators representing (orbital) angular momentum in quantum mechanics(Example A.4.1 with ~ = 1). Except for a factor of �i they also bear a strikingresemblance to the infinitesimal generators (A.3), (A.4), and (A.5) of angular mo-mentum in classical mechanics. This suggests that (J1, J2, J3) = (M23,M31,M12)should be “associated with” the components of angular momentum.

The operators describing M01,M02, and M03 are less familiar. These correspondto the generators of boosts in P"+ and are the operators associated with what is calledrelativistic angular momentum. This is a topic that generally does not find its wayinto most introductions to special relativity (in particular, it will not be found in[Nab4]) and any discussion of it here would take us rather far afield. For those whoare interested in pursuing this we can suggest Section 7.8 of [Gold] for the physi-cist’s perspective and Chapter VII of [Synge] for a geometrical treatment in muchthe same spirit as [Nab4]. We will take this rather subtle physics for granted andsimply “associate” (K1,K2,K3) = (M01,M02,M03) with relativistic angular mo-mentum.

Exercise 2.5.13. Regard

M23 = i(x3@2 � x2@3), M31 = i(x1@3 � x3@1), M12 = i(x2@1 � x1@2)

as operators on L2(R3) and let be a smooth element of L2(R3).

1. Prove the following commutation relations for these operators.


[M23,M31] = i M12, [M31,M12] = i M23, [M12,M23] = i M31

2. Suppose depends only on distance to the origin, that is, (x1, x2, x3) = (|x|).Show that M23 = M31 = M12 = 0 so that these operators depend only onthe angular coordinates in R3. This suggests rewriting the operators in sphericalcoordinates.

3. Introduce spherical coordinates ⇢, � and ✓ in R3 by

x1 = ⇢ sin � cos ✓, x2 = ⇢ sin � sin ✓, x3 = ⇢ cos �,

where 0 � < ⇡ and 0 ✓ < 2⇡. Show that

M23 = i (sin ✓ @�+cot � cos ✓ @✓), M31 = �i (cos ✓ @��cot � sin ✓ @✓), M12 = �i @✓.

Notice that none of these depend on ⇢.

4. Define M2 = (M23)2 + (M31)2 + (M12)2 on the smooth elements of L2(R3) andshow that

M2 = �✓ 1

sin �@�(sin � @�) +

1sin2�

@2✓

◆.

Note: Physicists generally write the operators M23,M31,M12 and M2 as Lx, Ly, Lzand L2, respectively.

5. Compare this with the usual expression for the Laplacian � in spherical coordi-nates on R3 to show that

� =1⇢2 @⇢(⇢

2@⇢) �1⇢2 M2

so that M2 is the “spherical part” of the Laplacian on R3.

6. Prove that

[M2,M23] = [M2,M31] = [M2,M12] = 0.

Remark 2.5.3. All of the operators M23,M31,M12 and M2 are essentially self-adjoint on the Schwartz space S(R3) and so have unique self-adjoint extensions toL2(R3). As usual, we will use the same symbols to denote the extensions. These cor-respond to quantum observables (the components of the orbital angular momentumand the squared magnitude of the total orbital angular momentum, respectively) sothe spectrum of any one of these operators contains the set of possible measuredvalues of the corresponding observable (Postulate QM2 of Appendix A.4). In par-ticular, one is interested in the “eigenvalue problem” for each of these operators.


There are a number of ways to approach such problems. One can simply solve thedi↵erential equations. Another approach via raising and lowering operators is en-tirely analogous to the procedure carried out for the harmonic oscillator in Section5.3 of [Nab5]. One or the other, or both, of these will be discussed in some detailin essentially any basic quantum mechanics book, although perhaps not at a levelof rigor that will satisfy a mathematician (see, for example, Chapter 14 of [Bohm]or Sections 3.5 and 3.6 of [Sak]). A rigorous proof of the result we need, but willsimply quote can be found in Chapter II, Section 7, of [Prug].

Because M2 commutes with each of M23,M31, and M12 (Exercise 2.5.13 (6))one can actually find simultaneous eigenfunctions for M2 and any one of the angu-lar momentum components, but not more than one since the Mi j do not commutewith each other (Exercise 2.5.13 (1)). Physicists interpret this to mean that one can“simultaneously measure” M2 and any single one of the angular momentum com-ponents (see pages 248-252 of [Nab5] for more on the notion of simultaneous mea-surability, which is more subtle than one might expect). We will, in fact, describe anorthonormal basis for L2(R3) consisting of eigenfunctions for both M2 and M12. Inparticular, it will follow from this that the complete spectrum of each of these op-erators (possible measured values of the associated observables) consists preciselyof the corresponding eigenvalues). To find such an orthonormal basis we begin bynoting that, if the 2-sphere S 2 and the ray (0,1) are given the measures sin � d� d✓and 4⇡⇢2d⇢, respectively, then the product measure is just the Lebesgue measure onR3. Moreover, there is a unique unitary map ⇡ : L2(S 2) ⌦ L2((0,1)) ! L2(R3)of the Hilbert space tensor product L2(S 2) ⌦ L2((0,1)) onto L2(R3) satisfying⇡( f ⌦ g)((�, ✓), ⇢) = f (�, ✓)g(⇢) for all f 2 L2(S 2), g 2 L2((0,1)), (�, ✓) 2 S 2 and⇢ 2 (0,1) (Chapter II, Theorem 6.9, of [Prug]). Consequently, if { f1, f2, . . .} is anorthonormal basis for L2(S 2) and {g1, g2, . . .} is an orthonormal basis for L2((0,1)),then { f j⌦gk : j, k = 1, 2, . . .} is an orthonormal basis for L2(S 2)⌦L2((0,1)) (ChapterII, Theorem 6.10, of [Prug]).

We will briefly describe how to obtain such an orthonormal basis of simultaneouseigenfunctions for M2 and M12. The eigenvalue problems we are interested in are

1sin �

@�(sin � @� ) +1

sin2�@2✓ = �� (2.40)

and

i @✓ = �µ (2.41)

and we will begin with (2.40). Separating variables (⇢, �, ✓) = R(⇢)Y(�, ✓) thisreduces to

R(⇢)✓ 1

sin �@�(sin � @�Y(�, ✓)) +

1sin2�

@2✓Y(�, ✓) + �Y(�, ✓)

◆= 0.

Consequently, if one finds solutions to the eigenvalue problem


1sin �

@�(sin � @�Y(�, ✓)) +1

sin2�@2✓Y(�, ✓) + �Y(�, ✓) = 0, (2.42)

then R(⇢) can be chosen arbitrarily. As it happens, (2.42) is an old and well-understood problem in partial di↵erential equations and mathematical physics. Todescribe its solutions we must introduce some equally old and well-known func-tions. For each

l = 0, 1, 2, . . .

and each

m = �l,�l + 1, . . . , l,

we define a function Ylm(�, ✓) as follows.

Ylm(�, ✓) = (�1)m2l + 1

4⇡(l � |m|)!(l + |m|)!

�1/2Pl |m|(cos �)eim✓, (2.43)

where Plm(u) are the associated Legendre functions of order m defined by the Ro-drigues’ formula

Plm(u) =(1 � u2)m/2

2ll!dl+m

dul+m (u2 � 1)l, m = 0, 1, 2, . . . .

The functions Ylm(�, ✓), which are clearly smooth on S 2, are called spherical har-monics and our interest in them arises from the following result (which is Theorem7.1, Chapter II, of [Prug]).

Theorem 2.5.1. For each l = 0, 1, 2, . . . and each m = �l,�l+ 1, . . . , l, the sphericalharmonic Ylm(�, ✓) satisfies

1sin �

@�(sin � @�Ylm(�, ✓)) +1

sin2�@2✓Ylm(�, ✓) + l(l + 1)Ylm(�, ✓) = 0. (2.44)

Furthermore, the functions�Ylm(�, ✓) : l = 0, 1, 2, . . . ,m = �l,�l + 1, . . . , l

form

an orthonormal basis for L2(S 2).

Consequently, if {Rj(⇢) : j = 0, 1, 2, . . . } is any orthonormal basis for L2((0,1)),then {Rj(⇢)Ylm(�, ✓) : j = 0, 1, 2, . . . , l = 0, 1, 2, . . . , m = �l,�l + 1, . . . , l } is anorthonormal basis for L2(R3) consisting of eigenfunctions of M2.

M2( Rj(⇢)Ylm(�, ✓) ) = l(l + 1)Rj(⇢)Ylm(�, ✓)

In particular, the eigenvalues of M2 are

l(l + 1), l = 0, 1, 2, . . . .


Exercise 2.5.14. Show that each Rj(⇢)Ylm(�, ✓) is also an eigenfunction of M12 witheigenvalue m.

Finally, we should point out that had we not chosen to work in units for which~ = 1 the eigenvalues (possible measured values) of M2 would have a factor of ~2

l(l + 1)~2

and those of M12 would have a factor of ~.

m~

Now that we have gained some intuition for what the physical interpretation ofthe generators of the Poincare algebra “should” be we need to draw some con-clusions from it regarding quantum systems that admit a unitary representation ofISL(2,C) and so are “relativistically invariant” in a sense determined by the rep-resentation (such systems are discussed in more detail in Section 2.7). Since itcosts no more to do so we will describe the plan in more generality by consider-ing an arbitrary matrix Lie group G with Lie algebra g and a unitary representation� : G ! U(H) of G on a complex, separable Hilbert space H. Ideally, we wouldlike to define a Lie algebra homomorphism of g into a Lie algebra of self-adjointoperators on H so that we can identify the images of the generators with observ-ables of a quantum system whose Hilbert space is H and then supply these with anappropriate physical interpretation.. There are at least two obvious di�culties withsuch a plan. The first is that observables are generally unbounded operators so thatall of the usual domain issues arise and the commutator of two observables need notbe defined on a subspace of H larger than the trivial one. Moreover, even if one canget around these domain issues, the sad fact is that, even for bounded operators, thecommutator of two self-adjoint operators is not self-adjoint. Fortunately, this lastdi�culty is rather easy to circumvent.

Exercise 2.5.15. Let A and B be bounded, symmetric operators on H so thathA , �i = h , A�i and hB , �i = h , B�i for all , � 2 H. Show that the com-mutator [A, B] is skew-symmetric, that is,

h[A, B] , �i = �h , [A, B]�i

for all A, B 2 H.

Skew-symmetric operators, however, are better behaved with respect to the forma-tion of commutators.

Exercise 2.5.16. Let C and D be bounded, skew-symmetric operators on H so thathC , �i = �h ,C�i and hD , �i = �h ,D�i for all , � 2 H. Show that [C,D] isalso skew-symmetric.


Moreover, there is a simple one-to-one correspondence between symmetric andskew-symmetric operators, even if they are unbounded.

Exercise 2.5.17. Let A be a symmetric operator on H with domain D(A) so thathA , �i = h , A�i for all , � 2 D(A). Show that iA is skew-symmetric on D(A).Show, conversely, that if A is skew-symmetric, then �iA is symmetric.

All of this extends to (essentially) self-adjoint and (essentially) skew-adjoint op-erators on H and suggests that, for the purposes of introducing a commutator struc-ture on observables, one should focus on the latter rather than the former. Then oneneed only cure the domain problems. This, however, can only be accomplished by anassumption about the operators of interest. We begin with a definition that addressesthese issues.

Let H be a complex, separable Hilbert space. A collection W of operators on His said to be a Lie algebra of operators on H if the following conditions are satisfied.

1. There exists a dense linear subspace D of H such that, for every W 2W,

a. D ✓ D(W),b. W(D) ✓ D,c. W is essentially skew-adjoint on D.

2. If W,W1 and W2 are in W and a 2 R, then there exist operators R, S ,T 2W suchthat, for every 2 D,

W1( ) +W2( ) = R( )

aW( ) = S ( )

and

W1(W2 ) �W2(W1 ) = T ( ).

We will write R, S and T as W1 + W2, aW and [W1,W2], respectively, with the un-derstanding that they may be defined only on D and are assumed to be in W.

Remark 2.5.4. Some caution is required since, for example, “W1+W2” as it is beingused here need not be the same as the sum of the operators W1 and W2 in that itmay have a di↵erent domain. The terminology not withstanding, a “Lie algebra ofoperators” need not be a “Lie algebra”.

Every element of W has a unique extension to a skew-adjoint operator on Hwhich we will denote by the same symbol. Multiplying any of these by �i thengives a self-adjoint operator on H.

What we would like now is an analogue for strongly continuous, unitary repre-sentations of a Lie group on an infinite-dimensional Hilbert space of the fact that


any representation of a matrix Lie group G on a finite-dimensional Hilbert spacegives rise to a representation of the corresponding Lie algebra g simply by di↵er-entiation at the identity. More precisely, and more generally, one has the followingwell-known result (if it is not-so-well-known to you, see Theorem 3.18 of [Hall]).

Theorem 2.5.2. Let G and H be matrix Lie groups with Lie algebras g and h,respectively, and suppose � : G ! H is a Lie group homomorphism. Then thereexists a unique real linear map d� : g! h such that

1. d�( [X,Y]g

) = [ d�(X), d�(Y) ]h

for all X,Y 2 g,

2. d�(X) = ddt�(etX)

��t=0 = limt!0

�(etX )� 1Ht for all X 2 g, and

3. �(eX) = ed�(X) for every X 2 g.

This result applies, in particular, to a representation � : G ! GL(V) of G on afinite-dimensional Hilbert space V to give a representation d� : g! gl(V) of the Liealgebra of G. Segal has proved an analogue of this result for strongly continuous,unitary representations of G on any complex, separable Hilbert space (Theorem 3.1of [Segal2]), but we will not need to appeal to this. We will proceed toward theapplication we have in mind in the following way.

Let � : G ! U(H) be a strongly continuous, unitary representation of the matrixLie group G on the complex, separable Hilbert space H. For each X in the Liealgebra g of G the 1-parameter subgroup t ! etX of G is mapped by � to a stronglycontinuous 1-parameter group t ! �(etX) of unitary operators on H. Accordingto Stone’s Theorem (Section VIII.4 of [RS1]) there exists a unique skew-adjointoperator d�(X) on H such that

�(etX) = exp (td�(X))

for each t 2 R. Here the exponential map exp is defined by the functional calcu-lus (Theorem 5.5.8 of [Nab5]) in the following way. Since td�(X) is skew-adjoint,�i (td�(X)) is self-adjoint and the functional calculus gives a unitary operator

exp (i [�i(td�(X))]).

Since i [�i(td�(X))] = td�(X) we take this to be the definition of exp (td�(X)).d�(X) is given by

d�(X) =ddt�(etX)

��t=0 = lim

t!0

�(etX) � t

and its domain is the set of all 2 H for which this limit in H exists.Next we must appeal to a rather deep theorem of Nelson [Nel1] which asserts

that there exists a dense linear subspace D� of H with D� ✓ D(d�(X))8X 2 g thatis invariant under each d�(X)


d�(X)(D�) ✓ D� 8X 2 g

and on which each d�(X) is essentially skew-adjoint. The route Nelson took to-ward this result is sketched on pages 292-295 of [Nab5] and an application to theHeisenberg algebra is described on pages 295-300 of [Nab5]. As usual we will usethe same symbol d�(X) for the unique skew-adjoint extension of d�(X). With D�

we can define the Lie algebra of operators W(D�) on H consisting of all skew-symmetric operators W on H for which D� ✓ D(W), W(D�) ✓ D�, and W isessentially skew-adjoint on D�. The map

d� : g!W(D�)

then satisfies

d�(X + Y) = d�(X) + d�(Y) 8X,Y 2 g and 8 2 D�,

d�(aX) = ad�(X) 8X 2 g 8a 2 R and 8 2 D�,

and

d�([X,Y]g

) = [d�(X), d�(Y)] 8X,Y 2 g and 8 2 D�.

d� is called a realization of g by skew-adjoint operators on H. For each X 2 g,�id�(X) is self-adjoint on H. We will describe a concrete example at the end ofSection 2.8.

Remark 2.5.5. It is not uncommon to see d� referred to as a “representation” of theLie algebra g. The terminology can be misleading since the set of operators d�(X)need not form a Lie algebra.

Now suppose that H is the Hilbert space associated to some quantum systemand G is a symmetry group of that system represented by � : G ! U(H) (seeRemark A.4.2). Then, for each X 2 g, �id�(X) is a self-adjoint operator on H andtherefore corresponds to some observable for the quantum system. If {X1, . . . , Xn}is a basis for g and if each of the self-adjoint operators �id�(Xj), j = 1, . . . , n, hasbeen associated with some physical quantity, then we will say that the symmetrygroup G has been assigned a physical interpretation.

Return now to the case in which G = P"+ is the Poincare group (or its universalcover ISL(2,C) which has the same Lie algebra p). The existence of a unitary repre-sentation � : P"+ ! U(H) expresses a form of relativistic invariance of the quantumsystem whose Hilbert space is H (this is discussed in more detail in Section 2.7).The Lie algebra p has ten generators Pµ and Mµ⌫ = �M⌫µ, µ, ⌫ = 0, 1, 2, 3, µ , ⌫,which we have already associated with physical quantities. Specifically, in a fixedadmissible frame of reference, P1, P2, and P3 are associated with the linear momen-tum in the x1, x2 and x3 directions, P0 with the energy, M23,M31 and M12 with the


components of angular momentum, and M01,M02, and M03 with the components ofrelativistic angular momentum. Recall that the Pµ, µ = 0, 1, 2, 3, transform under L"+as a 4-vector (Exercise 2.5.9) and the Mµ⌫ transform under L"+ as a 4-tensor (Exer-cise 2.35). Moreover, conjugating by elements of the Abelian translation groupR1,3

leaves both invariant. From this we conclude that the physical interpretations of thegenerators are the same in every admissible frame of reference. We are therefore ledto postulate that the corresponding self-adjoint operators �id�(P0),�id�(P1), . . .on H represent the same physical quantities in the relativistically invariant quan-tum theory whose Hilbert space is H. Thus, for example, �id�(P0) is the operatorrepresenting the total energy of the system, otherwise known as the Hamiltonian.

In the next section we will introduce the machinery necessary to extend theseconsiderations to operators arising from “products” of generators of the form PµPµwhich do not live in the Lie algebra, but rather in what is called the “universalenveloping algebra”.

2.5.4 Universal Enveloping Algebra and Casimir Invariants

In this section we need to define what are called the “Casimir invariants” for thePoincare algebra p. These objects do not live in the Lie algebra. They are certainquadratic functions of the generators of p and a Lie algebra does not have enoughstructure to make sense of a quadratic function of its elements. The idea is to con-struct a certain unital, associative algebra U(p) with its associated commutator Liealgebra structure (Example 1.1.4) and show that p can be embedded, as a Lie alge-bra, in U(p). One can then use the ambient multiplicative structure of U(p) to definethe required quadratic functions. U(p) is called the “universal enveloping algebra”of p. Every Lie algebra g has one and we will begin with a brief discussion of how itis defined as a universal object, why it exists and how it can be described concretely.A good reference for all of the details is Chapter III of [Knapp].

Remark 2.5.6. Although much of what we have to say is true of both real and com-plex Lie algebras, we will generally want the field of scalars to be algebraicallyclosed. For this reason we will consider only complex Lie algebras in this section.If the Lie algebra of interest at any particular moment happens to arise as the (real)Lie algebra of some Lie group we will therefore reserve the symbol g for the com-plexification of this real Lie algebra just as we wrote p for the complexification ofthe Lie algebra of the Poincare group in the previous section.

We let g denote a finite-dimensional, complex Lie algebra. A universal envelop-ing algebra for g is a complex, unital, associative algebra U (with its associated com-mutator Lie algebra structure) together with a Lie algebra homomorphism ◆ : g! Uwith the following property. If A is another complex, unital, associative algebra(with its associated commutator Lie algebra structure) and


� : g! A

is another Lie algebra homomorphism, then there exists a unique algebra homomor-phism

: U! A

such that (1U) = 1A and

� = � ◆.

It is not at all obvious, but the Poincare-Birkho↵-Witt Theorem (Theorem 3.8 of[Knapp]) implies that ◆ : g ! U is injective so we can, and will, identify g asa Lie algebra with its image ◆(g) in U. In particular, the bracket on g, however itwas initially defined, can be viewed as the commutator bracket assoociated with themultiplication on U.

As usual, defining an object by a universal property does not guarantee that it ex-ists, but it does guarantee that, if it exists, it must be unique. Thus, any two universalenveloping algebras for g must be isomorphic as unital, associative algebras and weare justified in denoting it U(g) and calling it the universal enveloping algebra of g.To settle the question of existence one must construct from g a unital, associativealgebra and a Lie algebra homomorphism ◆ from g into it that satisfies the universalproperty proposed in the definition. For this one begins with the tensor algebra

T(g) =1M

k=0

Tk(g)

of the vector space g, where T0(g) = C, T1(g) = g and, for k � 2,

Tk(g) = g⌦ k· · · ⌦ g

is the kth-tensor power of g (there is a review of the tensor algebra in Appendix Aof [Knapp]). This is a (graded) associative algebra with multiplication given by thetensor product ⌦ and unit element 1 2 T0(g) = C. We let J(g) denote the ideal inT(g) generated by all elements of the form

X ⌦ Y � Y ⌦ X � [X,Y]

for all X and Y in g = T1(g). The quotient T(g)/J(g) is then a (graded) unital, associa-tive algebra. Products in T(g)/J(g) are generally written without the multiplicationsign ⌦ and the inclusion of g in T(g) as a vector subspace induces a map denoted◆ : g! T(g)/J(g) that satisfies

◆([X,Y]) = ◆(X)◆(Y) � ◆(Y)◆(X)


for all X,Y 2 g. The universal properties of the tensor algebra then imply thatT(g)/J(g) and ◆ : g ! T(g)/J(g) satisfy the defining properties of the universalenveloping algebra U(g) (Proposition 3.3 of [Knapp]) and this establishes the exis-tence of U(g). In particular, ◆ is injective and therefore embeds g in T(g)/J(g) as aLie subalgebra when T(g)/J(g) is given its commutator Lie algebra structure.

Since ◆ embeds g in U(g) as a Lie algebra one generally suppresses all mentionof it and simply regards g as a Lie subalgebra of U(g). Since g then lives insidethe associative algebra U(g) we can form products of the elements of g and theseproducts will also live in U(g). In particular, if {X1, . . . , Xn} is a basis for g, then themonomials

X j1i1 · · · X

jkik (2.45)

with 1 k n, jl � 0 for 1 l k, and

i1 < · · · < ik (2.46)

are all in U(g). According to Theorem 3.8 of [Knapp] these monomials actuallyform a basis for U(g).

Exercise 2.5.18. Show that Lie algebra representations of g are in one-to-one corre-spondence with algebra representations of U(g). More precisely, prove each of thefollowing.

1. Every Lie algebra representation � : g! End(V) of g on a complex vector spaceV extends to a unique algebra representation : U(g)! End(V) of U(g) on V .

2. Every algebra representation : U(g) ! End(V) of U(g) on a complex vectorspace V is the extension of a unique Lie algebra representation of g on V .

This exercise applies, in particular, to the adjoint representation ad of g on gwhich therefore extends to an algebra representation of U(g) on g. We will denotethis extension

ad : U(g)! End(g)

and to call it the adjoint representation of U(g).

Remark 2.5.7. Although we will not make use of it we mention that there is another,more analytic description of the universal enveloping algebra for the Lie algebra ofa Lie group G. One can identify this Lie algebra with the left-invariant vector fieldson G. Now, a vector field can be thought of as a derivation on the smooth functiondefined on G, that is, as first order di↵erential operator. Composing such opera-tors generates left-invariant di↵erential operators of higher order on G. Proposition1.9, Chapter II, of [Helg] proves that the associative algebra generated by the left-invariant vector fields on G and the identity map is isomorphic to U(g).


Now we will turn to the special case of the Poincare algebra p. A basis for p isgiven by

X1 = P0, X2 = P1, X3 = P2, X4 = P3,

X5 = J1 = M23, X6 = J2 = M31, X7 = J3 = M12,

X8 = K1 = M10, X9 = K2 = M20, X10 = K3 = M30,

so that the monomials (2.45) in these generators span U(p). We are particularlyinterested in two specific elements of U(p). The first is denoted P2 and is defined asfollows. For each µ = 0, 1, 2, 3, define Pµ by Pµ = ⌘µ�P�. Then

P2 = PµPµ = P20 � P2

1 � P22 � P2

3. (2.47)

The second is denoted W2 and is defined as follows. Begin by defining the com-ponents Wµ 2 U(p) of what is called the Pauli-Lubanski vector by

Wµ =12✏µ⌫⇢�M⌫⇢P�, (2.48)

where M⌫⇢ = ⌘⌫�⌘⇢�M�� and ✏µ⌫⇢� is the Levi-Civita symbol (1 if µ⌫⇢� is an evenpermutation of 0123, -1 if µ⌫⇢� is an odd permutation of 0123 and 0 otherwise).Raising the index we define

Wµ = ⌘µ�W�

so that W0 = W0 and W j = �Wj, j = 1, 2, 3. Thus, for example,

W0 = W0 =12✏0⌫⇢�M⌫⇢P� =

12✏0123M12P3 +

12✏0213M21P3 +

12✏0321M32P1

+12✏0231M23P1 +

12✏0312M31P2 +

12✏0132M13P2

=12

(1)M12P3 +12

(�1)(�M12)P3 +12

(�1)(�M23)P1

+12

(1)M23P1 +12

(1)M31P2 +12

(�1)(�M31)P2

and so

W0 = W0 = M12P3 + M23P1 + M31P2.

Notice that

Wµ =12✏µ⌫⇢�M⌫⇢P�,

where


✏µ⌫⇢� = ⌘µ↵⌘⌫�⌘⇢�⌘��✏↵�� = �✏µ⌫⇢�.


1. W0 = W0 = J1P1 + J2P2 + J3P3

2. W1 = �W1 = J1P0 + K2P3 � K3P2

3. W2 = �W2 = J2P0 + K3P1 � K1P3

4. W3 = �W3 = J3P0 + K1P2 � K2P1

Exercise 2.5.20. Prove that

Mµ⌫PµP⌫ = 0. (2.49)

Hint: [Pµ, P⌫] = 0 for all µ, ⌫ = 0, 1, 2, 3.

Next we define

W2 = WµWµ = W20 �W2

1 �W22 �W2

3 . (2.50)

We will discuss the significance of P2 and W2 shortly, but first we will need a fewuseful relations (the brackets below refer to the commutator bracket in U(p)).

WµPµ = 0 (2.51)

[Wµ, P⌫] = 0 8µ, ⌫ = 0, 1, 2, 3 (2.52)

[Wµ,M↵�] = i (W↵⌘µ� �W�⌘µ↵) 8µ,↵, � = 0, 1, 2, 3 (2.53)

[Wµ,W⌫] = i ✏µ⌫⇢�W⇢P� 8µ, ⌫ = 0, 1, 2, 3 (2.54)

For the proof of (2.51) we write

WµPµ =12✏µ⌫⇢�M⌫⇢P�Pµ =

12

M⌫⇢(✏µ⌫⇢�P�Pµ).

Now, for each fixed ⌫ and ⇢, ✏µ⌫⇢� is skew-symmetric in � and µ, whereas P�Pµ issymmetric in � and µ (because [P�, Pµ] = 0). Consequently, the terms in the sum✏µ⌫⇢�P�Pµ cancel in pairs for each ⌫ and ⇢ and (2.51) is proved.

To prove (2.52) we first note that, if [ , ] is the commutator bracket of any asso-ciative algebra A, then [AB,C] = A[B,C]+ [A,C]B for all A, B,C 2 A (just expandboth sides). Now compute


[Wµ, P⌫] =12✏µ↵��[M↵�P�, P⌫] =

12✏µ↵��M↵�[P�, P⌫] +

12✏µ↵��[M↵�, P⌫]P�.

The first term is clearly zero since [P�, Pµ] = 0 for all �, µ = 0, 1, 2, 3. To see thatthe second term is also zero we first compute

[M↵�, P⌫] = ⌘�⌫[M↵�, P�] = ⌘�⌫i (⌘��P↵ � ⌘↵�P�) = i (�⌫�P↵ � �⌫↵P�).

Consequently,

[Wµ, P⌫] =12

i ✏µ↵��⌫�P↵P� �12

i ✏µ↵��⌫↵P�P�

=12

i ✏µ↵⌫�P↵P� �12

i ✏µ⌫��P�P�

= �i ✏µ⌫��P�P�

which is zero because ✏µ⌫�� is skew-symmetric in � and �, whereas P�P� is symmet-ric in � and �. We will leave (2.53) and (2.54) for you.


1. [Wµ,M↵�] = i (W↵⌘µ� �W�⌘µ↵)2. [Wµ,W⌫] = i ✏µ⌫⇢�W⇢P�

Although a bit more labor-intensive, similar arguments will establish

W2 = �12

Mµ⌫Mµ⌫P2 + M⇢�M⌫�P⇢P⌫. (2.55)

The elements P2 and W2 of U(p) are called Casimir invariants of p. The reasonwe care about them is that they commute with all of the generators of p in U(p), thatis,

[Pµ,P2] = 0 and [Mµ⌫,P2] = 0, µ, ⌫ = 0, 1, 2, 3 (2.56)

and

[Pµ,W2] = 0 and [Mµ⌫,W2] = 0, µ, ⌫ = 0, 1, 2, 3. (2.57)

To prove (2.56) we compute

[Pµ,P2] = [Pµ, P⇢P⇢] = ⌘⇢�[Pµ, P⇢P�] = ⌘⇢�[Pµ, P⇢]P� + ⌘⇢�P⇢[Pµ, P�] = 0

by (2.24). Next we compute, using (2.25) and (2.24),


[Mµ⌫,P2] = [Mµ⌫, P⇢P⇢] = ⌘⇢�[Mµ⌫, P⇢]P� + ⌘⇢�P⇢[Mµ⌫, P�]

= ⌘⇢��

i(⌘⌫⇢Pµ � ⌘µ⇢P⌫)�P� + ⌘

⇢�P⇢�

i(⌘⌫�Pµ � ⌘µ�P⌫)�

= i��⌫PµP� � ��µP⌫P� + �

⇢⌫P⇢Pµ � �⇢µP⇢P⌫

�

= i�

PµP⌫ � P⌫Pµ + P⌫Pµ � PµP⌫�

= i�

[Pµ, P⌫] + [P⌫, Pµ]�

= 0.

Exercise 2.5.22. Prove (2.57). Hint: Use (2.55).

You are no doubt wondering about the significance of (2.56) and (2.57). Whatis implied by the fact that P2 and W2 commute in U(p) with all of the generatorsof p? Although we will not be in a position to o↵er a fully satisfactory explanationbefore Section 2.8 it may be helpful to consider a few finite-dimensional analoguesof what we have done just to be aware of what we will try to mimic in the infinite-dimensional situation in which we find ourselves.

Let us suppose then that we have a matrix Lie group G with (complexified) Liealgebra g and a representation � : G ! GL(V) of G on some finite-dimensionalcomplex vector space V. By Theorem 2.5.2, � induces a representation d� : g !gl(V) of the Lie algebra on V. Now let {X1, . . . , Xn} be a basis for g. Then U(g)is generated as an associative algebra by {X1, . . . , Xn} (see (2.45) and the remarksfollowing it).

Exercise 2.5.23. Show that the Lie algebra representation d� : g ! gl(V) extendsto an algebra representation of U(g) on V, that is, to an algebra homomorphism ofU(g) into the algebra End(V) of endomorphisms of V.

We will use the same symbol

d� : U(g)! End(V)

for the algebra representation determined by d� : g! gl(V) . Now suppose we havesome element Z of U(g) that commutes with every generator Xj in U(g). Since everyelement of U(g) is a linear combination of the monomials (2.45), Z must commutewith everything in U(g), that is, Z is in the center Z(U(g)) of U(g). Since d� is analgebra homomorphism, d�(Z) 2 End(V) commutes with everything in the imageof d�. If we now assume that � : G ! GL(V) is irreducible, so that d� : g! gl(V)and also d� : U(g) ! End(V) are irreducible, we can appeal to the followingversion of Schur’s Lemma.

Theorem 2.5.3. (Schur’s Lemma) Let A be an associative algebra over C and V afinite-dimensional complex vector space. Suppose ⇢ : A! End(V) is an irreduciblerepresentation of A on V. If A 2 End(V) commutes with ⇢(a) 2 End(V) for everya 2 A, then A = � idV for some � 2 C.


Proof. For any a 2 A, A(⇢(a)(v)) = ⇢(a)(A(v)) for every v 2 V. Consequently, ifv is in Kernel (A), then ⇢(a)(v) is in Kernel (A). Therefore Kernel (A) is an invari-ant subspace for ⇢. Since ⇢ is assumed irreducible, Kernel (A) is either 0 or V. IfKernel (A) = V, then A = 0 so A = 0 idV and we are done. If Kernel (A) = 0,then A is invertible. Any invertible operator on a finite-dimensional complex vectorspace has an eigenvalue � which cannot be zero. Thus, there exists a nonzero � 2 Cwith (A � � idV)v = 0 for some nonzero v 2 V. Since A and � idV both commutewith every ⇢(a), so does A � � idV and therefore, by what we have shown above,Kernel (A�� idV) is either 0 or V. But Kernel (A�� idV) = 0 is impossible since v isa nonzero vector on which A�� idV vanishes. Consequently, Kernel (A�� idV) = Vso A = � idV as required. ut

The conclusion we draw from all of this is as follows. If � : G ! GL(V) is anirreducible representation of the Lie group G on a finite-dimensional, complex vec-tor space V and d� : U(g) ! End(V) is the induced representation of the universalenveloping algebra U(g) and if Z 2 U(g) commutes in U(g) with all of the generatorsof g, then d�(Z) acts as a scalar on V.

Of course, none of this is directly applicable to the infinite-dimensional situationdescribed in the previous section where we had only a realization of the Poincarealgebra p on a Hilbert space that is generally infinite-dimensional. In particular, ittells us nothing about the Casimir invariants P2 and W2. Nevertheless, we will findthat, once we have explicit formulas for the irreducible, unitary representations ofthe Poincare group in hand (Section 2.8), entirely analogous results will emerge andthese will provide the key to the physical interpretation of the representations.

2.6 Momentum Space

2.6.1 Orbits and Isotropy Groups

Classical mechanics, as we viewed it in Sections A.2 and A.3, begins with a configu-ration space M and represents the states of the system either by points in the tangentbundle T M, called the state space, or by points in the cotangent bundle T ⇤M, calledthe phase space. Elements of T M are pairs (x, vx), where x 2 M and vx is in thetangent space Tx(M) to M at x, representing a velocity. Elements of T ⇤M are pairs(x, ⌘x), where x 2 M and ⌘x is in the dual T ⇤x (M) of Tx(M), representing a conju-gate momentum. When the configuration space isRn the velocities can be identifiedwith points in Rn and the conjugate momenta with points in its dual (Rn)⇤. In ourpresent context the role of the configuration space is played by M. With a choice ofadmissible basis {e0, e1, e2, e3} this is identified with R1,3. With this as motivationwe define the momentum space P1,3 associated with {e0, e1, e2, e3} to be the vectorspace dual of R1,3.

2.6 Momentum Space 93

P1,3 = (R1,3)⇤

The dual basis forP1,3 will be written {e0, e1, e2, e3} and we will designate the pointsof P1,3 by p = p↵e↵, or simply by their coordinates (p0, p1, p2, p3) = (p0,p).

The natural L"+-action on R1,3 gives rise to an L"+-action on P1,3, called the con-tragredient action, and defined as follows. If p : R1.3 ! R is a linear functional inP1,3, then ⇤ · p is defined by

(⇤ · p)(x) = p(⇤�1 · x)

for every ⇤ 2 L"+. In coordinates relative to {e0, e1, e2, e3} this is given by

⇤ · p = (⇤�1)T p.

Recalling from Exercise 2.3.3 that (⇤�1)T is in L"+ whenever ⇤ is in L"+ we concludethat all of the following subsets of P1,3 are invariant under this L"+-action. For anym > 0 define

X+m = {p 2 P1,3 : p20 � p2

1 � p22 � p2

3 = m2, p0 > 0},X�m = {p 2 P1,3 : p2

0 � p21 � p2

2 � p23 = m2, p0 < 0},

Ym = {p 2 P1,3 : p20 � p2

1 � p22 � p2

3 = �m2}

and, for m = 0, define

X+0 = {p 2 P1,3 : p20 � p2

1 � p22 � p2

3 = 0, p0 > 0},X�0 = {p 2 P1,3 : p2

0 � p21 � p2

2 � p23 = 0, p0 < 0},

X00 = {0}.

X+m and X�m for m > 0 are called the positive and negative mass hyperboloids, re-spectively. With the exception of a few remarks now and then we will concentratealmost exclusively on X+m.

Remark 2.6.1. We should say a word about the terminology. X+m is called the positivemass hyperboloid and the reason for the hyperboloid designation is no doubt clear.However, the mass m is assumed positive for both X+m and X�m so positive must referto something else. To see what this might be we proceed as follows. Suppose ↵(⌧)is a timelike worldline parametrized by proper time and m is a positive real number.Then the pair (↵,m) is identified with a material particle whose worldline is ↵ andwhose mass is m. Since the 4-velocity ↵0(⌧) is a unit timelike vector at each point,

p = p(⌧) = m↵0(⌧)

satisfies

hp, pi = m2.


In admissible coordinates this is just

p20 � p2

1 � p22 � p2

3 = m2.

Consequently, the 4-momentum of the particle lives in Xm = X+m t X�m. Now wewould like to rewrite p in terms of the instantaneous speed of the particle relative tothe given frame of reference. We denote this

� = �(x0) =

s✓dx1

dx0

◆2+

✓dx2

dx0

◆2+

✓dx3

dx0

◆2

and define � = �(x0) = (1 � �2(x0))�1/2. We will omit the simple computations(which are available on pages 51-52 and 81-82 of [Nab4]) and just quote the resultwe need. Assuming p0 > 0,

p = (p0, p1, p2, p3) = m�✓1,

dx1

dx0 ,dx2

dx0 ,dx3

dx0

◆= m� (1, v),

where v is the ordinary velocity vector of the particle in the given frame. Now, since�(x0) is in (�1, 1) for each x0 we can write a convergent binomial series expansionfor �.

� = (1 � �2)�1/2 = 1 +12�2 +

38�4 + · · ·

If we write v = (dx1/dx0, dx2/dx0, dx3/dx0) = (v1, v2, v3) this gives

pi = mvi +12

mvi �2 +38

mvi �4 + · · · , i = 1, 2, 3,

and

p0 = m +12

m �2 +38

m �4 + · · ·

The term mvi in the expression for pi is the ith-component of the Newtonian momen-tum in the given frame and the remaining terms are the relativistic corrections. Onthe other hand, the appearance of the term 1

2 m �2 corresponding to the Newtoniankinetic energy in the expression for p0 leads us to refer to p0 as the total relativisticenergy of the particle and denote it

E = p0 = m� = m +12

m �2 +38

m �4 + · · ·

In particular, X+m is the surface in P1,3 containing the 4-momenta of particles withmass m > 0 and positive energy. This would lead one to identify X�m with the sur-face in P1,3 containing the 4-momenta of particles with negative energy. Classicallyone would simply ignore X�m as being unphysical. In quantum theory the situation


is not so simple, but the physical interpretation of particles with negative energyinvolves subtleties that we will have no need to get into here and this accounts forour particular interest in X+m.

We would like to point out one other consequence of the identification of E andp0. We will call p = (p1, p2, p3) the relativistic 3-momentum of the particle in thegiven frame of reference and will write kp k2 for p2

1 + p22 + p2

3. Then

E2 = kp k2 + m2. (2.58)

This is called the Einstein energy-momentum relation. When one uses traditionalunits of time rather than x0 = ct (2.58) becomes

E2 = kp k2c2 + m2c4. (2.59)

It is worth pointing out that when � = 0 and E = p0 > 0 this reduces to

E = mc2

which it is possible you have seen before.

In Section 2.8 we will require some fairly detailed information about P1,3 so wewill devote the remainder of this section to deriving what we need. We have alreadyseen that the positive mass hyperboloid X+m in P1,3 is invariant under the L"+-actiononP1,3. Now we will show that it is the complete orbit of any one of its points underthis action. In other words, L"+ acts transitively on X+m.

Let p = (p0, p1, p2, p3) be an arbitrary point in X+m (m > 0). Since X+m is invariantunder L"+, any element of the L"+-orbit of p is in X+m. We will show, conversely, thatany element q = (q0, q1, q2, q3) of X+m is in the orbit of p. Notice that it is enough tofind a ⇤ 2 L"+ with ⇤p = q since then (⇤�1)T 2 L"+ and (⇤�1)T · p = q. We first showthat there is an element of L"+ that sends (m, 0, 0, 0) to p. We will apply Exercise2.3.4. Take µ = (p2

0 �m2)1/2/m and ⌫ = p0/m. Notice that p20 �m2 = p2

1 + p22 + p2

3 =kp k2 so µ is real. Moreover, ⌫ > 0 since p 2 X+m and ⌫2 � µ2 = 1. With thesechoices ⇤µ,⌫ sends (m, 0, 0, 0) to (p0, kp k, 0, 0). Since the rotation group SO(3) actstransitively on the sphere of radius kp k in R3 (see pages 90-91 of [Nab2]) thereis a rotation R 2 R that carries (p0, kp k, 0, 0) to (p0, p1, p2, p3). Thus, R⇤µ,⌫ 2L"+ sends (m, 0, 0, 0) to (p0, p1, p2, p3), as required. Since p was arbitrary we haveshown that, for any p, q 2 X+m, there exist ⇤p,⇤q 2 L"+ with ⇤p(m, 0, 0, 0) = p and⇤q(m, 0, 0, 0) = q. Consequently, ⇤ = ⇤q⇤�1

p carries p to q. Every q in X+m is inthe L"+-orbit of p, as required. Analogous arguments starting with (�m, 0, 0, 0) showthat X�m is the complete orbit of any one of its points.

Exercise 2.6.1. Prove the same result for X+0 by taking µ = (p20 � 1)/2p0 and ⌫ =

(p20 + 1)/2p0 and looking at the image of (1, 1, 0, 0) under ⇤µ,⌫. Find an analogous

argument for X�0 .


Exercise 2.6.2. Prove the same result for Ym by taking µ = p0/m and ⌫ = (p20+m2)/m

and looking at the image of (0,m, 0, 0) under ⇤µ,⌫.

Since every point of P1,3 is on exactly one of X±m, Ym, X±0 or X00 these are, in fact, all

of the orbits of L"+ in P1,3.

Next we will need to determine the isotropy subgroup HP0 of P0 = (m, 0, 0, 0) 2X+m under the L"+-action. Thus, we are looking for all of the ⇤ 2 L"+ for which⇤ · P0 = P0, that is, for which (⇤�1)T P0 = P0. If ⇤ = (⇤↵�)↵, �=0,1,2,3, then

(⇤�1)T = ⌘⇤⌘ =

0BBBBBBBBBBBB@

⇤00 �⇤0

1 �⇤02 �⇤0

3�⇤1

0 ⇤11 ⇤1

2 ⇤13

�⇤20 ⇤2

1 ⇤22 ⇤2

3�⇤3

0 ⇤31 ⇤3

2 ⇤33

1CCCCCCCCCCCCA

Applying this to (m, 0, 0, 0) gives

0BBBBBBBBBBBB@

⇤00m

�⇤10m

�⇤20m

�⇤30m

1CCCCCCCCCCCCA

which is equal to 0BBBBBBBBBBBB@

m000

1CCCCCCCCCCCCA

if and only if the first column of (⇤�1)T is

0BBBBBBBBBBBB@

1000

1CCCCCCCCCCCCA

and this is the case if and only if (⇤�1)T is in the rotation subgroup R of L"+ (Theorem2.3.4). Consequently,

HP = R � SO(3).

The same argument shows that, for m > 0, the isotropy subgroup of (�m, 0, 0, 0) inX�m is R.

Remark 2.6.2. The isotropy subgroups for Ym (m > 0) can be obtained in a similarfashion, while those of X±0 are somewhat more involved. Since we will require onlythe X+m case we will simply refer those interested in seeing the results for Ym and X±0to pages 340-341 of [Vara].


We have shown that L"+ acts transitively on the left on X+m with isotropy subgroupR at (m, 0, 0, 0). Consequently, the homogeneous manifold L"+/R is di↵eomorphicto X+m (Theorem 1.1.6) which, in turn, is di↵eomorphic to R3 via the projection

⇡+ : X+m ! R3

⇡+(p0, p1,p2, p3) = (p1, p2, p3).

Since R3 is connected and SO(3) is connected, it follows that L"+ is also connected(Proposition 1.6.5 of [Nab2]). We can say more, however. In Section 1.4 we showedthat the natural projection ⇡ : L"+ ! L"+/R has the structure of a principal R-bundleover L"+/R.. But L"+/R � R3 and every principal bundle over any Rn is trivial so

L"+ � R3 ⇥ SO(3).

From this we conclude that, not only is L"+ connected, but that its fundamental groupis Z2 (Theorem 2.4.5, Theorem 2.4.10, and Appendix A of [Nab2]).

Let : SL(2,C) ! L"+ be the double covering map described in Section 2.4.Then SL(2,C) acts on P1,3 via the L"+-action, that is, by

A · p = (A) · p = ((A)�1)T p

for every A 2 SL(2,C) and every p 2 R1,3. Since is surjective the orbits of theSL(2,C)-action are the same as those of the L"+-action and SL(2,C) acts transitivelyon X+m. The isotropy group of any point in X+m is the inverse image of the rotationgroup R in L"+ under and, as we have seen in Section 2.4, this is isomorphicto SU(2). Thus, just as for L"+/R, the homogeneous manifold SL(2,C)/SU(2) isdi↵eomorphic to X+m which, in turn, is di↵eomorphic to R3.

Exercise 2.6.3. Show that SL(2,C) is di↵eomorphic to R3 ⇥ SU(2).

According to Theorem 1.1.6 one can describe a di↵eomorphism of SL(2,C)/SU(2)onto X+m as follows. Fix the point P0 = (m, 0, 0, 0) 2 X+m. The isotropy subgroup ofP0 is SU(2) so a di↵eomorphism

�P0 : SL(2,C)/SU(2)! X+m

is defined by

�P0 ([A]) = A · P0

for every [A] 2 SL(2,C)/SU(2). Notice that this is independent of the representativeA chosen for [A] because SU(2) fixes P0. The inverse di↵eomorphism carries p 2 X+mto the unique [A] 2 SL(2,C)/SU(2) with A ·P0 = p for each representative A of [A].

(�P0 )�1(p) = (�P0 )�1(A · P0) = [A] (2.60)


We would like to make a smooth selection !(p) of a representative of this [A] forp 2 X+m. Since the smooth principal SU(2)-bundle SL(2,C) ! SL(2,C)/SU(2) �R3 is trivial it has smooth global sections. Choose such a section

u : SL(2,C)/SU(2)! SL(2,C)

and define

! : X+m ! SL(2,C)

by

!(p) = (u � ��1P0

)(p). (2.61)

Then !(p) is a smooth function from X+m to SL(2,C) satisfying

!(p) · P0 = p (2.62)

for every p 2 X+m.

Remark 2.6.3. The map ! : X+m ! SL(2,C) depends, of course, on the choiceof the section u : SL(2,C)/SU(2) ! SL(2,C). Somewhat later we will want tomake a choice for u that produces an ! that is equivariant with respect to cer-tain SU(2)-actions on X+m and SL(2,C). We define actions of SU(2) on SL(2,C),SL(2,C)/SU(2), and X+m as follows. Let U 2 SU(2) and define, for A 2 SL(2,C),[A] 2 SL(2,C)/SU(2), and p = �P0 ([A]) 2 X+m,

U · A = UAU�1

U · [A] = [UA]U · p = U · �P0 ([A]) = �P0 (U · [A]) = �P0 ([UA]).

Exercise 2.6.4. Show that, with respect to these SU(2)-actions, ! is equivariant

!(U · p) = U · !(p)

if and only if u is equivariant

u(U · [A]) = U · u([A]).

Thus, we need only construct a section u : SL(2,C)/SU(2) ! SL(2,C) satisfy-ing

u( [UA] ) = U u([A]) U�1

for any U 2 SU(2). For this we use the fact that any A 2 SL(2,C) has a unique polardecomposition


A = PV,

where P is positive definite Hermitian and V is in SU(2). Notice that det P = 1 soP is in SL(2,C) and, moreover,

[A] = A SU(2) = (PV) SU(2) = P SU(2) = [P].

Now define u : SL(2,C)/SU(2)! SL(2,C) by

u([A]) = u([P]) = P.

Next we observe that

[UA] = (UA) SU(2) = (UPV) SU(2) = (UP) SU(2) = (UPU�1) SU(2) = [UPU�1]

Consequently, u([UA]) = u([UPU�1]) = P0, where P0V 0 is the polar decompositionof UPU�1. But since P is positive definite Hermitian it can be written in the formP = eB, where B is Hermitian, and therefore

UPU�1 = UeBU�1 = eUBU�1

which is also positive definite Hermitian. By uniqueness of the polar decompositionof UPU�1, P0 = UPU�1 and V 0 = id2⇥2 so

u([UA]) = UPU�1 = U u([A]) U�1

as required. With this u, ! is equivariant with respect with respect to the SU(2)-actions on X+m and SL(2,C).

2.6.2 Invariant Measures

Since L"+ acts transitively on X+m one can ask if there is a Borel measure on X+mthat is invariant under this action in the same way that the Lebesgue measure onRn is invariant under the action of the translation group. Indeed, there is such ameasure and, up to a positive multiple, it is unique. The uniqueness modulo positiveconstants is proved in the Appendix to Section IX.8 of [RS2]. We will show that onesuch measure µm is defined as follows. For any Borel set B ✓ X+m, ⇡+(B) is a Borelsubset of R3, where

⇡+ : X+m ! R3

⇡+(p0, p1,p2, p3) = (p1, p2, p3)

is the projection. Now define


µm(B) =Z

⇡+(B)

d3p2!p,

where

!p =

qm2 + kp k2 =

qm2 + p2

1 + p22 + p2

3

and d3p = dp1 dp2 dp3 denotes integration with respect to Lebesgue measure onR3. Notice that, if p = ⇡+(p), then !p = p0.

Remark 2.6.4. This definition is not as mysterious as it might seem, being just aspecial case of a general construction of surface measures induced on smooth hy-persurfaces in Rn by the Lebesgue measure on Rn (see pages 78-79, Section IX.9,of [RS2] and also Section 1.2, Chapter III, of [GS]).

We will show that µm is invariant under the action of L"+ on X+m by simply apply-ing the Change of Variables Formula which we record here.

Theorem 2.6.1. (Change of Variables Formula) Let U and V be open subsets ofRn and g a C1-di↵eomeorphism of U onto V with Jacobian determinant det (g 0).Suppose B ✓ U is a Borel set and f : V ! R is Lebesgue integrable. Then g(B) ✓ Vis a Borel set, ( f � g) | det (g 0) | is integrable on B and

Z

g(B)f dnx =

Z

B( f � g) | det (g 0) | dnx.

Denote by ⇤ also the di↵eomorphism of X+m onto itself corresponding to theaction of some element ⇤ of L"+. Also let s+ : R3 ! X+m denote the maps+(p) = s+(p1, p2, p3) = (!p, p1, p2, p3). Thus, s+ : R3 ! X+m and ⇡+ : X+m ! R3

are inverse di↵eomorphisms and g = ⇡+ �⇤ � s+ is a di↵eomorphism ofR3 ontoR3

that carries ⇡+(B) onto ⇡+(⇤(B)). We need to show that µm(⇤(B)) = µm(B). For thiswe take f (p) = 1

2!p. Notice that

( f � g)(p) =1

2!⇡+(⇤(!p,p)).

The remainder of the proof is simplified if we recall (Theorem 2.3.5) that every⇤ 2 L"+ can be written as the composition of two rotations and a boost so it will beenough to prove µm(⇤(B)) = µm(B) for boosts and rotations separately. Furthermore,all of the boosts are treated in exactly the same way so it will su�ce to consider

⇤ =

0BBBBBBBBBBBB@

cosh ✓ 0 0 sinh ✓0 1 0 00 0 1 0

sinh ✓ 0 0 cosh ✓

1CCCCCCCCCCCCA.


From this we obtain

⇤(!p,p) = ⇤(!p, p1, p2, p3)= (!p cosh ✓ + p3sinh ✓, p1, p2,!p sinh ✓ + p3cosh ✓)

and therefore

!⇡+(⇤(!p,p)) = !p cosh ✓ + p3sinh ✓.

Now, for the Jacobian g 0 we write the coordinate transformation of R3 deter-mined by g as

(p1, p2, p3)! (p1, p2,q

m2 + p21 + p2

2 + p23 sinh ✓ + p3 cosh ✓)

and then just compute the derivatives. The result is

g 0 =

0BBBBBBBBB@

1 0 00 1 0

p1 sinh ✓!p

p2 sinh ✓!p

p3 sinh ✓+!pcosh ✓!p

1CCCCCCCCCA.

Consequently,

det (g 0) =p3sinh ✓ + !pcosh ✓

!p=!⇡+(⇤(!p,p))

!p

which is positive because ⇤ is orthochronous. Substituting all of this into the Changeof Variables Formula (Theorem 2.6.1) gives

µm(⇤(B)) =Z

⇡+(⇤(B))

d3p2!p

=

Z

⇡+(B)

12!⇡+(⇤(!p,p))

!⇡+(⇤(!p,p))

!pd3p

=

Z

⇡+(B)

d3p2!p

= µm(B)

as required.

Exercise 2.6.5. Complete the proof by showing that µm(⇤(B)) = µm(B) when ⇤ isin the rotation subgroup of L"+.

Remark 2.6.5. The Hilbert space L2(X+m, µm) plays a particularly prominent role inthe construction of a rigorous model of the so-called Wightman axioms for scalarquantum field theory. We will return to this somewhat later (also see Section X.7 of[RS2]).

Since the action of SL(2,C) on X+m is defined in terms of the L"+-action via thedouble covering : SL(2,C) ! L"+, the measure µm is also invariant under the


SL(2,C)-action. Recalling the X+m is di↵eomorphic to the homogeneous manifoldSL(2,C)/SU(2) which admits an SL(2,C)-action defined by B · [A] = [BA] forall B 2 SL(2,C) and all [A] 2 SL(2,C)/SU(2), we would like to move µm to anSL(2,C)-invariant measure on SL(2,C)/SU(2).

Remark 2.6.6. Recall that, if (X,A,m) is a measure space, (Y,B) is a measurablespace and g : (X,A) ! (Y,B) is a measurable function, then the pushforwardmeasure g⇤(m) on (Y,B) is defined by

(g⇤(m))(B) = m(g�1(B)) 8B 2 B.

There is an abstract change of variables formula (Theorem C, Section 39, of [Hal1])that asserts the following. If f is any extended real-valued measurable function on(Y,B), then f is integrable with respect to g⇤(m) if and only if f � g is m-integrableand, in this case,

Z

Yf (y) d(g⇤(m))(y) =

Z

X( f � g)(x) dm(x). (2.63)

Furthermore, (2.63) holds in the stronger sense that, if either side is defined (even ifit is not finite), then the other side is defined and they agree.

Exercise 2.6.6. Let ��1P0

be the di↵eomorphism defined by (2.60). Show that

µ = (��1P0

)⇤(µm)

is an SL(2,C)-invariant measure on SL(2,C)/SU(2).

2.6.3 Momentum Space and the Character Group

The next item we require is the identification of P1,3 with the character group R1,3

of the additive (translation) group of R1,3 (character groups for Abelian Lie groupsare reviewed in Remark 1.5.2). This will simplify the application of the Mackeymachine in Section 2.8.

We begin by reviewing the relevant notation. Choose a fixed, but arbitrary ad-missible basis {e0, e1, e2, e3} for Minkowski spacetime M and thereby identify Mwith R1,3, the elements of which will be denoted x = (x↵) = (x0, x1, x2, x3) =(x0, x), y = (y↵) = (y0, y1, y2, y3) = (y0, y), . . . We will use the same symbol R1,3

for its additive group of translations. The dual basis for P1,3 = (R1,3)⇤ is denoted{e0, e1, e2, e3} and the elements of P1,3 will generally be written in the dual basisas p = (p↵) = (p0, p1, p2, p3) = (p0,p). The double covering of the proper, or-thochronous Lorentz group L"+ is written


: SL(2,C)! L"+

and SL(2,C) acts on R1,3 by defining

� : SL(2,C)! GL(R1,3)

by

�(A)(x) = A · x = (A)x,

where A 2 SL(2,C) and (A)x is the matrix product of (A) 2 L"+ with the (col-umn) vector x 2 R1,3. With this action we define inhomogeneous SL(2,C), denotedISL(2,C), to be the semi-direct product ofR1,3 (thought of as the translation group)and SL(2,C), that is,

ISL(2,C) = R1,3o� SL(2,C).

As for P"+ one generally omits the explicit reference to � in the notation and writes

ISL(2,C) = R1,3o SL(2,C).

The action of SL(2,C) on R1,3 induces a contragredient action

�⇤ : SL(2,C)! GL( (R1,3)⇤ )

of SL(2,C) on the vector space dual P1,3 = (R1,3)⇤ of R1,3 defined by

[�⇤(A)(�) ](x) = [A · �](x) = �(�(A�1)(x) ),

where A 2 SL(2,C), � : R1,3 ! R is in P1,3 = (R1,3)⇤, and x 2 R1,3. In terms ofcomponents relative to {e0, e1, e2, e3}

�⇤(A)(p) = A · p = (A�1)T p,

where T means transpose and (A�1)T p is the matrix product of (A�1)T and the(column) vector p.

Next we consider the character group R1,3 of the translation group R1,3 (seeRemark 1.5.2). This has nothing to do with the signature of the inner product soR1,3 is the same as R4. The action � of SL(2,C) on R1,3 also induces an action� of SL(2,C) on R1,3 as follows. Any ⇠ 2 R1,3 is a continuous homomorphismfrom the additive Abelian group R1,3 of translations to the multiplicative group S 1

of complex numbers of modulus one. For any A 2 SL(2,C) we define

[ �(A)(⇠) ](a) = [A · ⇠](a) = ⇠(�(A�1)(a) )

for every a 2 R1,3. In Remark 1.5.2 it is shown that every ⇠ in R1,3 can be writtenuniquely in the form


ei(p0 x0+p1 x1+p2 x2+p3 x3)

for some (p0, p1, p2, p3) 2 R4 so that

(p0, p1, p2, p3) 2 R4 7! ei(p0 x0+p1 x1+p2 x2+p3 x3) 2 R1,3

is a group isomorphism of the additive group R4 onto the character group R1,3.

Remark 2.6.7. It will be convenient later on to note that the characters in R1,3 canequally well be written uniquely as

ei(p0 x0�p1 x1�p2 x2�p3 x3)

simply by changing the signs of p1, p2 and p3.

We would like to show that R1,3 admits the structure of a real vector space andthat with this structure there is a linear isomorphism of (R1,3)⇤ onto R1,3 that inter-twines the actions of SL(2,C) on R1,3 and (R1,3)⇤.

Exercise 2.6.7. Define addition + and real scalar multiplication on R1,3 by

(⇠1 + ⇠2)(a) = ⇠1(a)⇠2(a)(c⇠)(a) = ⇠(ca) 8c 2 R

and show that, with these operations, R1,3 becomes a real vector space.

Exercise 2.6.8. Let � : R! S 1 be any nontrivial character of the additive groupR,say, �(x) = eipx, where p is a nonzero real number. Now define ' : (R1,3)⇤ ! R1,3

by

'(�) = � � �

for all � in (R1,3)⇤. Show that ' is a vector space isomorphism. Hint: For surjectivitynote that � : R ! S 1 is a covering space with �(0) = 1 and that R1,3 is simplyconnected. It follows that any element of R1,3 has a unique lift � : R1,3 ! R with�(0) = 0 (see, for example, Theorem 5.1 and Theorem 6.1 of [Gre]). Now show that� is linear.

Exercise 2.6.9. Show that, for every A in SL(2,C),

�(A) = ' � �⇤(A) � '�1.

With this we conclude that R1,3 can be identified with the vector space dual(R1,3)⇤ of R1,3, that is, with momentum space P1,3, in such a way that the induced

2.7 Spacetime Symmetries and Projective Representations 105

actions � and �⇤ of SL(2,C) on R1,3 and (R1,3)⇤, respectively, are equivalent. Wewill put this to use in Section 2.8.

2.7 Spacetime Symmetries and Projective Representations

Symmetries in classical mechanics were discussed in Section A.2 (Lagrangian pic-ture) and Section A.3 (Hamiltonian picture). The corresponding notion in quantummechanics was introduced in Section A.4 (see page 154). We recall that a symmetryof a quantum system with Hilbert space H is a bijection of the state space P(H)onto itself that preserves transition probabilities. By Wigner’s Theorem (Theorem1.3.1 and Theorem A.4.2), each of these arises from an operator on H that is eitherunitary or anti-unitary. Our objective in this section is to take the first step toward“relativistic quantum mechanics” by defining what it means for a quantum systemto be “relativistically invariant”. To put the matter simply we will take this to meanthat the Poincare group P"+ “acts by symmetries on the state space P(H)” of thequantum system. But to say that P"+ acts by symmetries on P(H) means simplythat there is a projective representation of P"+ on H, that is, a continuous group ho-momorphism ⇢ : P"+ ! Aut(P(H)) of P"+ into the automorphism group of P(H)(again, see Section 1.3). Di↵erent “types” of relativistic invariance correspond todi↵erent projective representations just as di↵erent types of rotational invariance(scalar, vector, tensor) correspond to di↵erent representations of SO(3) in classicalmechanics. It would be nice then to find all of the (irreducible) projective represen-tations of P"+ on H. In this section we will simply sketch how this can be reducedto the well-studied problem of finding the irreducible unitary representations of thedouble cover ISL(2,C) of P"+ on H.

The principal tool we require is a highly nontrivial result of Bargmann which as-serts that every projective representation of ISL(2,C) on H lifts to a unitary repre-sentation of ISL(2,C) on H. Such liftings of projective representations do not existin general and Bargmann’s Theorem depends crucially on the topology of ISL(2,C)(see Section 1.3). For the proof of the following result we refer to Theorem 14.3 of[VDB].

Theorem 2.7.1. (Bargmann’s Theorem) Let H be a complex, separable Hilbertspace. Every projective representation ⇢ : ISL(2,C) ! Aut(P(H)) of ISL(2,C) onH lifts to a unique unitary representation ⇢ : ISL(2,C)! U(H) of ISL(2,C) on H.

U(H)

% #

ISL(2,C) ! Aut(P(H))


Here the downward vertical arrow is the restriction to U(H) of the quotient mapQ : U(H) t U�(H)! Aut(P(H)) described in Proposition 1.3.2 so

⇢ = Q � ⇢.

Moreover, ⇢ is irreducible if and only if ⇢ is irreducible.

The following is Corollary 14.4 of [VDB], but the proof is simple and a niceapplication of Schur’s Lemma. Since we will need to see the specific constructionemployed in the proof we will record it here as well.

Theorem 2.7.2. Let H be a complex, separable Hilbert space. Every irreducible,unitary representation ⇢ : ISL(2,C) ! U(H) of ISL(2,C) on H naturally inducesan irreducible, projective representation ⇢ : P"+ ! Aut(P(H)) of the Poincare groupP"+ on H.

Proof. We have already seen thatR1,3oSL(2,C) is the universal cover ofR1,3

oL"+,that the kernel of the covering map 0 = idR1,3 ⇥ : R1,3

o SL(2,C) ! R1,3o L"+

is Z2 = {(0,±id2⇥2)} and that this kernel is precisely the center of R1,3o SL(2,C).

Now let ⇢ : R1,3o SL(2,C) ! U(H) be an irreducible, unitary representation of

R1,3o SL(2,C) on a complex, separable Hilbert space H. It follows from Schur’s

Lemma 1.2.1 that ⇢ restricted to Kernel (0) acts by scalars of modulus one. It isshown in Section 1.3 that ⇢ induces an irreducible, projective representation ⇢ :R1,3o SL(2,C)! Aut(P(H)) defined by

⇢(x, A)([ ]) = [⇢(x, A) ].

Since ⇢ acts by scalars of modulus one on Kernel (0), the projective representa-tion ⇢ is trivial on Kernel (0). Consequently, factoring by Kernel (0) gives an ir-reducible, projective representation ⇢ of the quotient, that is, of the Poincare groupP"+ = ISL(2,C)/Kernel (0). ut

The objects of interest in relativistic quantum mechanics are the irreducible,projective representations of the Poincare group P"+ = R1,3

o L"+ (see RemarkA.4.2). We will now show that the previous two theorems reduce the problemof finding these to that of enumerating the irreducible, unitary representations ofISL(2,C) = R1,3

o SL(2,C). Theorem 2.7.2 assures us that any irreducible, unitaryrepresentation of ISL(2,C) induces an irreducible, projective representation of P"+so we need only show that every irreducible, projective representation ⇢ of P"+ givesrise to an irreducible, unitary representation of ISL(2,C) that induces ⇢ in the senseof Theorem 2.7.2. Suppose then that ⇢ : R1,3

o L"+ ! Aut(P(H)) is an irreducible,projective representation of P"+. Composing with the double cover homomorphismR1,3

o SL(2,C) ! R1,3o L"+ gives an irreducible, projective representation ⇢ of

ISL(2,C). Bargmann’s Theorem 2.7.1 then implies that ⇢ has a unique lift to anirreducible, unitary representation ⇢ of ISL(2,C).

2.8 Positive Energy Representations of P"+ with m > 0 107

Exercise 2.7.1. Show that ⇢ induces ⇢ in the sense of Theorem 2.7.2. Hint: Followthe construction in the proof of Theorem 2.7.2

Our problem then is to enumerate the irreducible, unitary representations ofISL(2,C) and for this we can apply the “Mackey machine” described in Section1.5. In the next section we will describe that part of enumeration that is of interestto us here (the so-called positive energy representations with mass m > 0).

2.8 Positive Energy Representations of P"+

with m > 0

The Mackey machine is described in Section 1.5 and we will simply follow theinstructions enumerated on page 41 when G is ISL(2,C) = R1,3

oSL(2,C). The firstof these requires that we select an orbit of the SL(2,C)-action on the character groupdR1,3 and a point P0 in it. In Section 2.3 we identified dR1,3 with the vector space dualP1,3 = (R1,3)⇤ ofR1,3 and the SL(2,C)-action on dR1,3 with the contragredient actionof SL(2,C) on P1,3, that is,

A · p = (A�1)T p,

where : SL(2,C)! L"+ is the covering map (Section 2.4). It will therefore su�ceto find the orbits of this action on P1,3. Since (A�1)T is in L"+, these are the same asthe L"+-orbits in momentum space and these were described in Section 2.4. Noticethat these are all closed subsets of P1,3 so ISL(2,C) is a regular semi-direct product.For our purposes we will be interested only in the mass hyperboloids X+m, m > 0.

Remark 2.8.1. We have chosen to restrict our attention to X+m for reasons that arein the physics, not the mathematics (see Remark 2.6.1). We will eventually describethe physical interpretation of the results we derive here for X+m, but for the remainingorbits there are subtleties such as negative energy states that we prefer not to becomeentangled in at the moment. For those who would like to see the full story we referto [Simms] and [Vara].

We have shown in Section 2.4 that, for any point p 2 X+m, the isotropy subgroupof p for the L"+-action on X+m is isomorphic to the rotation subgroup R � SO(3) ofL"+. Consequently, the isotropy subgroup for the SL(2,C)-action is the pre-imageunder of R and this is isomorphic to SU(2) (Section 2.4).

Remark 2.8.2. Recall that every point p in X+m gives rise to a character ⇠p 2 dR1,3

defined by

p = (p0, p1, p2, p3) 2 X+m 7! ⇠p(x) = ei(p0 x0�p1 x1�p2 x2�p3 x3).


For the remainder of the discussion we will fix the “base point” P0 = (m, 0, 0, 0) 2X+m and the corresponding character

⇠0 = eimx0(2.64)

in dR1,3 so that by SU(2) we mean the subgroup of SL(2,C) that leaves ⇠0 fixed indR1,3 and leaves P0 fixed in X+m.

We have also shown in Section 2.4 that X+m admits a measure µm that is invariantunder the L"+-action and therefore also under the SL(2,C)-action. Recall that it isdefined in the following way. For any Borel set B ✓ X+m, the projection ⇡+(B) is aBorel subset of R3 and we define

µm(B) =Z

⇡+(B)

d3p2!p

=

Z

⇡+(B)

d3p2p

m2 + kpk2,

where d3p = dp1 dp2 dp3 denotes integration with respect to Lebesgue measure onR3. The pushforward measure

µ = (��1P0

)⇤(µm)

is then an SL(2,C)-invariant measure on SL(2,C)/SU(2) (see (2.60) and Remark2.6.6). This fulfills the second requirement of the Mackey machine (see page 41).

To set the Mackey machine in motion for ISL(2,C) we must now select a stronglycontinuous, irreducible, unitary representation of the isotropy group SU(2). In Ex-ample 1.2.1 we saw that these are precisely the spinor representations

D( j/2) : SU(2)! U(C(D( j/2))),

where j � 0 is an integer and C(D( j/2)) is the ( j + 1)-dimensional Hilbert space ofcarriers of D( j/2), that is, the space of 2 j-tuples ⇠A1A2···A j , A1, A2, . . . , Aj = 1, 2, ofcomplex numbers that are symmetric under permutations of A1A2 · · · Aj. We nowfix one of these representations D( j/2) for some j � 0..

Next we are to consider the principal SU(2)-bundle

⇡ : SL(2,C)! SL(2,C)/SU(2) � X+m � R3,

where the right action of SU(2) on SL(2,C) is right multiplication. This bundle istrivial because any principal bundle over any Rn is trivial and it follows that thevector bundle

⇡D( j/2) : SL(2,C) ⇥D( j/2) C(D( j/2))! SL(2,C)/SU(2) � X+m � R3,


associated to it by D( j/2) is also trivial. We have seen that the Hilbert space HD( j/2)

of sections of this vector bundle that are L2 with respect to the invariant measure µon SL(2,C)/SU(2) can be identified with equivalence classes (modulo equality upto sets of measure zero) of functions f : SL(2,C) ! C(D( j/2)) that are measurablewith respect to Haar measure on SL(2,C) and satisfy each of the following.

1. f (AU) = D( j/2)(U�1)( f (A)) for all U 2 SU(2) and for almost all A 2 SL(2,C).

2.R

SL(2,C)/SU(2) k f ([A] k2 dµ([A]) < 1, where k f ([A]) k = k f (A) kC(D( j/2)).

For the moment we will think of HD( j/2) as the space of such functions and thenwill switch back to sections when the need arises.

Exercise 2.8.1. Show thatZ

SL(2,C)/SU(2)k f ([A] k2 dµ([A]) =

Z

X+mk f (!(p)) k2

C(D( j/2)) dµm(p).

Hint: Remark 2.6.6.

For the next step in the Mackey procedure we recall (Section 1.4) that the repre-sentation D( j/2) of SU(2) on C(D( j/2)) induces a unitary representation of SL(2,C)on HD( j/2) which sends any A 2 SL(2,C) to the operator on HD( j/2) which left trans-lates an f 2 HD( j/2) by A, that is,

IndSL(2,C)SU(2) (D( j/2)) : SL(2,C)! U(HD( j/2)

)

is defined by

IndSL(2,C)SU(2) (D( j/2))(A)

�f�(A0) = f (A�1A0). (2.65)

When the context makes the intention clear we will try to relieve some of thenotational clutter by writing the action of A in SL(2,C) on f corresponding toIndSL(2,C)

SU(2) (D( j/2)) the way we write every other group action, that is, we will write(2.65) simply as

(A · f )(A0) = f (A�1A0). (2.66)

At this point we have induced a representation of SL(2,C) on HD( j/2) and nowwe handle the first factor of ISL(2,C) = R1,3

o SL(2,C). Then we will put themtogether to get a representation of ISL(2,C) itself. For every x in the additive groupR1,3 we define a unitary multiplication operator U(x) on HD( j/2) by

[U(x) f ](A0) = [(A0 · ⇠0)(x)] f (A0),


where ⇠0 is the character in the SL(2,C)-orbit of dR1,3 corresponding to the basepoint P0 = (m, 0, 0, 0) in X+m (see (2.64)). Note that as A0 varies over all of SL(2,C),A0 · ⇠0 varies over this entire orbit.

Finally, we put the two factors together and define, for every (x, A) 2 ISL(2,C)the unitary operator

U(x, A) = U(x) � IndSL(2,C)SU(2) (D( j/2))(A).

Thus,

[U(x, A) f ](A0) = [(A0 · ⇠0)(x)](A · f )(A0) = [(A0 · ⇠0)(x)] f (A�1A0). (2.67)

Technically, this completes the application of the Mackey machine to ISL(2,C), butwe will need to work a bit to put (2.67) into a usable form.

Notice that, since A0 ·⇠0 is a character, (A0 ·⇠0)(x) is a complex number of modulus1 for each A0 2 SL(2,C) and each x 2 R1,3 . We will now write out this phase factorexplicitly. For this we recall that (A0 · ⇠0)(x) = ⇠0((A0)�1 · x) = ⇠0((A0)�1x).

Exercise 2.8.2. Denote (A0) by ⇤ 2 L"+ and show that the 0th-component of(A0)�1x is

⇤00x0 � ⇤1

0x1 � ⇤20x2 � ⇤3

0x3.

Conclude that

(A0 · ⇠0)(x) = eim(⇤00 x0�⇤1

0 x1�⇤20 x2�⇤3

0 x3).

Now notice that, since⇤ = (A0) 2 L"+, p = (p0, p1, p2, p3) = (m⇤00,m⇤1

0,m⇤20,

m⇤30) satisfies (p0)2 � (p1)2 � (p2)2 � (p3)2 = m2 and so, if p↵ = ⌘↵�p�, then

(p0, p1, p2, p3) is in X+m for every A0 2 SL(2,C). Moreover, as A0 varies over all ofSL(2,C), these points cover all of X+m. Consequently,

(A0 · ⇠0)(x) = ei(p0 x0�p1 x1�p2 x2�p3 x3) = eip↵x↵ = eihp,xi,

where p is just m times the 0th-column of (A0). With this (2.67) becomes

[U(x, A) f ](A0) = eihp,xi f (A�1A0). (2.68)

Next we will think a bit more about the second factor f (A�1A0) and return to theview of HD( j/2) as sections of a vector bundle. Recall that f : SL(2,C) ! C(D( j/2))is a map from SL(2,C) to C(D( j/2)) that is equivariant with respect to the SU(2)-actions, that is, satisfies

f (AU) = D( j/2)(U�1)( f (A))


for all U 2 SU(2) and for almost all A 2 SL(2,C). As such, it determines a section

s f : SL(2,C)/SU(2)! SL(2,C) ⇥D( j/2) C(D( j/2))

of the vector bundle

⇡D( j/2) : SL(2,C) ⇥D( j/2) C(D( j/2))! SL(2,C)/SU(2)

given by

s f ([A0]) = [A0, f (A0)]

for all [A0] 2 SL(2,C)/SU(2) and any A0 2 [A0] (see Remark 1.4.1). Moreover, anysection s of this vector bundle is s f for one and only one such equivariant map f .

Notice that both the base SL(2,C)/SU(2) and the vector bundle space SL(2,C)⇥D( j/2)

C(D( j/2)) admit continuous SL(2,C)-actions given by

A · [B] = [AB]

and

A · [B, v] = [AB, v],

respectively, and that these commute with the projection ⇡D( j/2) . Now, we define anSL(2,C)-action on the sections in HD( j/2) as follows. For any section s 2 HD( j/2) andany A 2 SL(2,C) we take A · s to be the section defined by

(A · s)([A0]) = A · (s(A�1 · [A0])) = A · s([A�1A0])

for all [A0] 2 SL(2,C)/SU(2). The motivation for the definition is as follows. If s fis the section corresponding to f , then A · s is the section corresponding to f � LA,where LA is the di↵eomorphism of SL(2,C) onto itself defined by LA(A0) = A�1A0,that is,

(A · s f )([A0]) = s f�LA ([A0]) = [A0, f (A�1A0)]

for all A0 2 SL(2,C) (compare (2.68)).Since µ is an invariant measure on SL(2,C)/SU(2) this action of SL(2,C) on the

sections in HD( j/2) determines a unitary representation of SL(2,C) on HD( j/2) andone can show that this representation is strongly continuous. Note that we can writethis in terms of the induced action of SL(2,C) on the corresponding equivariantfunction f as

(A · s f )([A0]) = [A0, (A · f )(A0)]. (2.69)

As an operator on sections the representation U(x, A) therefore takes the form


[U(x, A)s]([A0]) = eihp,xi(A · s)([A0]) (2.70)

where we again point out that p = p(A0) is m times the 0th column of (A0). This islooking more like the expression we want for U(x, A), but there is still work to do.

The next thing to observe about the vector bundle ⇡D( j/2) : SL(2,C) ⇥D( j/2)

C(D( j/2)) ! SL(2,C)/SU(2) is that, since SL(2,C)/SU(2) � R3, it is trivialso that any section s of it can be identified with a function on SL(2,C)/SU(2)whose value at any [A] is a point in the fiber ⇡�1

D( j/2) ([A]) above [A]. Furthermore,each of these fibers can be identified with the finite-dimensional Hilbert spaceC(D( j/2)). Such an identification is obtained in the following way. The principalbundle ⇡ : SL(2,C) ! SL(2,C)/SU(2) is trivial for the same reason so wecan select some global section u : SL(2,C)/SU(2) ! SL(2,C). Then, for each[A] 2 SL(2,C)/SU(2), u([A]) 2 SL(2,C) so there is a unique ([A]) 2 C(D( j/2)) forwhich

s([A]) = [u([A]), ([A])].

Specifically, if s = s f corresponds to the equivariant map f : SL(2,C)! C(D( j/2)),then = f = f � u. Thus, given u, we can identify the section s with the function : SL(2,C)/SU(2) ! C(D( j/2)). However, this identification of s with a functionfrom SL(2,C)/SU(2) to C(D( j/2)) is not unique since it depends on the choice ofthe section u. Indeed, if u : SL(2,C)/SU(2) ! SL(2,C) is another global section,then for each [A] 2 SL(2,C)/SU(2) we can write u([A]) = u([A])U([A]) for someU([A]) 2 SU(2) and then, by definition of the equivalence relation defining thepoints of SL(2,C) ⇥D( j/2) C(D( j/2)), [u([A]),U([A]) · ([A])] = [u([A]), ([A])] so

s([A]) = [u([A]),U([A]) · ([A])].

In other words, is determined only up to the action of SU(2) given by the repre-sentation D( j/2).

Next we will need a few computational formulas. First we note that, when s = s f ,f can be recovered from . To see this we write

f (A0) = f (u([A0]) u([A0])�1A0| {z }2SU(2)

)

= D( j/2)�(A0)�1u([A0])�

f (u([A0]))

so

f (A0) = D( j/2)�(A0)�1u([A0])� ([A0]). (2.71)

Remark 2.8.3. Notice that u([A0])�1A0 is in SU(2) because its projection intoSL(2,C)/SU(2) is the identity.


From this we obtain

f (A�1A0) = D( j/2)✓(A�1A0)�1u([A�1A0])

◆ ([A�1A0])

= D( j/2)✓(A0)�1Au(A�1 · [A0])

◆ (A�1 · [A0])

= D( j/2)✓

(A0)�1u([A0])| {z }2SU(2)

u([A0])�1Au(A�1 · [A0]| {z }2SU(2)

)◆ (A�1 · [A0])

= D( j/2)✓(A0)�1u([A0])

◆D( j/2)

✓u([A0])�1Au(A�1 · [A0])

◆ (A�1 · [A0])

(2.72)

We have not yet defined an action of SL(2,C) on , but (2.71) suggests that wedefine A · in such a way that

(A · f )(A0) = D( j/2)�(A0)�1u([A0])�

(A · )([A0])

and (2.72) would then give

(A · )([A0]) = D( j/2)✓u([A0])�1Au(A�1 · [A0])

◆ (A�1 · [A0]). (2.73)

Thus,

f (A�1A0) = D( j/2)✓(A0)�1u([A0])

◆ �(A · )([A0])

�

=�

(A0)�1u([A0])� · � (A · )([A0])

�.

We claim that, with the SL(2,C)-action (2.73),

(A · s)([A0]) = [u([A0]), (A · )([A0])]

so that A · is the function associated with the section A · s. To see this we write suniquely as s = s f and compute

(A · s f )([A0]) = [A0, f (A�1A0)]

= [A0, ((A0)�1u([A0])) · ( (A · )([A0]) )]

= [A0 · ((A0)�1u([A0])), (A · )([A0])]= [u([A0]), (A · )([A0])].

As an operator on the representation U(x, A) therefore takes the form

[U(x, A) ]([A0]) = eihp,xi(A · )([A0])

or, in more detail,


[U(x, A) ]([A0]) = eihp,xiD( j/2)✓u([A0])�1Au(A�1 · [A0])

◆ (A�1 · [A0]). (2.74)

Finally, we would like to replace the homogeneous manifold SL(2,C)/SU(2)with the mass hyperboloid X+m. Composing with a di↵eomorphism of X+m ontoSL(2,C)/SU(2) we can regard s as a section of a vector bundle over X+m and as afunction on X+m. Specifically, if we let P0 = (m, 0, 0, 0), then

�P0 : SL(2,C)/SU(2)! X+m

defined by

�P0 ([A0]) = A0 · P0

is a di↵eomorphism.

Exercise 2.8.3. Show that A0 · P0 = p, where p is the same point of X+m that appearsin the exponential factor eihp,xi of [U(x, A) ]([A0]).

Then

⇡m = �P0 � ⇡D( j/2) : SL(2,C) ⇥D( j/2) C(D( j/2))! X+m

is a C0-Hilbert bundle over X+m,

sm = s � ��1P0

: X+m ! SL(2,C) ⇥D( j/2) C(D( j/2))

is the section of this vector bundle corresponding to s,

! = u � ��1P0

: X+m ! SL(2,C)

is the smooth section of the principal SU(2)-bundle �P0 � ⇡ : SL(2,C) ! X+m corre-sponding to u, and

m = � ��1P0

: X+m ! C(D( j/2))

is the function on X+m corresponding to .

Notice that, since s is an L2-section with respect to the invariant measure µ onSL(2,C)/SU(2) and since µ is the pushforward measure of µm on X+m by the di↵eo-morphism ��1

P0,

m 2 L2(X+m, µm,C(D( j/2))).

Since m = � ��1P0

we define the action of SL(2,C) on m by

A · m = (A · ) � ��1P0


for every A 2 SL(2,C). Then (A· m)(p) = (A· )([A0]), where p = �P0 ([A0]) = A0·P0.Furthermore, u([A0]) = u(��1

P0(p)) = !(p).

Exercise 2.8.4. Show that �P0 (A�1 · [A0]) = A�1 · p.

However, SL(2,C) acts on X+m by Lorentz transformations via the double cover-ing so, if we write ⇤A 2 L"+ for the image (A) of A, then what the last Exerciseshows is that

A�1 · [A0] = ��1P0

(⇤�1A · p).

Substituting all of this into (2.73) gives

(A · m)(p) = D( j/2)✓!(p)�1 A!(⇤�1

A · p)◆ m(⇤�1

A · p). (2.75)

Thus, as an operator on m 2 L2(X+m, µm,C(D( j/2))),

[U(x, A) m](p) = eihp,xi(A · m)(p)

= eihp,xiD( j/2)✓!(p)�1 A!(⇤�1

A · p)◆ m(⇤�1

A · p). (2.76)

This is the form in which one most often sees the representation U(x, A) written inthe physics literature.

Remark 2.8.4. Just as a reminder and to anticipate some of the physical terminologythat we will discuss more fully as we proceed let’s record the interpretation of thevarious objects in (2.76). (x, A) is an element of the double cover R1,3

o SL(2,C)of the Poincare group P"+ so x is interpreted as a translation on Minkowski space-time and A is an element of SL(2,C) acting on Minkowski spacetime by Lorentztransformations via the double covering : SL(2,C) ! L"+. U(x, A) is a uni-tary operator on L2(X+m, µm,C(D( j/2))), where X+m ✓ P1,3 is the mass hyperboloid,µm is an SL(2,C)-invariant measure on X+m, and C(D( j/2)) is the finite-dimensionalHilbert space of carriers of the spin j/2 representation of SU(2). We will refer toL2(X+m, µm,C(D( j/2))) as the 1-particle space. The elements of L2(X+m, µm,C(D( j/2)))will be interpreted as wave functions for a free material particle with 4-momentum psatisfying hp, pi = m2 so that m is interpreted as the mass of the particle. These wavefunctions take values in C(D( j/2)) which is C only when j = 0. The significance ofthe additional components when j > 0 will be explained in due course (Section ??).The point p = A0 · (m, 0, 0, 0) varies over all of X+m as A0 varies over SL(2,C) and!(p) is a mapping of X+m to SL(2,C) with the property that !(p) · (m, 0, 0, 0) = pfor each p 2 X+m. Physically, !(p) therefore corresponds to a Lorentz transforma-tion from a frame in which the particle’s 4-momentum is (p0, p1, p2, p3) to a framein which the 4-momentum is (m, 0, 0, 0), that is, to the rest frame of the particle.⇤A = (A) and ⇤�1

A · p is the contragredient action of ⇤�1A on X+m ✓ P1,3, that is,

⇤A · p = [(⇤�1A )�1]T p = ⇤T

A p = (A)T p. U itself is called the (Wigner) representation


of mass m and spin j/2. What the term “spin” has to do with the corresponding termin physics will be discussed in Section A.5.

Notice that, when x = 0 (no translation), (2.76) reduces to

[U(0, A) m](p) = D( j/2)✓!(p)�1 A!(⇤�1

A · p)◆ m(⇤�1

A · p) (2.77)

whereas, when A = I = idSL(2,C) (no Lorentz transformation),

[U(x, I) m](p) = eihp,xi m(p). (2.78)

The simplest case of (2.76) occurs when j = 0 since C(D(0)) = C and D(0) isthe trivial representation, that is, D(0)(U) = idC for every U 2 SU(2) (see Remark1.2.3). In this case,

[U(x, A) m](p) = eihp,xi m(⇤�1A · p) (2.79)

for all (x, A) 2 ISL(2,C). Most of our attention later on will focus on this spin zerocase. We will refer to a m satisfying (2.79) as classical, relativistic scalar field onX+m. and our ultimate goal is to quantize the physical system it describes.

We make one more general observation about (2.76). The carriers of the rep-resentation D( j/2) can be identified with 2 j-tuples (⇠A1···Aj )A1,...,Aj=1,2 in C2 j that areinvariant under all permutations of A1 · · · Aj (see Example 1.2.2). The dimension ofthis subspace of C2 j is j + 1 so each m has j + 1 components which we write as

A1···Ajm (p), A1, . . . , Aj = 1, 2,

where A1···Ajm (p) is invariant under all permutations of A1 · · · Aj. The e↵ect of the

ISL(2,C)-action is to yield a new set of components

A1···Ajm (p) = [U(x, A) m]A1···Aj (p).

If we denote the element !(p)�1 A!(⇤�1A · p) of SU(2) by U = (UA

B)A,B=1,2, then

A1···Ajm (p) = eihp,xiUA1

B1 · · ·UAjBj

B1···Bjm (⇤�1

A · p), (2.80)

where we sum over B1, . . . , Bj = 1, 2. We will refer to a m with component func-tions that transform under the ISL(2,C)-action according to (2.80) as a (contravari-ant) spinor field of rank j on X+m.

All of our e↵orts in this section have been directed toward the irreducible, uni-tary representation of the double cover ISL(2,C) of the Poincare group P"+. We haveseen in Section 2.7 how these determine the irreducible projective representations ofP"+ and that these are the objects that express relativistic invariance for quantum sys-tems. We should note that we have, in fact, also determined the irreducible, unitary


representations of P"+ itself. Certainly, every such representation gives rise to one ofthe representations of ISL(2,C) that we have found by simply composing with thecovering homomorphism : ISL(2,C)! P"+. On the other hand, a representation ofISL(2,C) will descend to a representation of P"+ precisely when U(x,�A) = U(x, A)for all (x, A) 2 ISL(2,C).

Exercise 2.8.5. Show that a Wigner representation of ISL(2,C) descends to a rep-resentation of P"+ if and only if it has integral spin, that is, if and only if j is even.

We will conclude this section by writing out in a bit more detail a few specialcases. First we will consider the e↵ect of a pure translation by x 2 R1,3 given by(2.78). The generators of translations on R1,3 are the elements Pµ, µ = 0, 1, 2, 3, ofthe Lie algebra p. Let’s consider, for example, the generator P1 of translations in thex1-direction in R1,3. Then, for any t 2 R,

eitP1 = etO1 = tO1

which is the translation in R1,3 by x = (0, t, 0, 0) (see (2.23)). Thus, for any p =(p0, p1, p2, p3) in X+m,

eihp,xi = e�itp1

and so, by (2.78),

[U((0, t, 0, 0), I) m](p) = e�itp1 m(p).

Thus, U((0, t, 0, 0), I), t 2 R, is a strongly continuous, 1-parameter group of unitarymultiplication operators on L2(X+m, µm,C(D( j/2))) so, by Stone’s Theorem (SectionVIII.4 of [RS1]), there is a unique self-adjoint operator P1 on L2(X+m, µm,C(D( j/2)))with U((0, t, 0, 0), I) = exp (itP1) and P1 is determined by

(iP1 m)(p) = limt!0

✓e�itp1 m(p) � m(p)t

◆= �ip1 m(p).

Consequently, P1 is just the multiplication operator

(P1 m)(p) = �p1 m(p)

on L2(X+m, µm,C(D( j/2))). The domain of P1 is the set of all m in L2(X+m, µm,C(D( j/2)))for which p1 m(p) is also in L2(X+m, µm,C(D( j/2))). Similarly,

[U((0, 0, t, 0), I) m](p) = e�itp2 m(p),

[U((0, 0, 0, t), I) m](p) = e�itp3 m(p),


[U((t, 0, 0, 0), I) m](p) = eitp0 m(p).

and the corresponding self-adjoint operators P2, P3, and P0 are multiplication by�p2,�p3 and p0, respectively (note that there is no minus sign in the exponentfor time translations). All of the self-adjoint operators Pµ, µ = 0, 1, 2, 3, share acommon, invariant, dense subspace D on which they are essentially self-adjoint(the smooth functions on X+m with compact support). On this subspace each of theoperators P2

µ is defined and essentially self-adjoint since it is just multiplication byp2µ. Also on this subspace the operator

P20 � P2

1 � P22 � P2

3

is multiplication by

p20 � p2

1 � p22 � p2

3 = hp, pi

and this is constant and equal to m2 on X+m. Writing Pµ = ⌘µ⌫P⌫ we will refer to

M2 = PµPµ = P20 � P2

1 � P22 � P2

3

as the (squared) mass operator on L2(X+m, µm,C(D( j/2))). Notice that M2 is the op-erator corresponding to the first Casimir invariant of p which lives, not in p, butin its universal enveloping algebra U(p). It acts by scalar multiplication and theeigenvalue is just the squared mass m2. The skew-adjoint operators correspondingto Pµ, µ = 0, 1, 2, 3, are

dU(Pµ) = iPµ, µ = 0, 1, 2, 3

so dU qualifies as a realization of the Lie subalgebra of p generated by the Pµ, µ =0, 1, 2, 3.

One would like to extend this to a realization of p itself. For this one needs toexamine the remaining generators of p. The images of these under U are given by(2.77) which, for convenience, we repeat here.

[U(0, A) m](p) = D( j/2)✓!(p)�1 A!(⇤�1

A · p)◆ m(⇤�1

A · p) (2.81)

Here ! : X+m ! SL(2,C) is any function with the property that !(p) · (m, 0, 0, 0) = pfor each p 2 X+m. In Remark 2.6.3 we showed that one can choose an ! that isequivariant with respect to the natural actions of SU(2) on X+m and SL(2,C) and wewill now assume that we have made such a choice. In particular, if X is any elementof the Lie algebra of the isotropy subgroup SU(2) of (m, 0, 0, 0) and t 2 R, then

!(etX · p) = etX !(p) e�tX .

Exercise 2.8.6. Show that, with such a choice for !, (2.81) gives


[U(0, etX) m](p) = D( j/2)(etX) m(e�tX · p) (2.82)

for any X 2 su(2).

We want to compute the self-adjoint operator A corresponding to the 1-parametergroup U(0, etX) of unitary operators on L2(X+m, µm,C(D( j/2))). The dimension of thespace C(D( j/2)) of carriers of the representation D( j/2) of SU(2) is j + 1 (Example1.2.1) so we can regard m(e�tX · p) as a ( j + 1) column vector of functions of t foreach p 2 X+m and D( j/2)(etX) as a ( j + 1) ⇥ ( j + 1) matrix of smooth functions of t. If m is a smooth element of L2(X+m, µm,C(D( j/2))) and if we compute the t-derivativesof m(e�tX · p) and D( j/2)(etX) componentwise and entrywise, respectively, then

[iA m](p) = limt!0

[U(0, etX) m](p) � m(p)t

=ddt

⇥D( j/2)(etX) m(e�tX · p)

⇤��t=0

= D( j/2)(I)ddt m(e�tX · p)

��t=0 +

ddt

⇥D( j/2)(etX)

⇤��t=0 m(p)

=ddt m(e�tX · p)

��t=0 +

ddt

⇥D( j/2)(etX)

⇤��t=0 m(p)

so that A splits into the sum of two operators

�iddt m(e�tX · p)

��t=0 � i

ddt

⇥D( j/2)(etX)

⇤��t=0 m(p). (2.83)


Exercise 2.8.7. Let : SL(2,C) ! L"+ be the covering map. Prove each of thefollowing.

e�itM23 = etM1 =

0BBBBBBBBBBBB@

1 0 0 00 1 0 00 0 cos t �sin t0 0 sin t cos t

1CCCCCCCCCCCCA= ( e(it/2)�1 )

e�itM31 = etM2 =

0BBBBBBBBBBBB@

1 0 0 00 cos t sin t 00 0 1 00 �sin t cos t 0


e�itM12 = etM3 =

0BBBBBBBBBBBB@

1 0 0 00 cos t �sin t 00 sin t cos t 00 0 0 1


Hint: Exercise 2.4.5

Consider, for example, the (complex) generator M12 = J3 = iM3 of rotationsabout the x3-axis (M23 and M31 are entirely analogous). We will denote the self-adjoint operator corresponding to U(0, e�itM12 ) by M12 and will denote the two op-erators in (2.83) by O12 and S 12, that is, M12 = O12 + S 12, where

[O12 m](p) = �iddt m(eitM12 · p)

��t=0 (2.84)

and

[S 12 m](p) = �iddt

⇥D( j/2)(e�itM12 )

⇤ ��t=0 m(p). (2.85)

Thus,

[O12 m](p) = �iddt

m(p0, p1cos t � p2sin t, p1sin t + p2cos t, p3)

� ��t=0

= �i✓p2@ m

@p1� p1

@ m

@p2

◆

so O12 is the unique self-adjoint extension of

�i✓p2

@

@p1� p1

@

@p2

◆

on L2(X+m, µm.C(D( j/2))). We will refer to the physical quantity represented by O12 asthe orbital angular momentum about the x3-axis corresponding to the representationU and the section !.


The operator S 12 depends, of course, on the spin s = j2 2 {0, 1

2 , 1,32 , . . .} of the

representation. When s = 0, C(D(0/2)) = C and D(0/2) is the trivial representation ofSU(2) that sends every U 2 SU(2) to idC so S 12 is identically zero.

�iddt

⇥D(0/2)(e�itM12 )

⇤ ��t=0 = 0 (s = 0)

When s = 12 , C(D(1/2)) = C2 and D(1/2) is the standard representation of SU(2)

that sends every U 2 SU(2) to U acting on C2 by matrix multiplication. If weidentify M3 with the basis element X3 of the Lie algebra so(3) and this, in turn, withthe basis element � i

2�3 of su(2) (see Remark 2.5.1), then

�iddt

⇥D(1/2)(e�itM12 )

⇤ ��t=0 = �i

ddt

e�it/2 0

0 eit/2

! ��t=0=

� 1

2 00 1

2

!(s =

12

)

Notice that the eigenvalues of this matrix are {� 12 ,

12 }. The action of S 12 on m =

1m

2m

!is therefore as follows.

[S 12 m](p) = � 1

2 00 1

2

! 1

m(p) 2

m(p)

!=

12

� 1

m(p) 2

m(p)

!(s =

12

)

Exercise 2.8.8. Show that, when s = 1,

�iddt

⇥D(2/2)(e�itM12 )

⇤ ��t=0 =

0BBBBBBBBBBBB@

1 0 0 00 0 0 00 0 0 00 0 0 �1

1CCCCCCCCCCCCA

(s = 1)

so that the eigenvalues are {�1, 0, 1}. Hint: See Exercise 1.2.8.

Continuing in this way one finds that if s = j2 , then �i d

dt⇥D( j/2)(e�itM12 )

⇤ ��t=0 has

eigenvalues⇢� j

2, � j

2+ 1, . . . , � j

2+ j =

j2

�

so that the spin of the representation is the largest eigenvalue of S 12. We will referto S 12 as the spin angular momentum about the x3-axis corresponding to the rep-resentation U and the section !. M12 = O12 + S 12 is the total angular momentumabout the x3-axis.


Remark 2.8.5. An element m of the 1-particle space L2(X+m, µm,C(D( j/2))) is inter-preted as the wave function of a free material particle of “mass” m and “spin” s = j

2 ,where “mass” and “spin” here refer to the terms as they are used in physics to denotecertain physically measured observables. One would like to understand the rationalebehind identifying these physical observables with the mathematical mass and spinparameters that we have introduced here. In the case of the mass we have seen abovethat this rationale is essentially just the relativistic relation between the Minkowskinorm of a particle’s 4-momentum and its mass.

p20 � p2

1 � p22 � p2

3 = m2

The corresponding rationale for spin will, of course, have to wait until we knowwhat the physicists mean by “spin” and so must be deferred to Section A.5. Oneparticular aspect of this rationale is worth briefly anticipating here however. We haveidentified the spin of a representation with the largest eigenvalue of the spin angularmomentum about the x3-axis and one might reasonably wonder what is so specialabout the x3-axis. The answer is quite simple, that is, nothing at all. Physically, the“spin” of, say, an electron about the x3-axis is thought of as the third componentof a “spin vector”, but this is a rather peculiar “vector” that could only exist in thequantum world in that its projection onto every axis is either 1

2 or � 12 (± ~2 if ~ is

not taken to be 1). One can see that this is at least consistent with the mathematicalnotion of spin that we have introduced. For example, if one repeats the argumentswe have just given in the s = 1

2 case with M12 replaced by the generator M23 ofrotations about the x1-axis one finds that

�iddt

⇥D(1/2)(e�itM23 )

⇤ ��t=0 = �i

ddt

cos (t/2) �i sin (t/2)�i sin (t/2) cos (t/2)

! ��t=0=

0 � 1

2� 1

2 0

!(s =

12

)

(Exercise 2.4.5) and this has eigenvalues ± 12 . Consequently, S 12 and S 23 have the

same eigenvalues and, in particular, the same largest eigenvalue. The spin about thex1-axis is the same as the spin about the x3-axis.

Exercise 2.8.9. Prove the same thing for S 31.

One can generalize to show that, when the representation of SU(2) is D( j/2), thens = j

2 is the largest eigenvalue of all of the operators S 12, S 23, and S 31 so that, inthis sense at least, our spin parameter has the property required of the physicist’s“spin” observable.

Appendix APhysical Background

A.1 Introduction

Although this volume was planned as a sequel to [Nab5] we would not want thisearlier work to be a sine qua non for this one. Toward this end we will try to removesome potential obstacles by providing brief synopses of a number of topics in clas-sical and quantum mechanics that we must depend upon here and that are discussedin greater detail in [Nab5].

A.2 Finite-Dimensional Lagrangian Mechanics

Section 2.2 of [Nab5] describes the Lagrangian formulation of classical particlemechanics in some detail with numerous examples. Here we will simply collecttogether a summary of those items we need in order to pursue our current objectivesand conclude with one example that, we hope, will solidify the ideas.

One begins with an n-dimensional smooth manifold M called the configurationspace and generally denotes a local coordinate system on M by (q1, . . . , qn). Wethink of M as the space of possible positions of the particles in the system. For twoparticles moving inR3, for example, M = R3⇥R3 = R6. The tangent bundle T M ofM (Remark 2.2.5 of [Nab5] or Section 1.25 of [Warn]) is called the state space andthe points (x, vx) in it represent possible states of the system with x 2 M representinga possible configuration of the particles in the system and vx 2 Tx(M) a possible rateof change of the configuration at x. The local coordinate functions q1, . . . , qn on Mtogether with their coordinate velocity vector fields @q1 , . . . , @qn determine naturalcoordinates (q1, . . . , qn, q1, . . . , qn) on T M. These are defined in the following way.For each (x, vx) 2 T M, define

qi(x, vx) = qi(x), i = 1, . . . , n.

and then write vx = vx[q1]@q1 (x) + · · · + vx[qn]@qn (x) = vx[qi]@qi (x) and define

123

124 A Physical Background

qi(x, vx) = vx[qi]. i = 1, . . . , n.

We will, on occasion, write qi(x, vx) simply as qi(vx).Any smooth real-valued function L : T M ! R on the state space T M is called a

Lagrangian on M. Such a function can be described locally in natural coordinates.We adopt the usual custom of writing such a local coordinate representation as

L(q1, . . . , qn, q1, . . . , qn) = L(q, q).

For t0 < t1 in R and a, b 2 M the path space C1a,b([t0, t1],M) is the space ofall smooth curves ↵ : [t0, t1] ! M with ↵(t0) = a and ↵(t1) = b. Every ↵ inC1a,b([t0, t1],M) has a unique lift to a smooth curve

↵ : [t0, t1]! T M

in the tangent bundle defined by

↵(t) = (↵(t), ↵(t)),

where ↵(t) denotes the velocity (tangent) vector to ↵ at t. The action functionalassociated with the Lagrangian L is the real-valued function

S L : C1a,b([t0, t1],M)! R

defined by

S L(↵) =Z t1

t0L(↵(t)) dt =

Z t1

t0L(↵(t), ↵(t)) dt.

For any ↵ 2 C1a,b([t0, t1],M) we define a (fixed endpoint) variation of ↵ to be asmooth map

� : [t0, t1] ⇥ (�✏, ✏)! M

for some ✏ > 0 such that

�(t, 0) = ↵(t), t0 t t1�(t0, s) = ↵(t0) = a, �✏ < s < ✏�(t1, s) = ↵(t1) = b, �✏ < s < ✏.

For any fixed s 2 (�✏, ✏) the map

�s : [t0, t1]! M

defined by

�s(t) = �(t, s)

A.2 Finite-Dimensional Lagrangian Mechanics 125

is an element of C1a,b([t0, t1],M). Then S L(�s) is a smooth real-valued function ofthe real variable s whose value at s = 0 is S L(↵). We say that ↵ 2 C1a,b([t0, t1],M) isa stationary point, or critical point of the action functional S L if

dds

S L(�s)��s=0 = 0

for every variation � of ↵. In this case we will call ↵ a stationary curve, or a criticalcurve for the action S L. According to the Principle of Stationary Action, also calledthe Principle of Least Action, the actual time evolution of the state of the systemwill take place along the lift of a stationary curve for S L. One should understand,of course, that this is not a mathematical theorem, but rather a physical assumptionthat happens to be well born out by experience. We will o↵er some motivation inExample A.2.1.

For curves ↵ that lie in some coordinate neighborhood in M one can write downexplicit equations that are necessary conditions for ↵ to be a stationary point of S L.Let ↵ 2 C1a,b([t0, t1],M) be a curve whose image lies in a coordinate neighborhoodwith coordinate functions q1, . . . , qn. The lift ↵ of ↵ is written in natural coordinatesas ↵(t) = (q1(t) . . . , qn(t), q1(t), . . . , qn(t)), where qi(t) is a notational shorthand forqi(↵(t)) and similarly qi(t) means qi(↵(t)). Then it is shown in Section 2.2 of [Nab5]that if ↵ is a stationary point of S L, then

@L@qi

�↵(t), ↵(t)

� � ddt

✓ @L@qi

�↵(t), ↵(t)

�◆= 0, 1 i n. (A.1)

These are the famous Euler-Lagrange equations which one often sees written sim-ply as

@L@qi �

ddt

✓ @L@qi

◆= 0, 1 i n. (A.2)

The Euler-Lagrange equations are satisfied along any stationary curve for S L.Notice that one can draw a rather remarkable conclusion just from the form in

which the Euler-Lagrange equations are written, namely, that if the Lagrangian Lhappens not to depend on one of the coordinates, say qi, then @L

@qi = 0 everywhereand (A.1) implies that, along any stationary curve, @L

@qi is constant. In more colloquialterms, @L

@qi is conserved as the system evolves.

@L@qi = 0 =) @L

@qi is conserved along any stationary curve.

In the physics literature

pi =@L@qi

is called the momentum conjugate to qi.


A number of concrete examples of such conservation laws in classical mechanicsare written out in Section 2.2 of [Nab5] and we will see a few of these as we proceed.We have here the simplest instance of one of the most important features of theLagrangian formalism, that is, the deep connection between the symmetries of aLagrangian (in this case, its invariance under translation of qi) and the existence ofquantities that are conserved during the evolution of the system. The next order ofbusiness is to describe a more general result giving rise to such conserved quantities.

If L : T M ! R is a Lagrangian on a smooth manifold M, then a symmetry of Lis a di↵eomorphism

F : M ! M

of M onto itself for which the induced map

T F : T M ! T M

given by

T F(x, vx) = (F(x), F⇤x(vx)),

where F⇤x is the derivative of F at x, satisfies

L � T F = L.

A symmetry (or, rather, its induced map on the state space) carries one state of thesystem onto another state at which the value of the Lagrangian is the same.

An “infinitesimal symmetry” of L is essentially a 1-parameter family of sym-metries arising from the 1-parameter group of di↵eomorphisms of a smooth vectorfield on M (Section 3.5 of [BG] or Section 5.7 of [Nab2]). The precise definition iscomplicated just a bit by the fact that not every vector field on M is complete, thatis, has integral curves defined for all t 2 R. In order not to cloud the essential issueswe will give the definition twice, once for vector fields that are complete and oncefor those that need not be complete (naturally, the first definition is a special case ofthe second).

Let L be a Lagrangian on a smooth manifold M. A complete vector field X onM is said to be an infinitesimal symmetry of L if each 't in the 1-parameter groupof di↵eomorphisms of X is a symmetry of L. Now we drop the assumption that Xis complete. For each x 2 M, let ↵x be the maximal integral curve of X throughx (Theorem 3.4.1 of [BG] or Theorem 5.7.2 of [Nab2]). For each t 2 R, let Dtbe the set of all x 2 M for which ↵x is defined at t and define 't : Dt ! M by't(x) = ↵x(t). By Theorem 5.7.4 of [Nab2], each Dt is an open set (perhaps empty)and, if Dt , ;, then 't is a di↵eomorphism of Dt onto D�t with inverse '�t. Nowwe will say that X is an infinitesimal symmetry of L if, for each t with Dt , ;,the induced map T't : TDt ! TD�t, defined by (T't)(x, vx) =

�'t(x), ('t)⇤x(vx)

�,

satisfies L � T't = L on TDt.


The content of our next result is that infinitesimal symmetries of L give rise toconserved quantities.

Theorem A.2.1. (Noether’s Theorem) Let L : T M ! R be a Lagrangian on asmooth manifold M and suppose X is an infinitesimal symmetry of L. Let q1, . . . , qn

be any local coordinate system for M with corresponding natural coordinatesq1, . . . , qn, q1, . . . , qn. Write X = X1@q1 + · · · + Xn@qn = Xi@qi . Then

Xi @L@qi = Xi pi = X1 p1 + · · · + Xn pn

is constant along every stationary curve in the coordinate neighborhood on whichq1, . . . , qn are defined.

Notice that if we have a Lagrangian L that is independent of one of the coordi-nates in M, say, qi, then certainly X = @qi is an infinitesimal symmetry. Since theonly component of X relative to @q1 , . . . , @qn is the ith and this is 1 we find that thecorresponding Noether conserved quantity is the same as the one we found earlier,namely, the conjugate momentum pi =

@L@qi

Infinitesimal symmetries often arise in the following way. Recall that a left actionof a Lie group G on M is a smooth map � : G ⇥ M ! M, usually written �(g, x) =g · x, that satisfies e · x = x8x 2 M, where e is the identity element of G, andg1 · (g2 · x) = (g1g2) · x for all g1, g2 2 G and all x 2 M. Given such an action onecan define, for each g 2 G, a di↵eomorphism �g : M ! M by �g(x) = g · x. If aLagrangian is given on M, then it may be possible to find a Lie group G and a leftaction � of G on M for which these di↵eomorphisms �g are all symmetries of L. Inthis case we refer to G as a symmetry group of L. If g is the Lie algebra of G, theneach nonzero element N of g gives rise to an infinitesimal symmetry XN defined ateach x 2 M by

XN(x) =ddt

(etN · x)��t=0.

Each of these in turn gives rise, via Noether’s Theorem, to a conserved quantity. Theconservation laws come from the Lie algebra of the symmetry group.

We will now try to illustrate all of these ideas with a concrete example. Manymore such examples are available in Section 2.2 of [Nab5].Example A.2.1. (Momentum, Angular Momentum, and Energy) For our configura-tion space we begin with M = Rn and choose global standard Cartesian coordinates(q1, . . . , qn) on Rn. The state space is then TRn = Rn ⇥ Rn and the correspondingnatural coordinates are q1, . . . , qn, q1, . . . , qn. Letting V(q1, . . . , qn) denote an arbi-trary smooth, real-valued function on Rn and m a positive constant, we take ourLagrangian to be


L(q1, . . . , qn, q1, . . . , qn) =12

mnX

i=1

(qi)2 � V(q1, . . . , qn).

When evaluated on the lift of some ↵ 2 C1a,b([t0, t1],Rn) this gives the kinetic minusthe potential energy of a particle of mass m moving along ↵ as a function of t. ThePrinciple of Stationary Action dictates that the actual trajectory of the particle is astationary curve for the action S L. To write down the Euler-Lagrange equations wenote that

@L@qi = �

@V@qi , i = 1, . . . , n

and

@L@qi = mqi, i = 1, . . . , n.

Thus, (A.2) becomes

�@V@qi �

ddt

(mqi) = 0, i = 1, . . . , n,

that is,

md2qi

dt2 = �@V@qi , i = 1, . . . , n

and these are just the components of Newton’s Second Law for a conservative force�@V/@qi, i = 1, . . . , n, with potential V . This is, of course, one of the primary moti-vations behind the Principle of Stationary Action.

Notice that if the potential V(q1, . . . , qn) happens not to depend on, say, the ith-coordinate qi, then, since pi =

@L@qi = mqi, we conclude that pi = mqi is con-

stant along the trajectory of the particle. Now, mqi is what physicists call the ith-component of the particle’s (linear) momentum. Thus, if the potential V is indepen-dent of the ith-coordinate, then the ith-component of momentum is conserved duringthe motion. The conjugate momentum is really momentum in this case. In partic-ular, for a particle that is not subject to any forces (a free particle), the potentialV(q1, . . . , qn) is constant so all of the momentum components are conserved. Onecan phrase this in the following way. If the potential V(q1, . . . , qn) and thereforethe Lagrangian L(q1, . . . , qn, q1, . . . , qn) = 1

2 mPn

i=1(qi)2 � V(q1, . . . , qn) is invariantunder translations in Rn, then momentum is conserved.

Spatial Translation Symmetry implies Conservation of (Linear) Momentum

We conclude by looking at a somewhat less obvious example of this sort of conser-vation law.

We will specialize to the case in which n = 3 and the potential is sphericallysymmetric, that is, V depends only on kqk = ( (q1)2 + (q2)2 + (q3)2 )1/2.


V(q1, q2, q3) = V(kqk)

Thus, we can write the Lagrangian as

L(q, q) =12

mkqk2 � V(kqk).

Now we will find a symmetry group of L. Recall that the rotation group SO(3)consists of all 3 ⇥ 3 real matrices A that are orthogonal (AT A = AAT = id3⇥3) andhave determinant one (det A = 1). SO(3) acts on R3 by matrix multiplication

�A(q) = Aq,

where Aq means matrix multiplication with q 2 R3 thought of as a column vector.Since A is invertible, �A is a di↵eomorphism of R3 onto R3. Moreover, since �Ais linear, its derivative at each point is the same linear map (multiplication by A)once the tangent spaces are canonically identified with R3. Thus, the induced mapon state space

T�A : TR3 = R3 ⇥R3 ! TR3 = R3 ⇥R3

is given by

(T�A)(q, q) = (�A(q), (�A)⇤q(q)) = (Aq, Aq).

This is again a di↵eomorphism of TR3 onto TR3. Moreover, since A is orthogonal,kAqk = kqk and kAqk = kqk so

L � T�A = L

and therefore SO(3) is indeed a symmetry group of L. One says simply that L isinvariant under rotation.

From this symmetry group we would now like to build infinitesimal symmetriesof L and compute some Noether conserved quantities. For this we need informationabout the Lie algebra so(3) of SO(3) and its exponential map. Now, so(3) consistsof the set of all 3 ⇥ 3, skew-symmetric, real matrices with entrywise linear opera-tions and matrix commutator as bracket (Section 5.8 of [Nab2]). The following isTheorem 2.2.2 of [Nab5], but the proof is on pages 393-395 of [Nab2].

Theorem A.2.2. Let N be an element of so(3). Then the matrix exponential etN isin SO(3) for every t 2 R. Conversely, if A is any element of SO(3), then there is aunique t 2 [0, ⇡] and a unit vector n = (n1, n2, n3) in R3 for which

A = etN = id3⇥3 + (sin t)N + (1 � cos t)N2,

where N is the element of so(3) given by


N =

0BBBBBBBB@

0 �n3 n2

n3 0 �n1

�n2 n1 0

1CCCCCCCCA .

Geometrically, one thinks of A = etN as the rotation of R3 through t radians aboutan axis along n in a sense determined by the right-hand rule from the direction of n.

Now fix an n and the corresponding N in so(3). For any q 2 R3,

t ! etNq

is a curve in R3 passing through q at t = 0 with velocity vector

ddt

(etNq)��t=0 = Nq.

Doing this for each q 2 R3 gives a smooth vector field XN on R3 defined by

XN(q) =ddt

(etNq)��t=0 = Nq.

Like any (complete) vector field on R3, XN determines a 1-parameter group of dif-feomorphisms

't : R3 ! R3, �1 < t < 1,

where 't pushes each point of R3 t units along the integral curve of XN that startsthere. This 1-parameter group of di↵eomorphisms is also called the flow of the vec-tor field. In the case at hand,

't(q) = etNq.

Notice that each 't, being multiplication by some element of SO(3), is a symmetryof L so XN is indeed an infinitesimal symmetry of L.

Next we will write out a few of these vector fields explicitly. Choose, for example,n = (0, 0, 1) 2 R3. Then

N =

0BBBBBBBB@

0 �1 01 0 00 0 0

1CCCCCCCCA

so

XN(q) = Nq =

0BBBBBBBB@

�q2

q1

0

1CCCCCCCCA .

This gives the components of XN relative to {@q1 , @q2 , @q3 } so the vector field XN isjust

X12 = q1@q2 � q2@q1 (A.3)


Taking n to be (1, 0, 0) and (0, 1, 0) one obtains, in the same way, the vector fields

X23 = q2@q3 � q3@q2 (A.4)

and

X31 = q3@q1 � q1@q3 . (A.5)

Exercise A.2.1. In the physics literature the matrices N corresponding to n =(1, 0, 0), (0, 1, 0) and (0, 0, 1) are denoted Lx, Ly and Lz, respectively. Show that theseform a basis for so(3) and satisfy the commutation relations

[Lx, Ly] = Lz, [Lz, Lx] = Ly, [Ly, Lz] = Lx,

where [ , ] denotes the matrix commutator.

To find the Noether conserved quantity corresponding to the infinitesimal sym-metry X12 in standard coordinates we compute

Xk12@L@qk = X1

12@L@q1 + X2

12@L@q2 + X3

12@L@q3 = �q2(mq1) + q1(mq2) = m(q1q2 � q2q1).

For X23 and X31 one obtains m(q2q3 � q3q2) and m(q3q1 � q1q3), respectively. Thus,along any stationary curve,

m [q1(t)q2(t) � q2(t)q1(t)]

m [q3(t)q1(t) � q1(t)q3(t)]

m [q2(t)q3(t) � q3(t)q2(t)]

are all constant. Notice that these are precisely the components of the cross product

r(t) ⇥ [m r(t)]

of the position and momentum vectors of the particle and this is what physicists callits (orbital) angular momentum (with respect to the origin). Thus, angular momen-tum is constant for motion in a spherically symmetric potential in R3.

Rotational Symmetry implies Conservation of Angular Momentum

Notice also that the constancy of the angular momentum vector along the trajectoryof the particle implies that the motion takes place entirely in a 2-dimensional planein R3, namely, the plane with this normal vector.

We will conclude with an example of a slightly di↵erent sort. We have defined aLagrangian to be a function on the tangent bundle T M, but it is sometimes conve-nient to allow it to depend explicitly on t as well, that is, to define a Lagrangian to


be a smooth map L : R ⇥ T M ! R. Then, for any path in the domain of a coordi-nate neighborhood on M, one would write L = L(t,↵(t), ↵(t)), t0 t t1, in naturalcoordinates. The action associated with this path is defined, as before, to be the inte-gral of L(t,↵(t), ↵(t)) over [t0, t1]. Stationary points for the action are also defined inprecisely the same way and one can check that the additional t-dependence has noe↵ect at all on the form of the Euler-Lagrange equations, that is, stationary curvessatisfy

@L@qk

�t,↵(t), ↵(t)

� � ddt

✓ @L@qk

�t,↵(t), ↵(t)

�◆= 0, 1 k n. (A.6)

Write the stationary curve in local coordinates as ↵(t) = (q1(t), . . . , qn(t)) and theLagrangian evaluated on the lift of this curve as

L(t, q1(t), . . . , qn(t), q1(t), . . . , qn(t)).

Computing dLdt from the chain rule and (A.6) gives

ddt

(L � piqi) =@L@t

along the stationary curve. If it so happens that L does not depend explicitly on t,then @L

@t = 0 so EL = piqi � L is conserved along the stationary curve. Notice thatfor the particle Lagrangian L(q1, . . . , qn, q1, . . . , qn) = 1

2 mPn

i=1(qi)2 � V(q1, . . . , qn)

EL = piqi � L =12

mnX

i=1

(qi)2 + V(q1, . . . , qn) (A.7)

which is the total energy (kinetic plus potential). We will phrase this in the followingway.

Time Translation Symmetry implies Conservation of Energy

Here then is a synopsis of what we have concluded about symmetries and con-servation laws for classical mechanical systems.

Symmetry Conservation Law

Spatial Translation Linear MomentumRotation Angular Momentum

Time Translation Total Energy

A.3 Finite-Dimensional Hamiltonian Mechanics 133

A.3 Finite-Dimensional Hamiltonian Mechanics

In Section 2.3 of [Nab5] the Hamiltonian picture of classical mechanics evolved outof the Lagrangian picture, but it is only the Hamiltonian view itself that is significantfor us at the moment and not the particular route by which one arrives at it. Exceptfor the occasional comment, such as those in Remarks A.3.1 and A.3.3 below, wewill take no heed of its Lagrangian origins here.

We will start with the simplest case and then generalize. One begins, as in theLagrangian case, with an n-dimensional smooth manifold M called the configura-tion space and generally denotes a local coordinate system on M by (q1, . . . , qn).We think of M as the space of possible positions of the particles in the system. Forexample, it is shown in Section 2.2 of [Nab5] that the configuration space of a rigidbody constrained to pivot about some fixed point is the manifold SO(3) and anysuch motion of the rigid body is represented by a continuous curve t 2 R ! A(t) 2SO(3) in SO(3). The cotangent bundle P = T ⇤M of M (Remark 2.3.2 of [Nab5] orSection 3.2 of [BG]) is called the phase space and the points (x, ⌘x) in it representstates of the system. Here x is a point in M and ⌘x 2 T ⇤x (M) is a covector at x, thatis, an element of the dual of the tangent space Tx(M) to M at x.

Remark A.3.1. In the Lagrangian picture described in Section A.2 the states ofthe system at described by points (x, vx) in the tangent bundle T M of M. Then xrepresents a possible configuration of the particles and vx a possible rate of changeof the configuration. Locally, in coordinates, (x, vx) = (q1, . . . , qn, q1, . . . , qn) and thetransition from T M to T ⇤M amounts to replacing the qi by the conjugate momentapi = @L/@qi (see Section 2.3 of [Nab5]). We trust that the context (Lagrangian orHamiltonian) will always make it clear whether the word “state” refers to a point inT M or T ⇤M; these should be thought of simply as two di↵erent ways of describingthe “physical state” of some underlying mechanical system.

T ⇤M admits a canonical 1-form ✓ defined in the following way. For any (x, ⌘x) 2T ⇤M, with x 2 M and ⌘x 2 T ⇤x (M), ✓(x,⌘x) is the linear functional defined onT(x,⌘x)(T ⇤M) by ✓(x,⌘x) = ⌘x � ⇡⇤(x,⌘x), where ⇡ is the projection of T ⇤M onto Mand ⇡⇤(x,⌘x) is its derivative at (x, ⌘x). Near each point of T ⇤M there are local co-ordinates (q1, . . . , qn, p1, . . . , pn), called canonical coordinates, relative to which✓ = pidqi = p1dq1 + · · · + pndqn. The 2-form ! = �d✓ is closed (d! = 0) andnondegenerate (◆X! = 0 ) X = 0, where ◆X! is the contraction of ! with thesmooth vector field X on T ⇤M, defined by (◆X!)(Y) = !(X,Y) for any smooth vec-tor field Y on T ⇤M). ! is called the canonical symplectic form on T ⇤M and, incanonical coordinates, it is given by ! = dqi ^ dpi.

One selects a distinguished, smooth real-valued function H : T ⇤M ! R,called the Hamiltonian and representing the total energy of the system beingmodeled, and defines from it and the symplectic form ! a vector field XH onT ⇤M, called the Hamiltonian vector field, by the requirement that ◆XH! = dH.In canonical coordinates, XH = (@H/@pi)@qi � (@H/@qi)@pi . The state of the sys-tem is then assumed to evolve with time from some initial state along the inte-


gral curve of XH through this initial state. Consequently, the evolution of the state(q1(t), . . . , qn(t), p1(t), . . . , pn(t)) with time satisfies Hamilton’s equations

qi =@H@pi

and pi = �@H@qi , i = 1, . . . , n.

The Hamiltonian flow, that is, the flow of the Hamiltonian vector field XH , thereforecontains all of the information about the evolution of the state of the system.

Remark A.3.2. Depending on the nature of H, Hamilton’s equations may have so-lutions defined only on some finite interval of t values so the Hamiltonian flow maybe only a local flow.

Example A.3.1. (Particle Motion inRn) The standard example models a single par-ticle of mass m > 0 moving in R3 under the influence of some conservative forceF = �rV . The configuration space is M = R3 and we denote by (q1, q2, q3) itsstandard Cartesian coordinates. Physically, M is identified with the space of possi-ble positions of the particle. Since R3 is contractible, its cotangent bundle can beidentified with T ⇤M = R3 ⇥R3 = R6. The Hamiltonian is to be the total energy ofthe particle, that is, the sum of its kinetic and potential energies, and we would liketo describe this in canonical coordinates (q1, q2, q3, p1, p2, p3).

Remark A.3.3. In Example A.2.1 we saw that the di↵erence of the kinetic and poten-tial energies is the Lagrangian L for this system and that the momentum conjugateto qi is @L/@qi = mqi. In light of Remark A.3.1 we have pi = mqi so the particle’skinetic energy can be written 1

2mkpk2.

The particle’s potential energy due to the influence of the conservative force �rV(q)is just V(q). The Hamiltonian H(q, p) is the sum of these.

H(q, p) =1

2mkpk2 + V(q) =

12m

(p21 + p2

2 + p23) + V(q1, q2, q3).

Relative to these canonical coordinates on R6 the canonical symplectic form is

! = dqi ^ dpi = dq1 ^ dp1 + dq2 ^ dp2 + dq3 ^ dp3

and the Hamiltonian vector field is therefore

XH(q, p) = (@H/@pi) @qi � (@H/@qi) @pi =1m

pi @qi � @V@qi @pi .

From this we read o↵ Hamilton’s equations

qi =1m

pi and pi = �@V@qi , i = 1, 2, 3.


Notice that substituting the first of these into the second gives Newton’s Second Law

mqi = �@V@qi , i = 1, 2, 3.

There is an obvious generalization of all of this to Rn for any n = 1, 2, 3, . . . and wewill record one particularly important example.

Take n = 1 so that the configuration space is M = R and the phase space isT ⇤M = R2. Writing q1 = q and p1 = p for the canonical coordinates and takingV(q) = m!2

2 q2, where ! is a positive constant, we obtain the Hamiltonian

H(q, p) =1

2mp2 +

m!2

2q2

for the classical harmonic oscillator of mass m and natural frequency! (see Chapter1 of [Nab5]) . The Hamiltonian vector field is

XH(q, p) =1m

p @q � m!2q @p

and Hamilton’s equations are

q =1m

p and p = �m!2q.

These combine to give

q + !2q = 0

which is easily solved to obtain

q(t) = A cos (!t + '),

where A is a non-negative constant, called the amplitude of the oscillator, and ' isa real constant called the phase. Notice that, because the Hamiltonian is quadratic,Hamilton’s equations are linear so the integral curves are defined for all t and theflow is global.

The essential data of Hamiltonian mechanics as we have discussed it thus farconsists of the phase space T ⇤M with its canonical symplectic form ! and a distin-guished real-valued function H on phase space (the canonical 1-form ✓ played onlya supporting role in that it gave rise to !). The rest of the formalism, which we nowreview, is all derived from these (all of the details are available in Section 2.3 of[Nab5]).

Each smooth, real-valued function f on the phase space T ⇤M is called a classicalobservable. The space C1(T ⇤M;R) of smooth, real-valued functions on T ⇤M, withits usual real vector space structure and pointwise multiplication, is referred to as thealgebra of classical observables. Each f 2 C1(T ⇤M;R) has a symplectic gradient


Xf defined by the requirement that ◆Xf! = d f .

Note: The symplectic gradient of f is also called the Hamiltonian vector field of f ,but we will reserve this term for the symplectic gradient XH of the Hamiltonian H.

In canonical coordinates, Xf = (@ f /@pi)@qi � (@ f /@qi)@pi . Each Xf preserves thesymplectic form ! in the sense that

LXf! = 0,

where LXf is the Lie derivative with respect to Xf . This follows from Cartan’s“magic” formula for the Lie derivative since

LX f! = (d◆Xf + ◆Xf d)! = d(d f ) + ◆Xf (d!) = 0 + ◆X f (0) = 0.

Equivalently, '⇤t! = ! for every 't in the (possibly local) 1-parameter group ofdi↵eomorphisms of Xf .

For f , g 2 C1(T ⇤M;R) we define their Poisson bracket { f , g} by { f , g} =!(Xf , Xg). One finds that, in canonical coordinates,

{ f , g} = @ f@qi

@g@pi� @ f@pi

@g@qi

and that the Lie bracket of two symplectic gradient vector fields Xf and Xg is thesymplectic gradient of the Poisson bracket of f and g, that is,

[Xf , Xg] = X{g, f } 8 f , g 2 C1(T ⇤M;R).

The Poisson bracket provides C1(T ⇤M;R) with the structure of a Lie algebra since

{ , } : C1(T ⇤M;R) ⇥C1(T ⇤M;R)! C1(T ⇤M;R)

is R-bilinear, skew-symmetric

{g, f } = �{ f , g} 8 f , g 2 C1(T ⇤M;R),

and satisfies the Jacobi Identity

{ f , {g, h}} + {h, { f , g}} + {g, {h, f }} = 0 8 f , g, h 2 C1(T ⇤M;R).

Moreover, C1(T ⇤M;R) is a Poisson algebra since the Poisson bracket satisfies theLeibniz Rule

{ f , gh} = { f , g}h + g{ f , h} 8 f , g, h 2 C1(T ⇤M;R).

In terms of the Poisson bracket Hamilton’s equations take the form


qi = {qi,H} and pi = {pi,H}, i = 1, . . . , n.

Moreover, for any f , g 2 C1(T ⇤M;R),

{ f , g} = Xg[ f ] = LXg f ,

so, in particular,

{ f ,H} = LXH f

is the rate of change of f along the integral curves of the Hamiltonian, that is, alongthe trajectories of the system. Consequently, f is conserved (constant along trajecto-ries) if and only if it Poisson commutes with the Hamiltonian, meaning { f ,H} = 0. Inparticular, the Hamiltonian itself is conserved and this is the form that conservationof energy takes in Hamiltonian mechanics.

We see then that conservation laws appear to arise rather di↵erently in the Hamil-tonian picture than in the Lagrangian and one might wonder if there is an analogue ofNoether’s Theorem A.2.1 relating conserved quantities to “symmetries” in Hamil-tonian mechanics. There is indeed, but the real beauty and significance of the resultwill not be so apparent if we continue to restrict our attention to the rather specialcase we have been considering to this point (the case in which the phase space isa cotangent bundle). For this reason we will now pause to describe the much moregeneral context in which the Hamiltonian picture lives most naturally. This done,we will return to the question of symmetries and conserved quantities.

The structure we have been discussing was built from essentially just three ba-sic ingredients: a phase space and, defined on it, a nondegenerate, closed 2-formand a distinguished smooth, real-valued function. As a result it is a simple matterto describe a very general, abstract mathematical structure that encompasses theHamiltonian picture of classical mechanics as a special case. This is worth doingfor many reasons related to quantization, representation theory, geometry, topology,and as a stepping stone to the infinite-dimensional version required to accommo-date classical field theory. Our description will be relatively brief, but for those whowish to see more of this we might recommend [Bern], the article Introduction to LieGroups and Symplectic Geometry by Robert Bryant in [FU], [AM], [AMR], [Arn2],[GS1], and [Ch].

We begin with a smooth manifold P of dimension 2n (the reason for assumingthe dimension is even will be clear momentarily); for simplicity we will assume alsothat P is connected. P will be called the phase space. A symplectic form on P is a2-form ! on P that is closed (d! = 0) and nondegenerate (◆V! = 0 ) V = 0). Thepair (P,!) is called a symplectic manifold. Bilinearity and nondegeneracy imply that! determines an isomorphism ![ from the space X(P) of smooth vector fields on Pto the space ⌦1(P) of smooth, real-valued 1-forms on P defined by

![(X) = ◆X! 8X 2 X(P).


The inverse of ![ is denoted !] : ⌦(P)! X(P).

Remark A.3.4. If a manifold P has a symplectic form ! defined on it, then P isnecessarily even dimensional. The reason is as follows. Suppose dim P = m. Forany p 2 P, !p can be identified with a skew-symmetric m⇥m matrix (!p) and non-degeneracy implies that this matrix is nonsingular. But then det (!p) = det (!p)T =(�1)m det (!p) by skew-symmetry and this is possible only if m is even.

P must also be orientable since the nondegeneracy of ! implies that !n = !^ n· · ·^! is a nonzero 2n-form on the 2n-dimensional manifold P, that is, a volume form,and the existence of a volume form is equivalent to orientability (Theorem 4.3.1 of[Nab3]). However, not every orientable, even dimensional manifold admits a sym-plectic form. An example is the 4-sphere S 4 and the reason is topological. Indeed,a symplectic form ! on S 4, being a closed 2-form, represents an element [!] ofthe second de Rham cohomology group H2

deRham(S 4). This element would have tobe nonzero since [!] = 0 ) [!]2 = [!2] = 0 and from this one would obtainR

S 4 !2 = 0 which contradicts the fact that !2 is a volume form on S 4. However,

H2deRham(S 4) is the trivial group (Exercise 5.4.3 of [Nab3]) so such an ! cannot ex-

ist.

Example A.3.2. Let M be any smooth, n-dimensional manifold. Then the cotan-gent bundle T ⇤M is a smooth, 2n-dimensional manifold which, as we have seen,admits a (canonical) symplectic form !. Thus, (T ⇤M,!) is a symplectic manifold.Notice that, for this example, ! is not only closed, but also exact (! = d(�✓),where ✓ is the canonical 1-form on T ⇤M). From the perspective of de Rham coho-mology this means that the 2-form ! is cohomologically trivial. This is certainlynot the case for a general symplectic form. Indeed, the ideas we just used to showthat S 4 does not admit a symplectic structure show also that the symplectic formon any compact symplectic manifold is cohomologically nontrivial, that is, repre-sents a nonzero second cohomology class. There are, incidentally, lots of compactsymplectic manifolds (see Example A.3.4).

Example A.3.3. Let V be a 2n-dimensional real vector space (like R2n or the tan-gent space to a 2n-dimensional manifold). Any basis {e1, . . . , e2n} for V determines anatural topology and di↵erentiable structure for V. Specifically, if {e1, . . . , e2n} is thedual basis, then p 2 V 7! (x1, . . . , x2n) = (e1(p), . . . , e2n(p)) 2 R2n is a bijection. Wesupply V with the unique topology for which this map is a homeomorphism. A dif-ferent choice of basis determines the same topology so these homeomorphisms arecharts on the topological space V. These charts overlap smoothly and so determinea di↵erentiable structure on V. The tangent space Tp(V) at any p 2 V is naturallyidentified with V itself. Specifically, each vp 2 Tp(V) is the velocity vector to thecurve t ! p + tv for some v 2 V and vp 7! v is an isomorphism. From now on wewill make this identification without further comment.

Now, let S : V ⇥ V ! R be a nondegenerate, skew-symmetric, bilinear form onV. For example, if h , i denotes the usual positive definite inner product on Rn andif we regard R2n as Rn ⇥Rn, then


S ( (x1, y1), (x2, y2) ) = hx1, y2i � hy1, x2i

defines such a bilinear form. Any such S determines a (constant) symplectic form! on the manifold V by defining, at each p 2 V,

!p(vp,wp) = S (v,w).

It would seem then that one can produce a wide variety of symplectic forms onV by simply making various choices for S . These hopes are dashed, however, bya theorem in linear algebra according to which these are, up to a change of linearcoordinates, all the same. The following is Proposition 1.3(ii) of [Bern].

Theorem A.3.1. (Linear Darboux Theorem) Let V be a 2n-dimensional real vec-tor space, S : V ⇥ V ! R a nondegenerate, skew-symmetric, bilinear formon V, and ! the corresponding symplectic form on V. Then there exists a basis{e1, . . . , en, en+1, . . . , e2n} for V with corresponding linear coordinates(q1, . . . , qn, p1, . . . , pn) such that

! = dqi ^ dpi = �d(pidqi).

It is remarkable that this result extends locally to arbitrary symplectic manifolds.For a very detailed proof of the following theorem of Darboux see Section 2.2 of[Bern].

Theorem A.3.2. (Darboux Theorem) Let (P,!) be a symplectic manifold of dimen-sion 2n. Then, at each point in P, there exists a local coordinate neighborhood Uwith coordinates (q1, . . . , qn, p1, . . . , pn) such that, on U,

! = dqi ^ dpi = �d(pidqi).

Remark A.3.5. The essential content of the Darboux Theorem is that all symplec-tic manifolds of the same dimension are locally the same. This contrasts rathermarkedly with Riemannian manifolds of the same dimension, for which the localstructure is determined by curvature. This gives symplectic geometry and topologyquite a di↵erent flavor from classical di↵erential geometry and topology (see, forexample, [MS]).

Example A.3.4. Let P be any orientable smooth surface (that is, orientable 2-dimensional manifold) and ! a volume (area) form on P. Then ! is a 2-form and itis closed because every 2-form on a 2-manifold is closed. It is also nondegenerate.To see this we assume that V is a vector field on P for which ◆V! = 0 and will showthat V(p) = 0 at each p 2 P. Since ! determines an orientation for P there exists, ona neighborhood U of p, a chart with coordinates (x, y) for which !(@x, @y) > 0 (seeTheorem 4.3.1 of [Nab3]). Then !(@y, @x) < 0 and !(@x, @x) = !(@y, @y) = 0 on U.


Now write V = Vx@x + Vy@y for some smooth functions Vx and Vy on U. Then, byassumption,

0 = !(Vx@x + Vy@y, @y) = Vx!(@x, @y)

at every point of U. Consequently, Vx = 0 on U. Similarly, Vy = 0 on U. In par-ticular, V(p) = Vx(p)@x(p) + Vy(p)@y(p) = 0 as required. Consequently, ! is asymplectic form on P and so every orientable surface is a symplectic 2-manifold.Since lots of these are compact (S 2 for example) we have fulfilled our promise toexhibit examples of compact symplectic manifolds (Example A.3.2).

Now suppose (P,!) is an arbitrary symplectic manifold of dimension 2n. TheDarboux Theorem implies that, at each point of P, there exists a local coordinateneighborhood U with coordinates (q1, . . . , qn, p1, . . . , pn) such that, on U,

! = dqi ^ dpi.

We call these canonical coordinates. The algebra C1(P;R) of smooth real-valuedfunctions on P with its usual real vector space structure and pointwise multiplica-tion is called the algebra of classical observables and each element of it is called aclassical observable. Select some element H of C1(P;R), christen it the Hamilto-nian and think of it as the total energy of a physical system whose phase space isbeing modeled by (P,!). The pair ((P,!),H) is then called a Hamiltonian system.

Since ! is nondegenerate any covector in T ⇤p(P) is !p(vp, · ) for some tangentvector vp 2 Tp(P) and every 1-form on P is !(V, · ) = ◆V! for some smooth vectorfield V on P. In particular, any smooth real-valued function f on P has a symplecticgradient X f defined by the requirement that ◆Xf! = d f , that is, !(Xf , · ) = d f ( · ).The symplectic gradient XH of the Hamiltonian is called the Hamiltonian vectorfield.

Remark A.3.6. Again we point out that Xf is often called the Hamiltonian vectorfield of f .

Exercise A.3.1. Show that, in local canonical coordinates,

Xf = (@ f /@pi)@qi � (@ f /@qi)@pi .

In particular, the integral curves of the Hamiltonian vector field XH are locally givenby functions (q1(t), . . . , qn(t), p1(t), . . . , pn(t)) that satisfy Hamilton’s equations

qi =@H@pi

and pi = �@H@qi , i = 1, . . . , n

and the rate of change of any classical observable f 2 C1(P;R) along these integralcurves is locally given by


XH[ f ] = LXH f =@H@pi

@ f@qi �

@H@qi

@ f@pi.

More generally, we define, for any f , g 2 C1(P;R), the Poisson bracket of f and gdetermined by ! by

{ f , g} = !(Xf , Xg)

and find that, locally in canonical coordinates,

{ f , g} = @ f@qi

@g@pi� @ f@pi

@g@qi .

Exercise A.3.2. Let (P,!) be an arbitrary symplectic manifold and let (q, p) =(q1, . . . , qn, p1, . . . , pn) be canonical coordinates on some open neighborhood U inP. Let !U denote the restriction of ! to U, that is, the pullback of ! to U by theinclusion map U ,! P. Show that (U,!U) is a symplectic manifold and that thecanonical coordinate functions q1, . . . , qn, p1, . . . , pn 2 C1(U;R) satisfy the classi-cal canonical commutation relations

{qi, q j} = {pi, p j} = 0 and {qi, p j} = �ij, i, j = 1, . . . , n, (A.8)

where �ij is the Kronecker delta.

One can now show that, in this more general context, all of the fundamentalproperties that we enumerated for the cotangent bundle remain valid.

Remark A.3.7. The proofs given in Section 2.3 of [Nab5] for the cotangent bun-dle are intentionally phrased in such a way as to carry over verbatim to arbitrarysymplectic manifolds.

Specifically, one can prove all of the following.

LXf! = 0 8 f 2 C1(P;R)

[Xf , Xg] = X{g, f } 8 f , g 2 C1(P;R)

Furthermore

{ , } : C1(P;R) ⇥C1(P;R)! C1(P;R)

is R-bilinear, skew-symmetric

{g, f } = �{ f , g} 8 f , g 2 C1(P;R),

satisfies the Jacobi Identity


{ f , {g, h}} + {h, { f , g}} + {g, {h, f }} = 0 8 f , g, h 2 C1(P;R),

and the Leibniz Rule

{ f , gh} = { f , g}h + g{ f , h} 8 f , g, h 2 C1(P;R).

Moreover, in terms of the Poisson bracket, Hamilton’s equations take the form

qi = {qi,H} and pi = {pi,H}, i = 1, . . . , n

and f 2 C1(P;R) is conserved (constant along the integral curves of the Hamilto-nian vector field) if and only if it Poisson commutes with the Hamiltonian, that is,if and only if

{ f ,H} = 0.

In particular, the Hamiltonian itself (that is, the total energy) is clearly conservedsince {H,H} = 0 follows from the skew-symmetry of the Poisson bracket. More-over, it follows from the Jacobi identity that the Poisson bracket of two conservedquantities is also conserved. Indeed,

{ f ,H} = {g,H} = 0) {{ f , g},H} = {g, {H, f }} + { f , {g,H}} = {g, 0} + { f , 0} = 0.

More generally, even if f is not conserved, the Poisson bracket keeps track of howit evolves with the system in the sense that, along an integral curve of XH ,

d fdt= { f ,H} (A.9)

(because XH[ f ] = �{H, f } = { f ,H}).

Now we are prepared to return to the question of symmetries and conservationlaws (see page 137). We begin with a few general definitions. Let (P1,!1) and(P2,!2) be two symplectic manifolds. A map F : P1 ! P2 is called a symplec-tic di↵eomorphism, or symplectomorphism, or, in the physics literature, a canonicalmap if it is a di↵eomorphism of P1 onto P2 that carries !1 onto !2 in the sense that

F⇤!2 = !1.

A smooth vector field X on a symplectic manifold (P,!) is called a symplectic vectorfield if it preserves the symplectic form in the sense that

LX! = 0.

If X is complete this is equivalent to the requirement that each 't, t 2 R, in the1-parameter group of di↵eomorphisms determined by X is a symplectic di↵eomor-phism. A vector field X on (P,!) is said to be Hamiltonian if it is the symplecticgradient of some f 2 C1(P;R). Every Hamiltonian vector field is therefore also a


symplectic vector field. That the converse is generally not true follows from the nextexercise.

Exercise A.3.3. Let X be a smooth vector field on (P,!). Prove each of the follow-ing.

1. X is symplectic if and only if the 1-form ◆X! is closed.2. X is Hamiltonian if and only if the 1-form ◆X! is exact.

When the first de Rham cohomology group of P is trivial (for example, when P =R2n), every closed 1-form is exact so a vector field X on (P,!) is symplectic if andonly if it is Hamiltonian. More generally, since P is locally di↵eomorphic to R2n

any symplectic vector field on P is locally Hamiltonian.

A symmetry of the Hamiltonian system ((P,!),H) is a symplectic di↵eomor-phism F : P ! P of P onto itself that preserves the Hamiltonian H in the sensethat

F⇤H = H,

that is,

H � F = H.

Thus, a symmetry is a di↵eomorphism of phase space that preserves both the sym-plectic form and the Hamiltonian. For the infinitesimal version we proceed as in theLagrangian case (see page 126). A smooth, complete vector field X on P is said tobe an infinitesimal symmetry of the Hamiltonian system ((P,!),H) if each 't, t 2 R,in the 1-parameter group of di↵eomorphisms of X is a symmetry of ((P,!),H); if Xis not complete then one modifies the definition exactly as in the Lagrangian case(page 126).

Infinitesimal symmetries generally arise from group actions of the followingtype. Let G be a (matrix) Lie group with Lie algebra g and suppose � : G ⇥ P! P,�(g, x) = �g(x) = g · x, is a smooth left action of G on P. If each of the di↵eomor-phisms �g : P ! P, g 2 G, is a symmetry of the Hamiltonian system ((P,!),H),then we refer to G as a symmetry group of ((P,!),H). In this case each nonzeroelement ⇠ of g gives rise to an infinitesimal symmetry X⇠ defined at each x 2 P by

X⇠(x) =ddt

(et⇠ · x)��t=0.

In order to write down a Hamiltonian version of Noether’s Theorem A.2.1 weintroduce an idea that is fundamental to modern symplectic geometry. For this wedenote by g⇤ the vector space dual of the Lie algebra g. This is a finite-dimensionalreal vector space so it has a natural topology and di↵erentiable structure. Considera smooth map


µ : P! g⇤.

Then for every x 2 P, µ(x) is a real-valued linear map on g

µ(x) : g! R

so that (µ(x))(⇠) is a real number for every ⇠ 2 g. We want to fix µ and ⇠ and regardthis as a real-valued function on P, that is, we define

hµ, ⇠i : P! R

by

hµ, ⇠i(x) = µ(x)(⇠).

Note: The notation is meant to suggest the natural pairing of g and g⇤ so that oneshould think of hµ, ⇠i as being defined by

hµ, ⇠i(x) = hµ(x), ⇠i.

The map hµ, ⇠i is smooth so it has a symplectic gradient Xhµ,⇠i. We will say that µ isa moment map, also called a momentum map, for the Hamiltonian system ((P,!),H)with respect to the symmetry group G if

Xhµ,⇠i = X⇠

for every ⇠ 2 g. The Hamiltonian version of Noether’s Theorem A.2.1 asserts thatif µ is a moment map for the Hamiltonian system ((P,!),H) with respect to thesymmetry group G, then hµ, ⇠i is conserved for every ⇠ 2 g.

Theorem A.3.3. (Noether’s Theorem: Hamiltonian Version) Let ((P,!),H) be aHamiltonian system, G a symmetry group of ((P,!),H) with Lie algebra g, andµ : P ! g⇤ a moment map for ((P,!),H) with respect to G. Then, for every ⇠ 2 g,the function

hµ, ⇠i : P! R

defined by

hµ, ⇠i(x) = µ(x)(⇠)

for every x 2 P is constant along the integral curves of the Hamiltonian vector fieldXH. In particular, each hµ, ⇠i Poisson commutes with the Hamiltonian H.

�hµ, ⇠i,H = 0


Proof. Let �(t) be an integral curve of XH . Then, by definition, �(t) = XH(�(t)) foreach t. We compute, for each fixed ⇠ 2 g,

ddt

⇥hµ, ⇠i(�(t))⇤= dhµ, ⇠i�(t)(�(t))

= dhµ, ⇠i�(t)(XH(�(t)))=

�◆Xhµ,⇠i!

��(t)(XH(�(t)))

= !(Xhµ,⇠i, · )�(t)(XH(�(t)))= !(X⇠, · )�(t)(XH(�(t)))= !�(t)(X⇠(�(t)), XH(�(t)))= �!�(t)(XH(�(t)), X⇠(�(t)))= �dH�(t)(X⇠(�(t)))

= �dH�(t)

✓ dds

(es⇠ · �(t))��s=0

◆

= � dds

�H(es⇠ · �(t))

� ��s=0

= � dds

�H(�(t))

� ��s=0

= 0

as required. ut

This result does not address the issue of actually finding moment maps or thequestion of their existence. As it happens a symmetry group for a Hamiltonian sys-tem need not have a moment map, although there are broad classes of such problemsfor which the existence of moment maps can be proved; for an introduction to thissee, for example, Chapter 4 of [AM]. We will conclude this section with just onesimple example that should at least clarify the origin of the terminology.

Example A.3.5. We construct a Hamiltonian system ((P,!),H) as follows. Let P =T ⇤R3 = R3 ⇥R3 be the tangent bundle of R3 with its canonical symplectic form

! = dqi ^ dpi = dq1 ^ dp1 + dq2 ^ dp2 + dq3 ^ dp3.

For the Hamiltonian we can take any smooth map H : T ⇤R3 ! R that satisfiesH(q + a, p) = H(q, p) for all (q, p) 2 T ⇤R3 and any a 2 R3. One possibility is thekinetic energy Hamiltonian

H(q, p) =kpk22m,

where m is a positive constant. Next we need to find a symmetry group for thisHamiltonian system. Take G to be the additive group R3. We think of G as thetranslation group of R3, that is, we define a left action � : G ⇥ P! P of G on P by


�(a, (q, p)) = (a + q, p)

for all (a, (q, p)) 2 G⇥P. Then each map �a : P! P defined by �a(q, p) = (a+q, p)is a di↵eomorphism and, by assumption,

�⇤aH = H.

Moreover,

�⇤a! = �⇤a(dqi ^ dpi) = �⇤a(dqi) ^ �⇤a(dpi)

= d(qi � �a) ^ d(pi � �a) = dqi ^ dpi

= !.

Consequently, each �a is a symmetry of the Hamiltonian system so G = R3 acts asa symmetry group.

We will identify the Lie algebra ofR3 withR3 (Exercise 1.1.1). The exponentialmap on the Lie algebra satisfies et⇠ = t⇠ for every ⇠ in g and every t 2 R (Exercise1.1.3). Consequently, the vector field X⇠ generated by ⇠ is given by

X⇠(q, p) =ddt

et⇠ · (q, p)��t=0 =

ddt

(q + et⇠, p)��t=0 =

ddt

(q + t⇠, p)��t=0 = ⇠

i @

@qi

��(q,p),

that is,

X⇠ = ⇠i @

@qi .

To find a moment map for this G-action on ((P,�),H) we need a smooth mapµ : P ! g⇤ for which Xhµ,⇠i = X⇠. But Xhµ,⇠i is characterized by ◆Xhµ,⇠i! = dhµ, ⇠i sowhat we need is

◆X⇠! = dhµ, ⇠i.

Exercise A.3.4. Show that ◆X⇠! = ◆X⇠ (dqi ^ dpi) is the 1-form

◆X⇠! = ⇠idpi = ⇠

1dp1 + ⇠2dp2 + ⇠

3 p3.

According to this exercise ◆X⇠! will be equal to dhµ, ⇠i if we define µ in such away that hµ, ⇠i(q, p) = ⇠i pi = ⇠1 p1 + ⇠2 p2 + ⇠3 p3 for each ⇠, that is,

(µ(q, p))(⇠) = ⇠i pi

and this defines the moment map µ : P ! g⇤. Notice that if we identify g⇤ with R3

via the coe�cients of the linear functional on g, then

A.4 Postulates of Quantum Mechanics 147

µ(q, p) = p

so that, in this case, the moment map, or momentum map, really is momentum andwe conclude, as we did in Section A.2, that spatial translation symmetry implies theconservation of momentum.

What we have tried to do in this section is merely suggest that the mathematicalstructure of classical Hamiltonian mechanics is but a special case of the much moregeneral and very elegant subject of symplectic geometry. We have made no attemptto capitalize on this by exploiting ideas from symplectic geometry to shed light onmechanics; this is an enormous subject and there are many superb introductions toit available (see, for example, [AM], [AMR], [Arn2] and [GS1]).

A.4 Postulates of Quantum Mechanics

Physical theories are expressed in the language of mathematics, but these mathe-matical models are not the same as the physical theories and they are not unique.Symplectic geometry provides one possible context in which to understand clas-sical particle mechanics, but not the only one. The Lagrangian formulation has arather di↵erent flavor than the Hamiltonian and the two are not equivalent, but eachencompasses all of classical Newtonian mechanics and each provides its own par-ticular insights. Appropriate mathematical contexts in which to formulate the prin-ciples of quantum mechanics can likewise be constructed in a number of di↵erentways, but we will focus our attention on just one of these which goes back to vonNeumann [v.Neu]; another approach due to Feynman is discussed in Chapter 8 of[Nab5]. Both are quite unlike anything one might naively anticipate from classicalmechanics, although there are precursors in classical statistical mechanics (Sections2.4 and 3.3 of [Nab5]). The roots of the mathematical formalism lie deep in theanalysis of the physical phenomena that the theory purports to describe (Chapter 4of [Nab5]). Section 6.2 of [Nab5] introduces nine Postulates that seem to captureat least the formal structure of quantum mechanics. Since our objective here is sub-stantially more modest we will limit ourselves to only four of these and will discussthem in somewhat less detail. You will notice that the term quantum system is usedrepeatedly, but never defined; the same is true of the term measurement. It is notpossible, nor would it be profitable, to try to define these precisely; they are definedby the assumptions we make about them in the postulates. Each of these postulatesdeserves a commentary on where it came from, what it is intended to mean, how itshould be interpreted, and what objections might be raised to it. However, since anattempt was made to address these issues in Section 6.2 of [Nab5] we will contentourselves here with just a few comments on each following the precise statement ofthe postulate. We will conclude this section with a brief description of the examplesto which we will need to refer in the main body of the text.


Postulate QM1

To every quantum system is associated a separable, complex Hilbert space H. Thestates of the system are represented by vectors 2 H with k k = 1 and, for anyc 2 C with |c| = 1, and c represent the same state.

In 1900 Max Planck [Planck] appended to time-honored concepts in classicalphysics what we would today call an ad hoc “quantization condition” in order tosolve the classically perplexing problem of the equilibrium distribution of electro-magnetic energy in a black box (this is described in some detail in Section 4.3of [Nab5]). For the next quarter of a century physicists devised ever more inge-nious such appendages to classical physics in order to solve the equally perplex-ing problem of atomic structure. These e↵orts had some limited success, but werenot directed by any underlying theoretical or mathematical model of the quantumworld. This changed in 1925-26 when Heisenberg [Heis1] and Schrodinger [Schro1]each proposed such models. On the surface these looked quite di↵erent. Heisen-berg’s formalism eventually came to be known as matrix mechanics since it was ex-pressed in terms of infinite arrays of complex numbers (see Section 7.1 of [Nab5]).Schrodinger’s approach, called wave mechanics, represented the state of a quantumsystem by a complex-valued wave function that was assumed to satisfy a certainpartial di↵erential equation, known today as the Schrodinger equation. Eventually itcame to be understood that these two are both physically and mathematically equiv-alent (see [Cas]). Naively, this equivalence can be understood in the following way.Schrodinger’s wave function is most naturally regarded as an element of L2(M),where M is the configuration space of the classical mechanical system whose quan-tum counterpart is under consideration (for example, the classical 2-body problemif one is interested in the hydrogen atom). But an orthonormal basis for L2 givesrise to an isometric isomorphism onto l2 so anything one might want to say aboutcomplex-valued square integrable functions can equally well be said in terms of in-finite arrays of complex numbers. Indeed, all separable, infinite-dimensional, com-plex Hilbert spaces are isometrically isomorphic so, when von Neumann set himselfthe task of constructing a mathematically rigorous, abstract setting for quantum the-ory, it was more natural to formulate it in terms of an arbitrary such Hilbert spaceH. For any particular quantum system the choice of H then became a matter ofconvenience and clarity.

The reason for assuming that the states are represented by unit vectors in H ismore subtle and will be addressed more thoroughly in Postulate QM3. Briefly, thesituation is as follows. In its early years (1925-1927) quantum physics was in arather odd position. Heisenberg formulated his matrix mechanics without knowingwhat a matrix is and without having a precise idea of what the entries in his rectan-gular arrays of complex numbers should mean physically (see Section 7.1 of [Nab5]for more on this). Nevertheless, the rules of the game as he laid them down predictedprecisely the spectrum of the hydrogen atom. Schrodinger formulated a di↵erentialequation for his wave function that yielded the same predictions, but no one had anyreal idea what the wave function was supposed to represent (Schrodinger himselfinitially viewed it as a sort of “charge distribution”). It was left to Max Born, and


then Niels Bohr and his school in Copenhagen, to supply the missing conceptualbasis for quantum mechanics.

Remark A.4.1. It is our good fortune that Born himself, in his Nobel Prize Lecturein 1954, has provided us with a brief and very lucid account of the evolution of hisidea and we will simply refer those interested in pursuing this to [Born1]. Interest-ingly, Born attributes to Einstein the inspiration for the idea, although Einstein neveracquiesced to its implications.

As we will see, Born postulated that Schrodinger’s wave function should be inter-preted as a probability amplitude. In particular, for a single particle moving alonga line, | (q)|2 is the probability density function for the particle’s position, that is,for any Borel set S in R,

RS | (q)|2dq is the probability that a measurement of the

particle’s position will result in a value in S . Since the particle is bound to be foundsomewhere in R,

RR| (q)|2dq = 1 so is a unit vector in L2(R). All of this will be

spelled out in more detail quite soon.Finally, we point out that, because unit vectors in H that di↵er only by a phase

factor describe the same state, one can identify the state space of a quantum sys-tem with the projectivization P(H) of H, that is, the quotient of the unit spherein H by the equivalence relation that identifies two points if they di↵er by a com-plex factor of modulus one. The mathematical structure of P(H) is spelled out inmore detail in Section 1.3, but for the moment we need only observe that, with thequotient topology, P(H) is a Hausdor↵ topological space. We will write for theequivalence class containing and refer to it as a unit ray in H. It is sometimes alsoconvenient to identify the state represented by the unit vector with the operator P

that projects H onto the 1-dimensional subspace of H spanned by (which clearlydepends only on the state and not on the unit vector representing it). In all candor,however, it is customary to be somewhat loose with the terminology and speak of“the state ” when one really means “the state ”, or “the state P ”. Since this isalmost always harmless, we will generally adhere to the custom.

Postulate QM2

For a quantum system with Hilbert space H, every observable is identified witha (generally unbounded) self-adjoint operator A : D(A) ! H on H and any pos-sible outcome of a measurement of the observable is a real number that lies in thespectrum �(A) of A.

Classically, one thinks of an observable associated with a physical system assomething specified by a real number that one can measure. For a single particlemoving in space one might think of a coordinate of the particle’s position, a com-ponent of its momentum, or its total energy. It would be more accurate, however,to fully identify an observable with a specific measurement procedure not only be-cause such things as position, momentum, and energy are defined operationally in


physics by specifying how they are to be measured, but also because familiar clas-sical concepts such as these do not always transition well into the quantum realm.The problem of measurement in quantum mechanics is very subtle and, as we willsee in Postulate QM3, quantum theory makes no predictions whatsoever regardingthe outcome of any single measurement even when the state of the system beingmeasured is known with certainty. Repeated measurements on identical systems inthe same state need not give the same result. The results of these measurementsare constrained, however, by Postulate QM2 because every quantum observable hasa specific set of possible measured values, that is, �(A). It turns out, for example,that a quantum harmonic oscillator must have a total energy that lies in a countable,discrete set of real numbers tending to infinity (see Section 7.4 of [Nab5]).

In the model we are in the process of constructing we require a functional ana-lytic object associated with the quantum Hilbert space H that encodes the essentialphysical content of this notion of a quantum observable. This essential content isthe observable’s set of possible measured values. Now recall that a closed, sym-metric operator A on H has a spectrum �(A) that consists entirely of real numbers.This suggests that one might try to identify a quantum observable such as the totalenergy with some appropriately chosen closed, symmetric operator whose spec-trum is equal to (or, at least, contains) the set of possible measured values. Such anidea has analogues in classical physics. In continuum mechanics, for example, thestress tensor of a 3-dimensional material body is a symmetric linear operator on R3

whose matrix in any Cartesian coordinate system is obtained from the stress com-ponents within the body in directions perpendicular to the coordinate planes andwhose eigenvalues are the principal stresses, that is, those that are independent ofthe coordinate system (see [Gurtin]).

Postulate QM2, however, specifies that an observable is represented by a self-adjoint operator whose spectrum contains the set of possible measured values. Now,every self-adjoint operator is closed and symmetric, but the converse is not true sothis is a strictly stronger requirement. The rationale behind this is largely mathe-matical rather than physical. The formalism of quantum mechanics depends cru-cially on two of the pillars of functional analysis, namely, the Spectral Theorem andStone’s Theorem, and both of these require self-adjointness (Section 5.5 of [Nab5]or Chapter IX, Section 9, and Chapter XI, Section 6, of [Yos]). Notice that if the setof possible measured values is unbounded (as it is for the harmonic oscillator, forexample), then the spectrum must be an unbounded subset of R and therefore theoperator itself must be unbounded.

Postulate QM3

Let H be the Hilbert space of a quantum system, 2 H a unit vector representinga state of the system and A : D(A) ! H a self-adjoint operator on H representingan observable. Let EA be the unique projection-valued measure on R associatedwith A by the Spectral Theorem and let {EA

� }�2R be the corresponding resolution ofthe identity. Denote by µ ,A the probability measure onR that assigns to every Borel


set S ✓ R the probability that a measurement of A when the state is will yield avalue in S . Then, for every Borel set S in R,

µ ,A(S ) = h , EA(S ) i = kEA(S ) k2.

If the state is in the domain of A, then the expected value of A in state is

hAi =Z

R

� dh , EA� i = h , A i

and its dispersion (variance) is

�2 (A) =

Z

R

(� � hAi )2 dh , EA� i = k (A � hAi ) k2 = kA k2 � hAi2 .

We should first be clear on how we will interpret the sort of probabilistic state-ment made in Postulate QM3. Given a quantum system in state and an observableA, quantum mechanics generally makes no predictions about the result of a singlemeasurement of A. Rather, it is assumed that the state can be replicated over andover again and the measurement performed many times on these identical systems.The probability of a given outcome might then be thought of intuitively as the rel-ative frequency of the outcome for a “large” number of repetitions. More precisely,the probability of a given outcome for a measurement of some observable in somestate is the limit of the relative frequencies of this outcome as the number of repeti-tions of the measurement in this state approaches infinity. Given this interpretationthe formulas for the expected value and the dispersion are just those borrowed fromprobability theory. In particular, the dispersion �2

(A) is zero precisely when a mea-surement of A in state will result in the expected value hAi with probability 1;this, of course, does not mean that every measurement of A in state will result inhAi . It follows from the Spectral Theorem that �2

(A) = 0 if and only if is aneigenvector of A with eigenvalue hAi (Section 6.2 of [Nab5]). As we mentionedearlier, this probabilistic interpretation of quantum mechanics is due to Max Bornand one can consult [Born1] to learn how the interpretation evolved in the early daysof quantum mechanics..

The assertion of Postulate QM3 that µ ,A(S ), which is defined physically, is givenby µ ,A(S ) = h , EA(S ) i is called the Born-von Neumann formula and is the linkbetween the mathematical formalism and the physics. It’s either true, or it’s not and,in principle, this can be decided in the laboratory. Choose a self-adjoint operatorA to represent the physical observable you have in mind and compute its spectralresolution. Then make many measurements of your physical observable in somefixed state to determine µ ,A. Now compare the observations µ ,A(S ) with thecomputations h , EA(S ) i. If they are the same, all is well; if not, you either madethe wrong choice for the operator A or the Born-von Neumann formula is wrong.The evidence would suggest that the formula is correct. To gain some appreciationof how all of this works in practice one might look at the concrete calculations of


µ ,A, hAi , and �2 (A) for various operators and states in Examples 6.2.2 - 6.2.7 of

[Nab5].Before moving on we would like to record a general result on dispersions of self-

adjoint operators that is related to the uncertainty relations of quantum mechanics.The following is Lemma 6.3.1 of [Nab5].

Lemma A.4.1. Let H be a separable, complex Hilbert space, A : D(A) ! H andB : D(B) ! H self-adjoint operators on H, and ↵ and � real numbers. Then, forevery 2 D([A, B]) = D(AB) \D(BA),

k (A � ↵) k2 k (B � �) k2 � 14

�� h , [A, B] i��2.

Now, if A and B represent observables and 2 D([A, B]) is a unit vector repre-senting a state and if we take ↵ and � to be the corresponding expected values ofA and B in this state, then it is shown in Section 6.3 of [Nab5] that Lemma A.4.1implies

�2 (A)�2

(B) � 14

�� h , [A, B] i��2.

This is called the Robertson Uncertainty Relation. It is more commonly expressedin terms of positive square roots � (A) and � (B) of the dispersions, that is, in termsof the standard deviations.

� (A)� (B) � 12

�� h , [A, B] i��.

In Section 6.3 of [Nab5] this is applied to the operators representing the position Qand momentum P of a single particle moving along a line to obtain

� (Q)� (P) � ~2, (A.10)

where ~ = h2⇡ is the reduced Planck constant. This is obviously a statistical statement

about position and momentum measurements for such a particle. It is quite common,and totally incorrect, to see it identified with the famous Heisenberg UncertaintyPrinciple which, as Heisenberg phrased it, states that

�q�p � ~2, (A.11)

where �q and �p are the “uncertainty”, or “inaccuracy” in the simultaneous mea-surements of position and momentum for a single particle. Section 6.3 of [Nab5]discusses in some detail the physical origin of Heisenberg’s Uncertainty Principle,its current experimental status and the fact that it has nothing whatsoever to do withthe inequality (A.10).


We will conclude our remarks on Postulate QM3 with a particularly significantspecial case. We let denote a unit vector in H representing some state. Now let �be another unit vector in H representing another state. Then | h , �i |2 is interpretedas the probability of finding the system in state � if it is known to be in state beforea measurement to determine the state is made; it is called the transition probabilityfrom state to state �. The complex number h , �i is called the transition amplitudefrom to �.

Postulate QM4

Let H be the Hilbert space of an isolated quantum system. Then there exists astrongly continuous 1-parameter group {Ut}t2R of unitary operators on H, calledevolution operators, with the property that, if the state of the system at time t = 0 is 0 = (0), then the state at time t is given by

t = (t) = Ut( 0) = Ut( (0))

for every t 2 R. By Stone’s Theorem (Theorem 5.5.10 of [Nab5]) there is a uniqueself-adjoint operator H : D(H) ! H, called the Hamiltonian of the system, suchthat Ut = e�itH/~, where ~ = h

2⇡ is the reduced Planck constant. Therefore

t = (t) = e�itH/~( 0) = e�itH/~( (0)).

Physically, the Hamiltonian H is identified with the operator representing the totalenergy of the quantum system.

We will view a quantum system as isolated if, as in the classical case, no matter orenergy can enter or leave the system and if, in addition, no measurements are madeon the system. One should be aware, however, that it is not at all clear that suchthings exist, nor is it clear precisely what we mean by a measurement . Rather thantry to define what a measurement is we will adopt Postulate QM4 as a definition ofwhat it means to say that measurements are not being performed on the system. Weshould point out also that the ~ is introduced here simply to keep the units consistentwith the interpretation of H as an energy (Remark 6.2.11 of [Nab5]).

The 1-parameter group {Ut}t2R of unitary operators is the analogue for an isolatedquantum system of the classical 1-parameter group {�t}t2R of di↵eomorphisms de-scribing the flow of the Hamiltonian vector field in classical mechanics; both “push”an initial state of the system onto the state at some future time. That the appro-priate analogue of the di↵eomorphism �t is a unitary operator Ut deserves somecomment. On the surface, the motivation seems clear. A state of the system is rep-resented by a 2 H with h , i = 1 so the same must be true of the evolvedstates (t), that is, we must have h (t), (t)i = 1 for all t 2 R. Certainly, this willbe the case if (t) is obtained from (0) by applying a unitary operator U sincehU( (0)),U( (0)) i = h (0), (0) i = 1. Notice, however, that this is also true if U


is anti-unitary since then hU( (0)),U( (0)) i = h (0), (0) i = 1. Physically, onewould probably also wish to assume that the time evolution preserves all transitionprobabilities | h , �i |2, but this is also the case for both unitary and anti-unitary op-erators. Since unitary and anti-unitary operators di↵er only by a factor of ±i, onemight be tempted to conclude that one choice is as good as the other. Physically,however, matters are not quite so simple (see [Wig3]). Furthermore, it is not so clearthat there might not be other possibilities as well, that is, maps of H onto H thatpreserve transition probabilities but do not arise from unitary or anti-unitary opera-tors.

That, in fact, there are no other possibilities is a consequence of a highly nontriv-ial result of Wigner ([Wig2]) that we described in Section 1.3. For convenience, wewill repeat the description here our current notation. We identify the state space ofour quantum system with the projectivization P(H) of H. For any ,� 2 P(H) wedefine the transition probability ( ,�) from state to state � by

( ,�) =|h , �i|2k k2k�k2

for any 2 and any � 2 �. Wigner defined a symmetry of the quantum systemwhose Hilbert space is H to be a bijection T : P(H) ! P(H) that preserves tran-sition probabilities in the sense that (T ( ),T (�)) = ( ,�) for all ,� 2 P(H).Notice that, although P(H) has a natural quotient topology, no continuity assump-tions are made. Any unitary or anti-unitary operator U on H induces a symmetryTU that carries any representative of to the representative U( ) of TU( ). WhatWigner proved was that every symmetry is induced in this way by a unitary or anti-unitary operator. The result is, in fact, a bit more general. There is a detailed proofof the following result in [Barg].

Theorem A.4.2. (Wigner’s Theorem on Symmetries) Let H1 and H2 be complex,separable Hilbert spaces and T : P(H1)! P(H2) a mapping ofP(H1) intoP(H2)satisfying

(T ( ),T (�))P(H2) = ( ,�)P(H1)

for all ,� 2 P(H1). Then there exists a mapping U : H1 ! H2 satisfying

1. U( ) 2 T ( ) if 2 2 P(H1),

2. U( + �) = U( ) + U(�) for all , � 2 H1, and

3. either

a. U(� ) = �U( ) and hU( ),U(�)iH2 = h , �iH1 , or

b. U(� ) = �U( ) and hU( ),U(�)iH2 = h , �iH1

for all � 2 C and all , � 2 H1.


Furthermore, if H1 and H2 are of dimension at least 2 and T ([ 1]) = T ([ 2]), where 1 and 2 are unit vectors in H1, then there is a unique such mapping U : H1 ! H2for which U( 1) = U( 2).

In particular, any symmetry of a quantum system with Hilbert space H arises froman operator on H that is either unitary or anti-unitary. The anti-unitary operatorscorrespond physically to discrete symmetries such as time reversal (they are theanalogue of reflections in Euclidean geometry).

Now we can justify the unitarity assumption in Postulate QM4. Suppose thatthe time evolution is described by an assignment to each t 2 R of a symmetry↵t : P(H)! P(H) and that t ! ↵t satisfies ↵t+s = ↵t �↵s for all t, s 2 R. Then, forany t 2 R, ↵t = ↵2

t/2. By Wigner’s Theorem, ↵t/2 is represented by an operator Ut/2that is either unitary or anti-unitary. Since the square of an operator that is eitherunitary or anti-unitary is necessarily unitary, every Ut must be unitary.

Remark A.4.2. We investigate the consequences of Wigner’s Theorem more thor-oughly in Section 1.3, but a few remarks here would seem to be in order. A bijectionT : P(H) ! P(H) preserving transition probabilities that is a homeomorphismwith respect to the quotient topology is called an automorphism of P(H) and thecollection of all such is clearly a group under composition. This is called the auto-morphism group of P(H) and is denoted Aut(P(H)). Wigner’s Theorem will permitus, in Section 1.3, to provide Aut(P(H)) with a natural topology with respect towhich it is a Hausdor↵ topological group. Now suppose that G is a Lie group. Acontinuous homomorphism of G into Aut(P(H)) is called a projective representa-tion of G on H. If H is the Hilbert space of some quantum system, then such aprojective representation assigns a symmetry to each element of G in such a waythat the group operations are respected and this leads us to refer to G as a symmetrygroup of the quantum system. When G is the Poincare group P"+, the existence ofsuch a projective representation is the natural expression of the “relativistic invari-ance” of the quantum system.

Notice that there is nothing special about t = 0 in Postulate QM4. If t0 is anyreal number, then (t0) = Ut0 ( (0)) so (t + t0) = Ut+t0 ( (0)) = Ut(Ut0 ( (0))) =Ut( (t0)) and therefore

(t) = ((t � t0) + t0) = Ut�t0 ( (t0)) = e�i(t�t0)H/~( (t0)).

Thus,

Ut�t0 = e�i(t�t0)H/~

propagates the state at time t0 to the state at time t for any t0, t 2 R. It follows fromthis that if (t) is thought of as a curve in H and (t0) is in the domain of H, then,by Stone’s Theorem, (t) is in the domain of H for all t 2 R and


i~d (t)

dt= H( (t)), (A.12)

where the derivative is defined by

d (t)dt= lim

t!t0

(t) � (t0)t � t0

(A.13)

and the limit is in H. Equation (A.12) is called the abstract Schrodinger equation.This, however, is not the way one generally sees the Schrodinger equation written

in the physics literature. When H is, for example, L2(Rn), then each state of thesystem is represented by a complex-valued, square integrable wave function (q)on Rn with L2-norm 1. In this case the Hamiltonian H is generally a di↵erentialoperator of the form

H = H0 + V(q) = � ~2

2m� + V(q), (A.14)

where V(q) is a real-valued function on Rn, called the potential, which acts onL2(Rn) as a multiplication operator, m is a positive constant, and � is the distri-butional Laplacian on L2(Rn).

Remark A.4.3. The operators H0 and V are both self-adjoint on L2(Rn), but gener-ally their sum is not. V must satisfy conditions that are su�cient to ensure that theoperator H is self-adjoint on L2(Rn) and finding such conditions is not at all trivial.There is a discussion of some results of this sort in Section 8.4.2 of [Nab5].

The time evolution of the wave function is then generally written (q, t) rather than t(q) or ( (t))(q) and the Schrodinger equation is written

i~@ (q, t)@t

= � ~2

2m� (q, t) + V(q) (q, t). (A.15)

The point we would like to stress, however, is that writing the Schrodinger equa-tion in the form (A.15) involves a fair amount of potentially hazardous notationalsubterfuge. The abstract Schrodinger equation (A.12) contains in it the t-derivatived (t)/dt which is defined as the limit in L2(Rn) of the familiar di↵erence quotient(see (A.13)). The partial derivative @ (q, t)/@t that appears in the traditional physi-cist’s form (A.15) of the Schrodinger equation is, on the other hand, defined as alimit in C of an equally familiar di↵erence quotient. In general, there is no reason tosuppose that the latter exists for 2 L2(Rn) and, even if it does, that it is in L2(Rn)for each t and, granting even this, that it is equal to d (t)/dt. Su�cient regularityassumptions on (q, t) will guarantee that all of these things are true (Exercise 6.2.1of [Nab5]), but even these may not be enough to ensure that, for each t, the distri-bution � is actually a function in L2(Rn) and, if this is not the case, we have nobusiness equating it in (A.15) to something that is a function in L2(Rn). All of thesedi�culties can be willed away by restricting attention to functions (q, t) that are


su�ciently regular, say, continuously di↵erentiable with respect to t and, for each t,a Schwartz function or smooth function with compact support onRn. The hope thenis that one can find su�ciently many such classical solutions to (A.15) to providean orthonormal basis for L2(Rn) for each t. Example 5.3.1 of [Nab5] shows that, atleast in the case of the harmonic oscillator, one’s hopes are not dashed.

We will conclude this section with a brief synopsis the salient features of thepicture of quantum mechanics we have painted thus far and then suggest a ratherdi↵erent way of looking at it that will bear a striking resemblance to the picture ofclassical Hamiltonian mechanics in Section A.3. A quantum system has associatedwith it a complex, separable Hilbert space H and a distinguished self-adjoint opera-tor H, called the Hamiltonian of the system. The states of the system are representedby unit vectors in H and these evolve in time from an initial state (0) accord-ing to (t) = Ut( (0)) = e�itH/~( (0)). As a result, the evolving states satisfy theabstract Schrodinger equation

i~d (t)

dt= H( (t)). (A.16)

Each observable is identified with a self-adjoint operator A that does not changewith time. Neither the state vectors nor the observables A are accessible to directexperimental measurement. Rather, the link between the formalism and the physicsis contained in the expectation values hAi = h , A i. Knowing these one can con-struct the probability measures µ ,A(S ) = h , EA(S ) i and these contain all of theinformation that quantum mechanics permits us to know about the system.

We would now like to look at this from a slightly di↵erent point of view. As thestate evolves so do the expectation values of any observable. Specifically,

hAi (t) = h (t), A (t)i = hUt( (0)), AUt( (0))i = h (0), [U�1t AUt] (0)i,

because each Ut is unitary. Now, define a (necessarily self-adjoint) operator

A(t) = U�1t AUt

for each t 2 R. Then

hAi (t) = hA(t)i (0)

for each t 2 R. The expectation value of A in the evolved state (t) is the sameas the expectation value of the observable A(t) in the initial state (0). Since all ofthe physics is contained in the expectation values this presents us with the option ofregarding the states as fixed and the observables as evolving in time. From this pointof view our quantum system has a fixed state and the observables evolve in timefrom some initial self-adjoint operator A = A(0) according to

A(t) = U�1t AUt = eitH/~Ae�itH/~. (A.17)


This is called the Heisenberg picture of quantum mechanics to distinguish it fromthe view we have taken up to this point, which is called the Schrodinger picture.Although these two points of view appear to di↵er from each other rather trivially,the Heisenberg picture occasionally presents some significant advantages and wewill now spend a moment seeing what things look like in this picture.

We should first notice that, when A is the Hamiltonian H itself, Stone’s Theoremimplies that each Ut leaves D(H) invariant and commutes with H so

H(t) = U�1t HUt = eitH/~He�itH/~ = H 8t 2 R.

The Hamiltonian is constant in time in the Heisenberg picture. For other observ-ables this is generally not the case, of course, and one would like to have a di↵er-ential equation describing their time evolution in the same way that the Schrodingerequation describes the time evolution of the states in the Schrodinger picture. Wewill describe such an equation in the case of observables represented by boundedself-adjoint operators in the Schrodinger picture.

Remark A.4.4. This is a very special case, of course, so we should explain therestriction. In the unbounded case, a rigorous derivation of the equation is substan-tially complicated by the fact that, in the Heisenberg picture, the operators (andtherefore their domains), are varying with t so that the usual domain issues for un-bounded operators also vary with t. Physicists have the good sense to ignore allof these issues and just formally di↵erentiate (A.17), thereby arriving at the verysame equation that appears in our theorem below. Furthermore, it is not hard toshow that, if A is unbounded and A(t) = U�1

t AUt, then, for any Borel function f ,f (A(t)) = f (A)(t) = U�1

t f (A)Ut so that one can generally study the time evolu-tion of A in terms of the time evolution of the bounded functions of A and these arebounded operators. In particular we recall that, from the point of view of physics, allof the relevant information is contained in the probability measures h , EA(S ) i sothat, in principle, one requires only the time evolution of the (bounded) projectionsEA(S ).

The following is Theorem 6.4.1 of [Nab5].

Theorem A.4.3. Let H be a complex, separable Hilbert space, H : D(H) ! H aself-adjoint operator on H and Ut = e�itH/~, t 2 R, the 1-parameter group of unitaryoperators determined by H. Let A : H ! H be a bounded, self-adjoint operator onH and define, for each t 2 R, A(t) = U�1

t AUt. If and A(t) are in D(H) for everyt 2 R, then A(t) satisfies the Heisenberg equation

dA(t)dt

= � i~

⇥A(t),H

⇤ , (A.18)

where the derivative is the H-limit


dA(t)dt

= lim�t!0

A(t + �t) � A(t)�t

�

and [A(t),H] is the commutator of A(t) and H on D(H) , that is, [A(t),H] =A(t)(H ) � H(A(t) ) for every 2 D(H).

Let’s simplify the notation a bit and write (A.18) as

dAdt= � i~

⇥A,H

⇤. (A.19)

Now compare this with the equation (A.9)

d fdt= { f ,H}

describing the time evolution of a classical observable in the Hamiltonian picture ofmechanics. The analogy is striking and suggested to Paul Dirac [Dirac1] a possibleavenue from classical to quantum mechanics, that is, a possible approach to thequantization of classical mechanical systems. The idea is that classical observablesshould be replaced by self-adjoint operators and the Poisson bracket { , } by thequantum bracket

�, ~

= � i~

⇥,⇤.

Let’s spell this out in a bit more detail. Dirac’s suggestion was to find a linear mapfrom the classical observables f , g, . . . to the quantum observables F,G, . . . with theproperty that

{ f , g} 7! {F,G}~

= � i~

[F,G].

If one further stipulates that the constant function 1 should map to the identity op-erator I then this implies that the classical canonical commutation relations (A.8)

{qi, q j} = {pi, p j} = 0 and {qi, p j} = �ij, i, j = 1, . . . n (A.20)

map to the quantum canonical commutation relations

{Qi,Qj}~

= {Pi, Pj}~ = 0 and {Qi, Pj}~ = �ijI, i, j = 1, . . . n, (A.21)

where Qi and Pi are the images of qi and pi, respectively, for i = 1, . . . , n. Writtenin terms of commutators, (A.21) becomes

[Qi,Qj] = [Pi, Pj] = 0 and [Qi, Pj] = i~�ijI, i, j = 1, . . . n. (A.22)

Presumably the image under this map of the classical Hamiltonian would be theappropriate quantum Hamiltonian and with this in hand the analysis of the quan-


tum system could commence. The extent to which Dirac’s program can actually becarried out is discussed in some detail in Section 7.2 of [Nab5].

Example A.4.1. We will conclude with a brief synopsis of the quantum systemdescribing a single particle of mass m moving in R3 under the influence of a time-independent potential V(q) = V(q1, q2, q3). The corresponding problem for motionin R is treated in considerable detail in [Nab5] and we will provide references forthose results whose extension from one to three spatial dimensions is not routine.

Remark A.4.5. We should point out that, in this example, we will be ignoring animportant quantum mechanical property of elementary particles called spin. We willtake up this subject and see what modifications of our present discussion are requiredin Section A.5.

The Hilbert space H of our quantum system is taken to be L2(R3), that is, theHilbert space of (equivalence classes of) complex-valued, square integrable func-tions on R3 with respect to Lebesgue measure. The dynamics of the system isgoverned, via the Schrodinger equation, by the Hamiltonian H which must be aself-adjoint operator on L2(R3) representing the total energy of the system. The to-tal energy is the sum of the kinetic energy and the potential energy so H will bethe sum of two self-adjoint operators. The kinetic term is always the same and isdefined in the following way. Begin by looking at the subspace C10 (R3) of L2(R3)consisting of smooth, complex-valued functions with compact support onR3 (or theSchwartz space S(R3) of rapidly decreasing complex-valued functions on R3). Onthis subspace the ordinary Laplacian � is defined and essentially self-adjoint (seepages 395-396 of [Nab5]) so the same is true of

H0 = �~

2

2m�.

The minus sign ensures that H0 is a positive operator, that is,

hH0 , i � 0

for all 2 C10 (R3) (or S(R3)). The unique self-adjoint extension is denoted withthe same symbols and its domain is

D(H0) = { 2 L2(R3) : � 2 L2(R3)},

where � now means the distributional Laplacian defined by taking Fourier trans-forms (see Sections 8.4.1 and 8.4.2 of [Nab5]). H0 is the kinetic energy term in ourHamiltonian.

Remark A.4.6. The motivation for adopting H0 as the kinetic energy operator isthat it is the canonical quantization of the classical kinetic energy (see Chapter 7 of[Nab5]).


The potential V is assumed to be a real-valued measurable function on R3 and,as an operator on L2(R3), it acts by multiplication ((V )(q) = V(q) (q)). It is de-fined and essentially self-adjoint on C10 (R3) (or S(R3)) and its unique self-adjointextension, still denoted V , has domain

D(V) = { 2 L2(R3) : V 2 L2(R3)}.

This is the potential energy term in the Hamiltonian. One would now like to definethe Hamiltonian H by H = H0+V . However, H0 and V are both unbounded operatorson L2(R3) and there is no a priori reason to suppose that their domains intersect inanything more than 0 2 L2(R3). Moreover, even if the intersection of their domainshappens to be a dense linear subspace, it is not true that the sum of two unbounded,self-adjoint operators is self-adjoint. In order to guarantee self-adjointness one mustimpose additional restrictions on V . Many such conditions are known and some ofthese are described in Section 8.4.2 of [Nab5]. We will state only the one that isactually proved in [Nab5] and refer those interested in seeing more to Chapter X of[RS2].

Theorem A.4.4. Let V be a real-valued, measurable function on R3 that can bewritten as V = V1 + V2, where V1 2 L2(R3) and V2 2 L1(R3). Then

H = H0 + V = � ~2

2m� + V

is essentially self-adjoint on C10 (R3) and self-adjoint on D(H0).

With the Hilbert space L2(R3) and a self-adjoint Hamiltonian H in hand onenext introduces self-adjoint operators that will act as the observables of the system.Initially, at least, one would look for observables that can be regarded as quantumanalogues of the appropriate observables for the corresponding classical system. Inthe case at hand (a particle of mass m moving in R3) these would include the posi-tion, energy, momentum and angular momentum. The energy, of course, is just theHamiltonian H. Choosing appropriate operators to represent position, momentum,angular momentum or any other observable of interest is called quantization andthis is not a process that admits a simple algorithmic synopsis. Those interested ina brief look behind the scenes at what is involved may want to refer to Chapter 7 of[Nab5]. Here we will simply record the operators of interest to us at the moment.

We fix an orthonormal basis forR3 and denote the corresponding coordinates byq1, q2 and q3. For each j = 1, 2, 3 we define the jth-coordinate position operator Qj

on L2(R3) by

(Qj )(q) = (Qj )(q1, q2, q3) = q j (q1, q2, q3) (A.23)

(motivation for the corresponding definition in one spatial dimension is provided inRemark 6.2.8 of [Nab5]). Then Qj is defined and self-adjoint on


D(Qj) = { 2 L2(R3) : q j (q1, q2, q3) 2 L2(R3)}. (A.24)

Example 5.2.6 of [Nab5] gives two proofs of the corresponding statement for motionin one spatial dimension that can easily be adapted to the 3-dimensional context inwhich we currently find ourselves; alternatively, Section 4.5, Chapter III, of [Prug]provides a di↵erent argument that proves self-adjointness in any number of spatialdimensions.

Next we would like to define, for each j = 1, 2, 3, the jth-coordinate momentumoperator Pj on L2(R3). One begins by showing that the operator

Pj = �i~@

@q j (A.25)

is defined and essentially self-adjoint on C10 (R3) (or S(R3)) and taking the momen-tum operator, denoted with the same symbol, to be its unique self-adjoint extension(motivation for the corresponding definition in one spatial dimension is provided inRemark 6.2.15 of [Nab5]). The domain D(Pj) is best viewed in the following way.The Fourier transform F : L2(R3)! L2(R3) is a unitary operator on L2(R3). For in C10 (R3) (or S(R3)) one has Pj = (F�1(~Qj)F)�. Consequently,

Pj = F�1(~Qj)F (A.26)

on C10 (R3) (or S(R3)). Since each of these is dense in L2(R3), (A.26) is true every-where on L2(R3). In particular, Pj is unitarily equivalent to Qj and the domain ofPj is

D(Pj) = F�1(D(Qj)). (A.27)

One often sees (A.26) taken as the definition of Pj in which case the self-adjointnessof Pj follows from its unitary equivalence to Qj (Lemma 5.2.5 of [Nab5]).

Finally, we would like to write out the operators on L2(R3) that represent thequantum analogues of the classical components of orbital angular momentum. Themotivation is to be found in the infinitesimal generators (A.3), (A.4) and (A.5) ofclassical angular momentum. Specifically, we begin by defining operators M23,M31

and M12 on C10 (R3) (or S(R3)) by

M23 = Q2P3 � Q3P2 = i ~✓

q3 @

@q2 � q2 @

@q3

◆(A.28)

M31 = Q3P1 � Q1P3 = i ~✓

q1 @

@q3 � q3 @

@q1

◆(A.29)

M12 = Q1P2 � Q2P1 = i ~✓

q2 @

@q1 � q1 @

@q2

◆(A.30)


where Pj = ⌘ jkPk = �Pj = i~ @@q j for j = 1, 2, 3.

Remark A.4.7. See Example 2.5.1 and what follows for more on these operators.Note, however, that there we have taken ~ = 1.

The operators M23,M31 and M12 are essentially self-adjoint on C10 (R3) (orS(R3)) and their unique self-adjoint extensions, denoted with the same symbols, arethe components of the orbital angular momentum operators in quantum mechanics.

Now recall that the Fourier transform F on Rn satisfies each of the following forany 2 S(Rn) and any j = 1, . . . , n.

F✓ @

@q j ◆(p) = i p jF( )(p)

F� � i q j

�(p) =

@

@p jF( )(p)

and is a unitary operator of L2(Rn, dnq) onto L2(Rn, dn p). Any operator A onL2(Rn, dnq) can therefore be regarded as an operator on L2(Rn, dn p), specifically,the operator FAF�1.

Exercise A.4.1. Write the Fourier transform F( ) of as and prove each of thefollowing.

1. [(FQjF�1) ](p) = i @@p j (p), j = 1, 2, 3

2. [(FPkF�1) ](p) = ~pk (p), k = 1, 2, 3

3. [(FQjF�1)(FPkF�1) ](p) = �i~pk

@@p j (p), j, k = 1, 2, 3, j , k

Exercise A.4.2. Show that, as operators on momentum space L2(R3, d3 p), the op-erators M23,M31, and M12 take the same form as on L2(R3, d3q), that is,

M23 = i ~✓

p3@

@p2� p2

@

@p3

◆, (A.31)

M31 = i ~✓

p1@

@p3� p3

@

@p1

◆, (A.32)

M12 = i ~✓

p2@

@p1� p1

@

@p2

◆. (A.33)

As we mentioned in Remark A.4.5 the discussion of particle motion in quantummechanics in Example A.4.1 took no account of the quantum mechanical propertyof spin; more precisely, the discussion is valid only for particles of spin zero andwe do not yet know what this means. In the next section we will try to remedy this(more information is available in Chapter 9 of [Nab5]).


A.5 Spin

We will try to provide some sense of what this phenomenon of spin is and howthe physicists have incorporated it into their mathematical model of the quantumworld. There is nothing like quantum mechanical spin in classical physics. Thereis, however, a classical analogy. The analogy is inadequate and can be misleading iftaken too seriously, but it is the best we can do so we will begin by briefly describingit.

Imagine a spherical mass m of radius a moving through space on a circular orbitof radius R � a about some point O and, at the same time, spinning around anaxis through one of its diameters (to a reasonable approximation, the Earth doesall of this). Due to its orbital motion, the mass has an angular momentum L =r ⇥ (mv) = r ⇥ p, where r is the position vector from O and v = r is the velocity,which we call its orbital angular momentum.. The spinning of the mass around itsaxis contributes additional angular momentum that one calculates by subdividingthe spherical region occupied by the mass into subregions, regarding each subregionas a mass in a circular orbit about a point on the axis, approximating its angularmomentum, adding all of these and taking the limit as the regions shrink to points.The resulting integral gives the angular momentum due to rotation. This is calledthe rotational angular momentum, is denoted S, and is given by

S = I!,

where I is the moment of inertia of the sphere and ! is the angular velocity (! isalong the axis of rotation in the direction determined by the right-hand rule fromthe direction of the rotation). If the mass is assumed to be uniformly distributedthroughout the sphere (in other words, if the sphere has constant density), then anexercise in calculus gives

S =25

ma2!.

The total angular momentum of the sphere is L + S.Now let’s suppose, in addition, that the sphere is charged. Due to its orbital mo-

tion the charged sphere behaves like a current loop and any moving charge gives riseto a magnetic field. If we assume that our current loop is very small (or, equivalently,that we are viewing it from a great distance) the corresponding magnetic field is thatof a magnetic dipole (see Sections 14-5 and 34-2, Volume II, of [FLS]). All we needto know about this is that this magnetic dipole is described by a vector µL calledits orbital magnetic moment that is proportional to the orbital angular momentum.Specifically,

µL =q

2mL,

where q is the charge of the sphere (which can be positive or negative). Similarly,the rotational angular momentum of the charge gives rise to a magnetic field that is

A.5 Spin 165

also that of a magnetic dipole and is described by a rotational magnetic moment µSgiven by

µS =q

2mS.

The total magnetic moment µ is

µ = µL + µS =q

2m(L + S).

The significance of the magnetic moment µ of the dipole is that it describes thestrength and direction of the dipole field and determines the torque

⌧ = µ ⇥ B

experienced by the magnetic dipole when placed in an external magnetic field B. Ifthe magnetic field B is uniform (that is, constant), then its only e↵ect on the dipoleis to force µ to precess around a cone whose axis is along B in the same way thatthe axis of a spinning top precesses around the direction of the Earth’s gravitationalfield (see Figure A.1 and Section 2, Chapter 11, of [Eis]). Notice that this precessiondoes not change the projection µ · B of µ along B.

Fig. A.1 Precession

If the B-field is not uniform, however, there will be an additional translationalforce acting on the mass which, if m is moving through the field, will push it o↵ thecourse it would have followed if B had been uniform. Precisely what this deflectionwill be depends, of course, on the nature of B and we will say a bit more about thisin a moment.

Now we will describe the famous Stern-Gerlach experiment (a schematic ofwhich is shown in Figure A.2). We are interested in whether or not the electronhas a rotational magnetic moment and, if so, whether or not its behavior is ade-


quately described by classical physics. What we will do is send a certain beam ofelectrically neutral atoms through a non-uniform magnetic field B and then let themhit a photographic plate to record how their paths were deflected by the field. Theatoms must be electrically neutral so that the deflections due to the charge do notmask any deflections due to magnetic moments of the atoms. In particular, we can’tdo this with free electrons. The atoms must also have the property that any magneticmoment they might have could be due only to a single electron somewhere withinit. Stern and Gerlach chose atoms of silver (Ag) which they obtained by evaporatingthe metal in a furnace and focusing the resulting gas of Ag atoms into a beam aimedat a magnetic field.

Fig. A.2 Stern-Gerlach Experiment

Remark A.5.1. Silver is a good choice, but for reasons that are not so apparent. Aproper explanation requires some hindsight (not all of the information was avail-able to Stern and Gerlach) as well as some quantum mechanical properties of atomsthat we have not discussed here. Nevertheless, it is worth saying at least once sinceotherwise one is left with all sorts unanswered questions about the validity of theexperiment. So, here it is. The stable isotopes of Ag have 47 electrons, 47 protonsand either 60 or 62 neutrons so, in particular, they are electrically neutral. Since themagnetic moment is inversely proportional to the mass and since the mass of theproton and neutron are each approximately 2000 times the mass of the electron, onecan assume that any magnetic moments of the nucleons will have a negligible e↵ecton the magnetic moment of the atom and can therefore be ignored. Of the 47 elec-trons, 46 are contained in contained in closed, inner shells (energy levels) and these,it turns out, can be represented as a spherically symmetric cloud with no orbital or

A.5 Spin 167

rotational angular momentum (this is not at all obvious). The remaining electron isin what is termed the outer 5s-shell and an electron in an s-state has no orbital angu-lar momentum (again, not obvious). Granting all of this, the only possible source ofany magnetic moment for a Ag atom is a rotational angular momentum of its outer5s-electron. Whatever happens in the experiment is attributable to the electron andthe rest of the silver atom is just a package designed to ensure this.

We will first see what the classical picture of an electron with a rotational mag-netic moment would lead us to expect in the Stern-Gerlach experiment and will thendescribe the results that Stern and Gerlach actually obtained (a more thorough, butquite readable account of the physics is available Chapter 11 of [Eis]). For this wewill need to be more specific about the magnetic field B that we intend to send theAg atoms through. Let’s introduce a coordinate system in Figure A.2 in such a waythat the Ag atoms move in the direction of the y-axis and the vertical axis of sym-metry of the magnet is along the z-axis so that the x-axis is perpendicular to bothof these. The magnet itself can be designed to produce a field that is non-uniform,but does not vary with y, is predominantly in the z-direction, and is symmetric withrespect to the yz-plane. The interaction between the neutral Ag atom (with magneticmoment µ) and the non-uniform magnetic field B provides the atom with a potentialenergy �µ · B so that the atom experiences a force

F = r(µ · B) = r( µxBx + µyBy + µzBz ).

For the sort of magnetic field we have just described, By = 0 and Bz dominates Bx.From this one finds that the translational motion is governed primarily by

Fz ⇡ µz@Bz

@z

(see pages 333-334 of [Eis]). The conclusion we draw from this is that the displace-ments from the intended path of the silver atoms will be in the z-direction (up anddown in Figure A.2) and the forces causing these displacements are proportional tothe z-component of the magnetic moment. Of course, di↵erent orientations of themagnetic moment µ among the various Ag atoms will lead to di↵erent values of µzand therefore to di↵erent displacements. Moreover, due to the random thermal ef-fects of the furnace, one would expect that the silver atoms exit with their magneticmoments µ randomly oriented in space so that their z-components could take onany value in the interval [�|µ|, |µ| ]. As a result, the expectation based on classicalphysics would be that the deflected Ag atoms will impact the photographic plate atpoints that cover an entire vertical line segment (see the segment labeled “Classicalprediction” in Figure A.2).

Remark A.5.2. Writing q = �e for the charge of the electron and m = me for itsmass we find that


Fz ⇡ µz@Bz

@z= � e

2meS z

@Bz

@z

so that the deflection of an individual Ag atom is a measure of the component S z ofS in the direction of the magnetic field gradient.

This, however, is not at all what Stern and Gerlach observed. What they foundwas that the silver atoms arrived at the screen at only two points, one above andone the same distance below the y-axis (again, see Figure A.2). The experiment wasrepeated with di↵erent orientations of the magnet (that is, di↵erent choices for thez-axis) and di↵erent atoms and nothing changed. We seem to be dealing with a verypeculiar sort of “vector” S. The classical picture would have us believe that, howeverit is oriented in space, its projection onto any axis is always one of two things.Needless to say, ordinary vectors in R3 do not behave this way. What we are reallybeing told is that the classical picture is simply wrong. The property of electrons thatmanifests itself in the Stern-Gerlach experiment is in some ways analogous to whatone would expect classically of a small charged sphere rotating about some axis,but the analogy can only be taken so far. It is, for example, not possible to make anelectron “spin faster (or slower)” to alter the length of its projection onto an axis.This projection is always the same; it is a characteristic feature of the electron. Whatwe are dealing with is an intrinsic property of the electron, like its mass me, that doesnot depend on its motion (or anything else); for this reason it is often referred to asthe intrinsic angular momentum of the electron, but, unlike its classical counterpart,it is quantized, that is, can take only two discrete values.

Not only the electron, but every particle (elementary particle, atom, molecule,etc.) in quantum mechanics is supplied with some sort of intrinsic angular momen-tum. We will briefly describe the general situation (for more details see, for example,Chapter 11 of [Eis], or Chapters 14 and 17 of [Bohm]). The basic idea is that theseparticles exhibit behaviors that mimic what one would expect of angular momen-tum, but that cannot be accounted for by any orbital motion of the particle and, fora given particle, are always the same. To quantify these behaviors every particle isassigned a spin quantum number s. The allowed values of s are

0,12, 1,

32, 2,

52, . . . ,

n � 12, . . . ,

where n = 1, 2, 3, 4, 5, . . .. Intuitively, one might think of n as the number of dotsthat appear on the photographic plate if a beam of such particles is sent through aStern-Gerlach apparatus. According to this scheme an electron has spin 1

2 (n = 2).For a particle with spin 0 there is just one dot, that is, there is no deflection atall. Particles with half-integer spin 1

2 ,32 ,

52 , . . . are called fermions, while those with

integer spin 0, 1, 2, . . . are called bosons. Fermions and bosons have very di↵erentphysical characteristics and play very di↵erent roles in particle physics (for moreon this see Chapter 9 of [Nab5]). Among the elementary fermions, particles of spin12 are by far the principal players. Indeed, one must look long and hard to find anelementary particle of higher half-integer spin. The best know examples are the so-

A.5 Spin 169

called � baryons which have spin 32 , but you dare not blink if you’re looking for one

of these since their mean lifetime is about 5.63⇥ 10�24 seconds. Among the bosons,the very recently observed Higgs boson has spin 0, whereas the conjectured, but notyet observed graviton has spin 2. The photon has spin 1, but it is massless and sodoes not quite fit into the m > 0 picture we have been discussing.

We have seen that the classical vector S used to describe the rotational angularmomentum does not travel well into the quantum domain where it simply does notbehave the way one expects a vector to behave. Nevertheless, it is still convenientto collect together the quantities S x, S y and S z, measured, for example, by a Stern-Gerlach apparatus aligned along the x-, y- and z-axes, and refer to the triple

S = (S x, S y, S z)

as the spin vector. Quantum theory decrees that, for a particle with spin quantumnumber s, the only allowed values for the “components” S x, S y and S z are

�s~, �(s � 1)~, . . . , (s � 1)~, s~. (A.34)

In particular, for a spin 12 particle such as the electron there are only two possible

values so, for example,

S z = ±~

2.

With this synopsis of the general situation behind us we will return to the par-ticular case of spin 1

2 . We know that the classical picture of the electron as a tinyspinning ball cannot describe what is actually observed so we must look for anotherpicture that can do this. Whatever this picture is it must be a quantum mechanicalone so we are looking for a Hilbert space H and some self-adjoint operators on itto represent the observables S x, S y and S z. Previously we represented the state ofthe electron by a wave function (x, y, z) that is in L2(R3), but we now know thatthe state of a spin 1

2 particle must depend on more that just x, y, and z since thisalone cannot tell us which of the two paths an electron is likely to follow in a Stern-Gerlach apparatus; we say “likely to” because we can no longer hope to know morethan probabilities. What we would like to do is isolate some appropriate notion ofthe “spin state” of the particle that will provide us with the information we needto describe these probabilities. Now, we know that the only possible values of S zare ± ~2 . By analogy with the classical situation one might view this as saying thatthe spin vector S can only be either “up” or “down”, but nothing in-between. Thissuggests that we consider wave functions

(x, y, z,�) (A.35)

that depend on x, y, z, and an additional discrete variable � that can take only twovalues, say, � = 1 and � = 2 (or, if you prefer, � = up and � = down). Then| (x, y, z, 1) |2 would represent the probability density for locating the electron at(x, y, z) with S z =

~

2 and similarly | (x, y, z, 2) |2 is the probability density for locat-


ing the electron at (x, y, z) with S z = � ~2 . Stated this way it sounds a little strange,but notice that this is precisely the same as describing the state of the electron withtwo functions 1(x, y, z) = (x, y, z, 1) and 2(x, y, z) = (x, y, z, 2) and this is whatwe will do. Specifically, we will identify the wave function of a spin 1

2 particle witha (column) vector

1(x, y, z) 2(x, y, z)

!,

where 1 and 2 are in L2(R3) andZ

R3( | 1(x, y, z) |2 + | 2(x, y, z) |2 ) dµ = 1

because the probability of finding the electron somewhere with either S z =~

2 orS z = � ~2 is 1. The Hilbert space is therefore

H = L2(R3;C2) � L2(R3) ⌦ C2.

Now we must isolate self-adjoint operators on H to represent the observablesS x, S y and S z. Since these observables represent an intrinsic property of a spin 1

2particle, independent of x, y and z, we will want the operators to act only on thespin coordinates 1 and 2 and the action should be constant in (x, y, z). Thus, we aresimply looking for 2⇥2 complex, self-adjoint (that is, Hermitian) matrices or, statedotherwise, self-adjoint operators on C2. Since the only possible observed values are± ~2 , these must be the eigenvalues of each matrix. There are, of course, many suchmatrices floating around and we must choose three of them. Our choice is motivatedby the desire to keep spin angular momentum and orbital angular momentum onthe same formal footing since, classically at least, they really are the same thing.We accomplish this by insisting that the operators satisfy the same commutationrelations as those of the orbital angular momentum.

Recall that the generators M23 = J1,M31 = J2, and M12 = J3 in p can be realizedas the components of the orbital angular momentum operators with ~ = 1 and thatthese satisfy the commutation relations (2.27), that is,

[J j, Jk] = i✏ jkl Jl, j, k = 1, 2, 3.

Including a factor of ~ in each operator these become

[J j, Jk] = i~✏ jkl Jl, j, k = 1, 2, 3.

Thus, we need to find operators S 1, S 2, and S 3 on C2 each of which has eigenvalues± ~2 and that satisfy

[S j, S k] = i~✏ jklS l, j, k = 1, 2, 3. (A.36)

As it happens, this is quite easy. We let � j, j = 1, 2, 3, be the Pauli spin matrices(Exercise 1.2.6) and define

A.5 Spin 171

S j =~

2� j, j = 1, 2, 3. (A.37)

Exercise A.5.1. Show that S 1, S 2, and S 3 given by (A.37) satisfy the required con-ditions.

Exercise A.5.2. Define the operator S2 on C2 by S2 = S 21 + S 2

2 + S 23 and show that

S2 =✓12

◆✓12+ 1

◆~

2 idC2 .

Note: There actually is a point to writing 34 in this peculiar way as we shall soon see.

In order to make some direct contact with the material at the end of Section 2.8we recommend the following exercise.

Exercise A.5.3. Let D(1/2) be the spin 12 representation of SU(2) on C(D(1/2)) = C2

and let M23,M31, and M12 be the generators of rotations in p. Show that

i~ddt

⇥D(1/2)(e�itM23 )

⇤��t=0 = S 1,

i~ddt

⇥D(1/2)(e�itM31 )

⇤��t=0 = S 2,

i~ddt

⇥D(1/2)(e�itM12 )

⇤��t=0 = S 3.

Hint: Remark 2.5.1 and Exercise 2.4.5.

The S j =~

2� j, j = 1, 2, 3, are operators on C2, but they give rise to operatorsS j = idL2(R3) ⌦ S j, j = 1, 2, 3, on the Hilbert space

L2(R3) ⌦ C2 � L2(R3) ⌦ C(D(1/2)).

Specifically,

S 1

1(x, y, z) 2(x, y, z)

!=~

2�1

1(x, y, z) 2(x, y, z)

!=~

2

2(x, y, z) 1(x, y, z)

!

S 2

1(x, y, z) 2(x, y, z)

!=~

2�2

1(x, y, z) 2(x, y, z)

!=~

2

�i 2(x, y, z)i 1(x, y, z)

!

S 3

1(x, y, z) 2(x, y, z)

!=~

2�3

1(x, y, z) 2(x, y, z)

!=~

2

1(x, y, z)� 2(x, y, z)

!


We call S j, j = 1, 2, 3, the spin operators for spin 12 particles, although the termi-

nology is often applied to the S j, j = 1, 2, 3, themselves. The physical observablescorresponding to these operators are the analogue of the classical spin angular mo-mentum and are referred to either as the spin components or the intrinsic angularmomentum components of the spin 1

2 particle. These all have the same eigenvalues± ~2 and the largest of these eigenvalues ~2 is the spin of the particle (the term “spin12 ” tacitly assumes a factor of ~ or a choice of units in which ~ = 1). In light of Ex-ercise A.5.3 this coincides with the spin parameter for D(1/2) introduced in Section2.8 when the units are chosen so that ~ = 1.

The scheme is precisely the same for particles of arbitrary spin s = j/2. Thesehave wave functions with 2s + 1 components corresponding to the 2s + 1 “dots” onthe Stern-Gerlach apparatus so that the Hilbert space is H = L2(R3) ⌦ C2s+1. Thespin operators S 1, S 2, S 3 on C2s+1 are again required to satisfy the commutationrelations (A.36) of angular momentum and have eigenvalues that are the observedvalues (A.34) of the spin components. One then notices that C(D(s)) � C2s+1 andthat a set of such matrices is given by

S 1 = i~ddt

⇥D(s)(e�itM23 )

⇤��t=0,

S 2 = i~ddt

⇥D(s)(e�itM31 )

⇤��t=0,

S 3 = i~ddt

⇥D(s)(e�itM12 )

⇤��t=0.

We will have no need to write these out explicitly, but for those who would like tosee S 1, S 2, and S 3 in one more case we recommend the following exercise for spins = 1.

Exercise A.5.4. Define 3 ⇥ 3 matrices S 1, S 2, and S 3 by

S 1 =~p2

0BBBBBBBB@

0 1 01 0 10 1 0

1CCCCCCCCA S 2 =

~p2

0BBBBBBBB@

0 �i 0i 0 �i0 i 0

1CCCCCCCCA S 3 = ~

0BBBBBBBB@

1 0 00 0 00 0 �1

1CCCCCCCCA

1. Show that S 1, S 2, and S 3 satisfy the commutation relations (A.36).2. Show that each S j, j = 1, 2, 3, has eigenvalues �~, 0, ~.3. Show that

S2 = S 21 + S 2

2 + S 23 = (1)(1 + 1)~2 idC3 .

4. Check at least one of the following.

S 1 = i~ddt

⇥D(1)(e�itM23 )

⇤��t=0,

A.5 Spin 173

S 2 = i~ddt

⇥D(1)(e�itM31 )

⇤��t=0,

S 3 = i~ddt

⇥D(1)(e�itM12 )

⇤��t=0.

We point out that, generalizing Exercise A.5.2 and Exercise A.5.4 (3), one findsthat, for any s,

S2 = S 21 + S 2

2 + S 23 = s(s + 1)~2 idC2s+1

The corresponding spin operators on the Hilbert space

H = L2(R3) ⌦ C2s+1 � L2(R3) ⌦ C(D(s))

are then

S j = idL2(R3) ⌦ S j, j = 1, 2, 3.

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Index

([v], [w])P(H), 26Ad, 11Adg, 11Aut(G), 1Aut(P(H)), 26Aut(g), 11H2, 59H⇠0 , 40Hx0 , 10IndG

H(�), 32L(↵), 56LO⇠0 ,�

, 40N o✓ H, 33P ⇥� H, 30R = (R↵

�)↵,�=0,1,2,3, 53S 1, 2[ , ]C, 5P2, 88W2, 89⇤ = (⇤↵�)↵,�=0,1,2,3, 47, 52⇤A, 60⇤i(✓), 54L"+, 4, 47, 53P"+, 47, 58⌃(x1, x2, x3), ⌃(x1, x2, x3), . . ., 44AC, 5C(D( j/2)), 23CN (x0),C±N (x0), 50CT (x0),C±T (x0), 50D( j/2), 23E(x0), 51H�, 32M, 43, 48O, O, . . ., 44O⇠0 , 40Ox0 , 10S(x0, x1, x2, x3), S(x0, x1, x2, x3), . . ., 46

T, T±, 50U(H), 14U(g), 86U�(H), 27h , i, 49C⇤, 2P(H), 24R⇤, 1R1,3, 51⌘ = (⌘↵�)↵,�=0,1,2,3, 47, 49⌘�1 = (⌘↵�)↵,�=0,1,2,3, 49N, 38 : SL(2,C)! L"+, 61, 63iso(3), 36p, 72A

T, 3

⇡P, 24�g, 10� j, j = 1, , 2, 3, 18⌧(v), 55{e0, e1, e2, e3}, 48ad, 11adX , 111-parameter group of di↵eomorphisms, 1301-particle space, 1154-acceleration, 574-velocity, 57

abstract Schrodinger equation, 156action functional, 124adjoint action, 12adjoint representation of U(g), 87adjoint representation of g, 12admissible basis, 51admissible observer, 44algebra of classical observables, 135, 140angular momentum

185

186 Index

classical, 131intrinsic, 168orbital, 164rotational, 164total, 164

angular momentum operatororbital, 120spin, 121total, 121

angular momentum operatorsorbital, 163

anti-unitary map, 26anticommutation relations, 19associated Legendre functions, 80associative algebra, 8associative algebra

commutator bracket, 9commutator Lie algebra structure, 9ideal, 8ideal generated by S , 9subalgebra, 8unit, 8unital, 8

automorphism group of P(H), 26, 155automorphism group of a Lie group, 1

is a Lie group, 37automorphism of P(H), 26, 155automorphism of Lie groups, 1

Bargmann’s Theorem, 105base space, 29boost, 54Born-von Neumann formula, 151boson, 168bracket, 4bundle space, 29

canonical 1-form, 133canonical commutation relations

for classical mechanics, 141for quantum mechanics, 159

canonical coordinates, 133, 140canonical map, 142canonical symplectic form, 133Cartan’s magic formula, 136Casimir invariants, viiiCasimir invariants of p, 90causal automorphism, 46causality assumption, 46Change of Variables Formula, 100character, 38character group, 38

ofRk, 39classical canonical commutation relations, 141

classical harmonic oscillator Hamiltonian, 135classical observable, 135, 140Closed Subgroup Theorem, 2commutation relations, 19iso(3), 37

commutator bracket, 9commutator Lie algebra structure, 9configuration space, 123, 133conjugate momentum, 125conjugation representation, 18conservation law

classical mechanics, 126conservation of energy, 137, 142conserved quantity, 125, 137

and the Poisson bracket, 142angular momentum, 131energy, 142

contragredient action, 93, 103covector, 133covering group, 13covering map, 13, 63covering space, 13critical curve, 125critical point, 125

Darboux Theoremfinite dimensional, 139linear, 139

derivation, 11dispersion, 151double cover

universal, 63dual group, 38duration of a timelike vector, 55

Einstein energy-momentum relation, 95Einstein summation convention, ixelsewhere, 51energy, 153equivalent projective representations, 28Euler-Lagrange equations, 125, 132

and Newton’s Second Law, 128event, 43, 49evolution operator, 153expectation value, 151expected value, 151exponential map, 9

fermion, 168fiber, 29Fizeau procedure, 45flow of a vector field, 130frame of reference, 46free group action, 10

Index 187

free particle, 128

general linear group, 2generators of a Lie algebra, 36generators of boosts, 66generators of rotations, 37, 66generators of translations, 37GL(n,C), 2GL(n,R), 2global section

of a principal H-bundle, 30graviton, 169group action, 10group representation

induced, 29semi-direct products, 33

Haar measure, 1Hamilton’s equations, 134, 136, 142Hamiltonian, 133, 140

classical harmonic oscillator, 135on L2(Rn), 156quantum, 153

Hamiltonian flow, 134Hamiltonian Noether Theorem, 144Hamiltonian system, 140Hamiltonian vector field, 133, 136, 140, 142harmonic oscillator

classical Hamiltonian, 135Heisenberg equation, 158Heisenberg picture, 158Heisenberg Uncertainty Principle, 152Hermitian inner product, 3Higgs boson, 169Hilbert bundle, 30, 31

section, 31homogeneous manifold, 12homogeneous space, 12

ideal, 8ideal generated by S , 9indefinite inner product, 4induced representation, 29, 32, 42infinitesimal symmetry, 126, 143inhomogeneous rotation group, 35inhomogeneous SL(2,C), 64, 103internal semi-direct product, 34intertwine, 15intrinsic angular momentum, 168, 172invariant under rotation, 129irreducible representation, 15ISO(3), 35isolated quantum system, 153isomorphic Lie algebras, 5

isomorphism of Lie groups, 1isomorphism of projectivizations, 26isotropy subgroup, 10

Jacobi Identity, 136, 141Jacobi identity, 4

kinetic energy operator, 160

Lagrangian, 124symmetry of, 126

left action, 10left invariant vector field, 5Leibniz Rule, 136, 142Leibniz rule, 11Lie algebra, 4

complex, 4complexification, 5exponential map on, 9isomorphism, 5of a Lie group, 5of GL(n,C), 7of GL(n,R), 7of O(n), 7of O(p, q), 8of SO(n), 7of SO(p, q), 8of SU(n), 8of the Lorentz group, 65of U(n), 8

Lie algebra automorphism, 5Lie algebra cohomology, 28Lie algebra homomorphism, 5Lie algebra isomorphism, 5Lie algebra of operators, 82Lie algebra representation, 5Lie group, 1

closed subgroups of, 2continuous homomorphisms of, 2isomorphism, 1representation, 1

Lie groups, 1Lie subalgebra, 5lift of a projective representation, 28light travel time, 45Linear Darboux Theorem, 139little group, 40local section

of a principal H-bundle, 30locally isomorphic Lie groups, 7Lorentz group

general, 4proper, 4proper, orthochronous, 4

188 Index

Lorentz inner product, 49Lorentz transformation

general, 52orthochronous, 53proper, 52special, 55

Mackey machine, vii, 37, 41and ISL(2,C), 107

Mackey’s Theorem, 41magnetic dipole, 164magnetic moment

orbital, 164rotational, 165total, 165

mass hyperboloid, 93mass operator, 118matrix Lie group, 3matrix mechanics, 148measure

pushforward, 102measurement, 147, 153Minkowski inner product, 49Minkowski spacetime, 48moment map, 144momentum, 128

conjugate, 125momentum map, 144momentum operator, 162momentum space, viii, 92

natural coordinates on T M, 123Noether’s Theorem, 127, 144null cone, 50

future, 50past, 50

null curve, 56future directed, 56past directed, 56

null vector, 49

O(n), 3O(p, q), 3observable

classical, 135quantum, 149

orbit, 10orbit space, 30orbital angular momentum, 164orbital magnetic moment, 164orthogonal group, 3orthogonal transformation of M, 52orthogonal vectors in M, 50orthonormal basis for M, 49

past directed, 50path space, 124Pauli spin matrices, 18Pauli-Lubanski vector, 88phase space

in classical mechanics, 133in Hamiltonian mechanics, 137

photon, 169Poincare algebra, 72Poincare group, 58Poincare-Birkho↵-Witt Theorem, 86Poisson algebra, 136Poisson bracket, 136, 141Poisson commutes, 137position operator, 161positive energy, 94Postulates of Quantum Mechanics

QM1, 148QM2, 149QM3, 150QM4, 153

potential, 156spherically symmetric, 128

potential energy operator, 161principal H-bundle

trivial, 29principal bundle, 29, 30principal H-bundle, 29Principle of Least Action, 125Principle of Stationary Action, 125probability amplitude, 149projection, 29

of a principal bundle, 29projective representation, vii, 24, 28

lift, 28projectivization of H, 149projectivization of a Hilbert space, 24proper time function, 57proper time length, 56proper time separation, 55pushforward measure, 102

quantization, 161quantum bracket, 159quantum canonical commutation relations, 159quantum field, viiquantum symmetry group, 155quantum system, 147

state space, 149symmetry of, 26, 154

quasi-invariant measure, 32

ray in a Hilbert space, 24realization of a Lie algebra, 84

Index 189

reducible representation, 15regular semi-direct product, 40relatiivistic invariance, viiirelativistic angular momentum, 77relativistic scalar field, 116Relativity Principle, 48representation

and left action, 14carriers, 14dimension, 14induced, 32, 42invariant subspace, 14, 15irreducible, 15of a group, 14reducible, 15spinor, 23trivial, 21Wigner, 116

rest frame, 115Reversed Schwartz Inequality, 55Reversed Triangle Inequality, 56right action, 10Robertson Uncertainty Relation, 152Rodrigues’ formula, 80rotation group, 3, 129

exponential map on, 129Lie algebra of, 129

rotation subgroup of L"+, 54rotational angular momentum, 164rotational magnetic moment, 165

scalar fieldrelativistic, 116

scalar field on X+m, 116Schrodinger equation, 148

abstract version, 156physicist’s version, 156

Schrodinger picture, 158Schur’s Lemma, 15

for associative algebras over C, 91infinite-dimensional version, 15

sectionglobal, 30local, 30of a Hilbert bundle, 31

semi-direct product, 33, 35internal, 34regular, 40representations, 33

semi-direct product of Lie groups, 37semi-orthogonal group, 3signature of Minkowski spacetime, ixSL(2,C)

conjugation representation, 20

standard representation, 20SL(n,C), 3SL(n,R), 3SO(3)

geometrical description, 10SO(n), 3SO(p, q), 4space of an admissible observer, 51spacelike curve, 56spacelike vector, 49special linear group, 3special Lorentz transformation, 55

hyperbolic form, 55special orthogonal group, 3special semi-orthogonal group, 4special unitary group, 3special unitary group SU(2), 17spherical harmonics, 80spherically symmetric potential, 128spin, 172

of an SU(2) representation, 24spin 1

2 , 168spin angular momentum operator, 121spin components, 172spin model ofR3, 19spin operators, 172spin quantum number, 168spin vector, 169spinor

contravariant, 23rank m, 23SL(2,C), 24SU(2), 23

spinor field on X+m, 116spinor map, 63spinor representation

SU(2), 23state

classical mechanical system, 123, 133in quantum mechanics, 148

state space, 123in quantum mechanics, 149

stationary curve, 125stationary point, 125Stern-Gerlach experiment, 165structure constants, 7structure group, 29SU(2)

defining representation, 17representation

spin, 24spinor representation, 23standard representation, 17

SU(n), 3

190 Index

SU(2), 17irreducible representations, 17

subalgebra, 8surface measure, 100symmetry, 126

quantum system, 26, 154symmetry group, 143

of a Lagrangian, 127physical interpretation, 84quantum, 155

symplectic di↵eomorphism, 142symplectic form, 137

canonical, 133symplectic gradient, 136, 140symplectic manifold, 137

compact, 138is even dimensional, 138is orientable, 138

symplectic vector field, 142symplectomorphism, 142

time axis, 51time cone, 50

future, 50past, 50

time orientation, 50, 51timelike curve, 56

future directed, 56past directed, 56

timelike vector, 49timelike worldline, 56total angular momentum, 164total energy

classical mechanics, 133in quantum mechanics, 153symplectic geometry, 140

total magnetic moment, 165total orbital angular momentum, 78

total relativistic energy, 94total space, 29transition amplitude, 153transition probability, 26, 153transitive group action, 10trivial principal H-bundle, 29trivial representation, 14, 21twin paradox, 56

U(n), 3uncertainty relations, 152

Robertson, 152unit ray, 25, 149unitarily equivalent representations, 15unitary group, 3unitary group representation, 14unitary representation, 14

strongly continuous, 14weakly continuous, 14

universal covering group, 63universal double cover, 63universal enveloping algebra, viii, 85

basis, 87

variance, 151

wave function, 148wave mechanics, 148weight of spinor representation, 23Wigner representation, 116Wigner representation of mass m and spin j/2,

116Wigner rotation, 67Wigner symmetry, 26, 154Wigner’s Theorem on Symmetries, 26, 154worldline, 43worldline of a material particle, 56

Positive Energy Representations of the Poincare...

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