Appendix A The Symmetry Representation Theorem · 2016. 4. 6. · the projective representations...

Appendix A Symrne try Rcpresentat ion Theorem 9 1

reference to symmetry principies, hecat~se whatever one thinks the syrnrnelry group of nature map be, there is alwcsys another group whose colasequences are iden~ical except Jhr the rrbsence uf supersrlr~ction rules.

Appendix A The Symmetry Representation Theorem

This appendix presents the proof of the fundamental theorem of wiper2 that any symmetry transformation can be represented on the Hilbert space of physical states by an operator that is either linear and unitary or antilinear and antiunitary. For our present purposes, the property of symmetry transformations on which we chiefly rely is that they are ray transformations T that preserve transition probabilities, in the sense that if and Y!2 are state-vectors belonging to rays 21 and 92 then any state-vectors Yi and Y!; belonging to the transformed rays T.@] and T9f2 satisfy

We also require that a symmetry transformation should have an inverse that preserves transition probabilities in the same sense.

To start, consider some complete orthonormal set of state-vectors YJk belonging to rags &, with

and let YL be some arbitrary choice of statevectors belonging to the transformed rays T9k. From Eq. (2.A.11, we have

I{'Y;,T;)I~ = ~(vkry!)~~ = ski.

But (Yk, Y'k) is automatically real and positive, so this requires that it should have the value unity, and therefore

i t is easy to see that these transformed states V T i also form a complete set, for if there were any non-zero state-vector Y f that was orthogonal to all of the Yb, then the inverse transform of the ray to which Yf belongs would consist of non-zero state-vectors Y" for which, for all k :

which is impossible since the Vk were assumed t o form a complete set. We must now establish a phase convention for the states YL. For this

purpose, we single out one of the Y k , say Y1, and consider the state-vectors

92 2 Relativistic Quantum Mechalalcs

belonging to some ray Yk, with k f 1. Any state-vector Ti belonging to the transformed ray T Y k may be expanded in the state-vectors Yi,

From Eq. (2.A. 1) we have 1

and for 1 # k and l + 1 :

For any given k, by an appropriate choice of phase of the two state- vectors Yk and Y i we can clearly adjust the phases of the two non-zero coefficients ckk and ckl so that both coefficients are just 1 1 8 . From now on, the state-vectors TL and Y ; chosen in this way will be denoted UTk and UYk. As we have seen,

1 1 U - [ v k + vI] = UYk = - [UTk + U y l ] . Js $ (2.A.5)

However, it still remains to define UY for general state-vectors Y. Now consider an arbitrary state-vector Y! belonging to an arbitrary ray 9, and expand it in the Y k :

Any state Y" that belongs to the transformed ray T W may similarly be expanded in the complete orthonormal set UYk:

The equality of ( Y k , y)12 and I(UYLk, yf)12 tells us that for all k (including k = 1):

1412 = lcLl2, (2.A.8) while the equality of i(Yk, Y)1 and LIT^, Y!')12 tells us that for all k # 1 :

The ratio of Eqs. (2.A.9) and (2.A.8) yields the formula

which with Eq. (2.A.8) also requires

Appendix A Symmetry Representation Theorem

and therefore either

or else

Furthermore, we can show that the same choice must be made for each k . (This step in the proof was omitted by Wigner.) To see this, suppose that for some k , we have Ck/C1 = Ci /C; , while for some I # k, we have instead CI/CI = (C;/C;)*. Suppose also that both ratios are complex, so that these are really different cases. (This incidentally requires that k # 1 and I # 1, as well as k # I.) We will show that this is impossible.

Define a state-vector (D = [V1 + Yk + Yi]. Since all the ratios of the ,. 3 coefficients in this state-vector are real, we must get the same ratios in any state-vector @' belonging to the transformed ray;

where a is a phase factor with lcll = 1. But then the equality of the transition probabilities I(@, CY)I and I(@', lyl ) l requires that

and hence

This is only possible if

or, in other words, if

Hence either Ck/C1 or CI/C1 must be real for any pair k , 1, in contra- diction with our assumptions. We see then that for a given symmetry transformation T applied to a given state-vector CkYlk, we must have either Eq. (2.A. 12) for all k, or else Eq. (2.A.13) for all k.

Wigner ruled out the second possibility, Eq. (2.A.13), because as he showed any symmetry transformation for which this possibility is realized would have to involve a reversal in the time coordinate, and in the proof he presented he was considering only symmetries like rotations that do

94 2 Relativistic Quantum Mechanics

not affect the direction of time. Here we are treating symmetries involving time-reversal on the same basis as all other symmetries, so we will have to consider that, for each symmetry T and state-vector Ck CkVk, either Eq, (2.A.12) or Eq. (2.A.13) may apply. Depending on which of these alternatives is realized, we will now define UY to be the particular one of the state-vectors Y' belonging to the ray T B with phase chosen so that either C1 = C; or CI = c;', respectively. Then either

or else

It remains to be proved that for a given symmetry transformation, we must make the same choice between Eqs. (2.A.14) and (2.A.15) for arbitrary values of the coefficients Ck. Suppose that Eq. (2.A. 14) applies for a state-vector Ck AkYk while Eq. (2.A.15) applies for a state-vector Ck BkVk. Then the invariance of transition probabilities requires that

or equivalently 1m (A;Al) lm (B,'BI) = 0 . (2.A. 16)

.k I

We cannot rule out the possibility that Eq. (2.A.14) may be satisfied for a pair of state-vectors Ck AkYk and Ck &YE; belonging to different rays. However, for any pair of such state-vectors, with neither Ak nor Bk all of the same phase (so that Eqs. (2.A.14) and (2.A.15) are not the same), we can always find a third state-vector '& Ck Yk for which*

and also

* If Eor some pair k, l both A; A, and B; Br arc complex, then ct~onsc all C s to vanish excepl Tor Ck and C l , and chouse these twn coefficients to hsvc dineren1 phases. If Aidr i s cumplex but B'BI is real for some pair k , l , then there must be some othcr pair m,n (possibly a h h either m or n b u ~ nnl both equal to k or I ) tor which B i B , ic complex. If a h A i A , is complex, then choose all Cs to vanish except for I;, and Cn, and choose these two coefficients Lo have diffcrcnt phase. If AkA, is rcal, Lhen choose all Cs to vanish except for CL, Cf ,C'm, and C,, and choose thcsc four coefficients all to have djfTercnt phases. The case whcrc BiB1 is complex but A;Al i s r e d is handled in just the samc way.

Appendix A Symmetry Representation Theorem 9 5

As we have seen, it follows from Eq. (2.A.17) that the same choice between Eqs. (2.A. 14) and (2.A.15) must be made for Ck AkYk and Ck CkYk, and it follows from Eq. (2.A.18) that the same choice between Eqs. (2.A.14) and (2.A.15) musl be made for Ck BkYk and Ck CkYk, SO the same choice between Eqs. (2.A.14) and (2.A.15) must also be made for the two state- vectors Ck AkVk and Ck BkYk with which we started. We have thus shown that for a given symmetry transformation T either a11 state-vectors satisfy Eq. (2.A.14) or else they all satisfy Eq. (2.A. 15).

I t is now easy to show that as we have defined it, the quantum mechan- ical operator U is either linear and unitary or antilinear and antiunitary. First, suppose that Eq. (2.A.14) is satisfied for all state-vectors Ck CkYk. Any two stale-vectors Y and @ may be expanded as

and so, using Eq. (2.A.141,

Using Eq. (2.A. 14) again, this gives

so U is linear. Also, using Eqs. (2.A.2) and (2.A.3), the scalar product of the transformed states is

and hence

so U is unitary. The case of a symmetry that satisfies Eq. (2.A.15) for all state-vectors

may be dealt with in much the same way. The reader can probably supply the arguments without help, but since antilinear operators may be unfamiliar, we shall give the details here anyway. Suppose that Eq. (2.A- 15) is satisfied for all state-vectors CkYlk. Any two state-vectors Y and @ may be expanded as before, and so:

96 2 Relativistic Qualatuna Mechanics

Using Eq. (2.A.15) again, this gives

so U is antilinear. Also, using Eqs. (2.A.2) and (2.A.31, the scalar product of the transformed states is

and hence

(UY, UO) = (Y!, a))',

Appendix B Group Operators and Homotopy Classes

In this appendix we shall prove the theorem stated in Section 2.7, that the phases of the operators U ( T ) for finite symmetry transformations T may be chosen so that these operators form a representation of the symmetry group, rather than a projective representation, provided (a) the generators of the group can be defined so that there are no central charges in the Lie algebra, and (b) the group is simply connected. We shall also comment on the projective representations encountered for groups that are not simply connected, and their relation to the homotopy classes of the group.

To prove this theorem, let us recall the method by which we construct the operators corresponding to symmetry transformations. As described in Section 2.2, we introduce a set of real variables Oa to parameterize these transformations, in such a way that the transformations satisfy the composition rule (2.2.15):

We want to construct operators U(T(B) ) = U[O] that satisfy the corresponding conditionm

To do this, we lay down arbitrary 'standard' paths O$(s) in group parameter space, running from the origin to each point 8, with OE(0) = 0 and @;(I) = 8". and define U s [ s ) along each such path by the differential

Square brackets arc used here to distinguish U operators construckd as functions of the group parameters from those expressed as functions of Ihe group transformations themselves.

4. Galilean Invariance

4.1 Generators of infinitesimal transformations

The freedom of choice for reference frames includes more than rotations: one can displace the origin, translate it by a constant vector; or one can let that translation grow proportionally with time; the two frames are in relative motion at constant velocity. We'll consider only relative speeds that are small on the scale of the speed of light; see Problems 4-3 and 4-4 for other circumstances. Then time has an absolute significance (Galilean*-Newtonian relativity) apart from the freedom of displacing its origin. The infinitesimal transformations of these types are displayed by the space-time changes

t=t-8t, if = r -lir ,

with lir = + liw x r + liv t , (4.1.1)

where 8t is a constant, as are the vectors liw, liv. The accompanying unitary operator is

u = 1 + iG ( 4.1.2)

where, now

G = . P + liw . J + liv . N - lit H + litp , ( 4.1.3)

and we want to recognize that we always have the freedom of a phase trans-formation. The names for the generators are derived from classical mechanics:

P: linear momentum vector, J: (already familiar) angular momentum vector, H: energy; Hamiltonian (or Hamiltont operator), N: no classical name, perhaps booster?

But now we have to notice something. If we write U = 1 + iG, it is clear that G is dimensionless - it is given by pure numbers. But P, the product

*Galileo GALILEI (1564-1642) tSir William Rowan HAMILTON (1805-1865)

J. Schwinger, Quantum Mechanics© Springer-Verlag Berlin Heidelberg 2001

184 4. Galilean Invariance

of length [L] by momentum [M L/T], or [M L2/T] - or equally well -8tH: time [T] times energy [M L2/T2], not to mention 8w .J: angle (dimensionless) times angular momentum [M L2/T) - has dimensions, those of action. It is clear that up to now we have been employing natural atomic units, not the arbitrary units of macroscopic physics. So, if we wish to use the latter, we must include a conversion factor:

(4.1.4)

where fi, the unit of action, is (21f)-1 times Planck's* constant h. Experiment tells us that

h fi = - = 1.05457 X 10-27 erg sec = 0.658212 e V fs

21f ( 4.1.5)

(1 eV = 1.602177 x 10-12 erg, electron-volt; 1 fs = 1O-15S, femto-second). It is important to recognize that the order in which these transformations,

even infinitesimal ones, are made is important, in general. To use a familiar situation consider rotations. Compare 1,2:

with 2,1:

r --+ r - 81w x r --+ r - 81w x r - 82 w x (r - 81w x r) = r - 81 w x r - 82w x r + 82w x (81 W x r)

The result of performing 1,2 and then the inverse of 2,1 is

r --+ r + 82w x (81w x r) -81w x (82w x r) , ...

= r - (81w x 82w) x r

= r - 8[12JW x r ,

i. e., another rotation described by

=(hw x (r x b2W)

( 4.1.6)

(4.1.7)

(4.1.8)

(4.1.9)

From the viewpoint of unitary transformations we are saying that U2 U1 i-UI U2 and

(4.1.10)

which for infinitesimal transformations becomes

'Max Karl Ernst Ludwig PLANCK (1858-1947)

4.1 Generators of infinitesimal transformations 185

or

(4.1.12)

since

U2 U1 = (1 + ) (1 + )

i 1 = 1 + fi(G 2 + Gd - li2G2G1 ( 4.1.13)

and

(4.1.14)

And so we have

( 4.1.15)

Now the only possibility for the scalar O[12Ji.p is a multiple of OlW' 02W, which is symmetrical in 1 and 2, not antisymmetrical. Hence o[12Ji.p = O. Then, written as

1 in [J, Ow . J] = Ow x J , (4.1.16)

we recognize the characterization of a vector under rotations. This immediately tells us that the analogous considerations for the vectors

P, N, and J will yield

[p, Ow . J] = Ow x P ,

1 iii [N, Ow . J] = ow x N ,

whereas, for the scalar H,

1 - [H Ow . J] = 0 . iii '

How about translations? As

(4.1.17)

(4.1.18)

(4.1.19)


indicates, we have

(4.1.20)

and

1 in [(h€· P, 82€· P] = M[12]<P , (4.1.21)

where the only possibility of 8[12]<P ex 81 €· 82 € shows that

(4.1.22)

or

and PxP=O. (4.1.23)

Similarly,

[Nk,Nd = 0, N x N = O. (4.1.24)

But when we come to

1 in[8� . P,8v· N] = M<p = M8� · 8v (4.1.25)

(dimension of M: mass) we can no longer conclude that 8<p = 0 since two different vectors are involved. So

(4.1.26)

With regard to transformations that include time displacement, consider

t -+ t - 81 t -+ t - 81 t - 82t , r -+ r - 81vt -+ r - (hvt - (hv(t - 81t) ,

so that (1,2) x (2,1)-1 leaves us with a net displacement

which will have no counterpart in displacements or rotations. So

or

1 =8v8t·P+M<p

In

-+0

1 in[N,H] = -P,

(4.1.27)

(4.1.28)

(4.1.29)

( 4.1.30)


whereas

1 -[iSw . J -6t H] = 0 in ' and

1 -[is£.· P -6tH] = 0 in ' (4.1.31)

imply

[J,H] =0 and [P,H]=O. (4.1.32)

The commutators involving J are the response to rotations, distinguishing vectors and scalars. Now let's look at the P commutators, the response to translations. From the P equation in (4.1.17) we get

1 in [J, 6£.· P] = iSJ = is£. x P , (4.1.33)

and since

(4.1.34)

also

(4.1.35)

and of course

( 4.1.36)

Both J and N show a response to translation which can be expressed by a vector R such that

1 is,,R = iii [R, is£. . P] = is£. ,

1 in [Rk' Pt] = iSkl . (4.1.37)

So

(4.1.38)

and we write

J=RxP+S, ( 4.1.39)

where the components of S commute with those of P,

(4.1.40)


Since R is a vector we must have

1 in[R,8w. J] = 8w x R

1 = in [R, 8w x R· P + 8w . S] (4.1.41)

which is certainly satisfied if

or R x R = 0 and [Rk, Bd = 0 . ( 4.1.42)

Also, since N generates a displacement proportional to t it must contain Pt, or

N=Pt-MR. (4.1.43)

In particular, for t = 0, N = - M R, and R x R = 0 follows from N x N = O. Inasmuch as Rand P are vectors, so is

L=RxP (4.1.44)

and, in view of

1 1 in[L,8w. J] = in[L,8w. L] = 8w xL (4.1.45)

one has

Lx L = inL (4.1.46)

which implies that

S x S = inS. (4.1.47)

We see that

J=L+S (4.1.48)

is the decomposition into external or orbital angular momentum L, and in-ternal or spin angular momentum S.

We have now recognized that the system as a whole is described by po-sition vector R, momentum vector P, which for each direction in space con-stitute a q,p set of operators:

(4.1.49)

Accordingly all these operators have continuous spectra and have a classical limit.


Notice also that

R·L=R·RxP=RxR·P=O (4.1.50)

which means that a rotation about the direction R has no effect, has zero quantum number,

c5 ( I = i ( I c5w . L = 0 if c5w ex: R . ( 4.1.51)

But zero is an integer and therefore all possible values of I in L 2 ' = 1(1 + 1)!l,2 are integers,

1= 0,1,2, .... (4.1.52)

Now look at the information we have about H:

[J,H] =0, [P,H]=O, 1 ifj,[N,H] = -P. ( 4.1.53)

The first says that H is a scalar, the second, according to

(R'lp = 1

(4.1.54)

(R components are compatible) says

( 4.1.55)

H does not depend on R; the third is

1 - [Pt - M R H] = - P in· , ( 4.1.56)

or

in' M (4.1.57)

But, according to (P components are compatible, too)

(4.1.58)

we have

( 4.1.59)

or

p 2

H = 2M + Hint with V pHint= 0 . (4.1.60)


4.2 Hamilton operator for a system of elementary particles

For us an elementary particle is defined as one without internal energy, or at least with inaccessible internal energy under the given circumstances. For atomic structure discussions the elementary particles are electrons and nuclei. For nuclear physics discussions, they are protons and neutrons, and so on.

Let each elementary particle be described by independent variables r a,

Pa, Sa and mass mao Then we construct P, J, N additively

P= LPa, a

J = L(ra X Pa + Sa) = R X P + S , a

N= L(Pat-mara) =Pt-MR, (4.2.1) a

where

M=Lma , (4.2.2) a a

and indeed

= a b

= L r;; = 8kl . (4.2.3) a

=1 =dkl

We write

L r a X Pa = L [R + (r a - R)] X Pa a a

= R X P + L (r a - R) X (Pa - P) , (4.2.4) a ' y ,

internal variables

since

L ma (r a - R) = 0 and L (Pa - P) = 0 , (4.2.5) a a

and get

S = L [(ra - R) X (Pa - P) + Sa] (4.2.6) a

for the total internal angular momentum.

Problems 191

If the constituents were isolated from each other we would have

(4.2.7)

More general we write

( 4.2.8)

with

(4.2.9)

where V, the potential interaction energy, is a scalar function of the internal variables and the Sa and possibly others.

Problems

4-1 Verify explicitly that L = R x P obeys the angular momentum com-mutation relations (4.1.46). Can you think of a reason, based on the vector structure of L, for the fact that any component of Lin has only integer values?

4-2 Show that L·S commutes with J, L2, and S2. Then find the eigenvalues of L· s.

4-3 Einsteinian * relativity: Replace the first line in (4.1.1) by

_ 1 1 t = t - -JED - 2Jv . r ,

c c

where c is the speed of light, and the Galilean form is formally recovered in the limit c -+ 00 if (l/c)<5to -+ Jt is understood. Show that the commutators are the same, with two exceptions:

and

4-4 In consequence of these modified commutation relations, what needs to be altered in the equations introducing Rand S?

* Albert EINSTEIN (1879-1955)


4-5 Photons have only spin angular momentum + 1 or -1 along their direc-tion of motion. (Incidentally, helicity is a more fitting term than spin under these circumstances.) A light beam is deflected through the angle B. To what extent can you anticipate the dependence of the deflected beam's intensity on angle from the spin properties of a photon? [Hint: Recall Problem 3-5.]

Chapter 9

Space-time symmetry transformations

In the last chapter, we set up a vector space which we will use to describe the state of a system of physicalparticles. In this chapter, we investigate the requirements of space-time symmetries that must be satisfiedby a theory of matter. For particle velocities small compared to the velocity of light, the classical laws ofnature, governing the dynamics and interactions of these particles, are invariant under the Galilean groupof space-time transformations. It is natural to assume that quantum dynamics, describing the motion ofnon-relativistic particles, also should be invariant under Galilean transformations.

Galilean transformation are those that relate events in two coordinate systems which are spatially rotated,translated, and time-displaced with respect to each other. The invariance of physical laws under Galileantransformations insure that no physical device can be constructed which can distinguish the di↵erence be-tween these two coordinate systems. So we need to assure that this symmetry is built into a non-relativisticquantum theory of particles: we must be unable, by any measurement, to distinguish between these coor-dinate systems. More generally, a symmetry transformation is a change in state that does not change theresults of possible experiments. We formulate this statment in the form of a relativity principle:

Definition 16 (Relativity principle). If | (⌃) i represents the state of the system which refers to coordinatesystem ⌃, and if a(⌃) is the value of a possible observable operator A(⌃) with eigenvector | a(⌃) i, alsoreferring to system ⌃, then the probability Pa of observing this measurement in coordinate system ⌃ mustbe the same as the probability P 0a of observing this measurement in system ⌃0, where ⌃0 is related to ⌃ bya Galilean transformation. That is, the relativity principle requires that:

P 0a = |h a(⌃0) | (⌃0) i|2 = Pa = |h a(⌃) | (⌃) i|2 . (9.1)

In quantum theory, transformations between coordinate systems are written in as operators acting onvectors in V. So let

| (⌃0) i = U(G) | (⌃) i , and | a(⌃0) i = U(G) | a(⌃) i , (9.2)

where U(G) is the operator representing a Galilean transformation between ⌃0 and ⌃. Then a theorem byWigner[1] states that:

Theorem 16 (Wigner). Transformations between two rays in Hilbert space which preserve the same proba-bilities for experiments are either unitary and linear or anti-unitary and anti-linear.

Proof. We can easily see that if U(G) is either unitary or anti-unitary, the statement is true. The reverseproof that this is the only solution is lengthy, and we refer to Weinberg [?][see Weinberg, Appendix A, p.91] for a careful proof.

The group of rotations and space and time translations which can be evolved from unity are linear unitarytransformations. Space and time reversals are examples of anti-linear and anti-unitary transformations. Wewill deal with the anti-linear symmetries later on in this chapter.

83

9.1. GALILEAN TRANSFORMATIONS CHAPTER 9. SYMMETRIES

a

v t

F’

F

R

X(t)

X’(t’)

Figure 9.1: The Galilean transformation for Eq. (9.1).

We start this chapter by learning how to describe Galilean transformations in quantum mechanics, andhow to classify vectors in Hilbert space according to the way they transform under Galilean transformations.In the process, we will obtain a description of matter, based on the irreducible representations of the Galileangroup, and use this information to build models of interacting systems of particles and fields.

The methods of finding unitary representations for the Galilean group in non-relativistic mechanics issimilar to the same problem for the Poincare group in relativistic mechanics. The results for the Poincaregroup are, perhaps, better known to physicists and well described in Weinberg[?, Chapter 2], for example.It turns out, however, that the group structure of the Galilean group is not not as simple as that of thePoincare group. The landmark paper by Bargmann[2] on unitary projective representations of continuousgroups contains theorems and results which we use here. Ray representations of the Galilean group are alsodiscusses by Hamermesh[?][p. 484]. We also use results from several papers by Levy-Leblond[3, 4, 5, 6] onthe Galilei group. In the next section, we show that Galilean transformation form a group.

9.1 Galilean transformations

A Galilean transformation includes time and space translation, space rotations, and velocity boosts of thecoordinate system. An “event” in a coordinate frame ⌃ is given by the coordinates (x, t). The same event isdescribed by the coordinates (x0, t0) in another frame ⌃0, which is rotated an amount R, displaced a distancea, moving at a velocity v, and using a clock running at a time t0 = t + ⌧ , with respect to frame ⌃, as shownin Fig. 9.1. The relation between the events in ⌃ and ⌃0 is given by the proper Galilean transformation:

x0 = R(x) + vt + a , t0 = t + ⌧ , (9.3)

with R a proper real three-dimensional orthogonal matrix such that detR = +1. We regard the transfor-mation (9.3) as a relationship between an event as viewed from two di↵erent coordinate frames. The basicpremise of non-relativistic quantum mechanics of point particles is that it is impossible to distinguish be-tween these two coordinate systems and so this space-time symmetry must be a property of the vector spacewhich describes the physical system. We discuss improper transformations in Section 9.7.

c� 2009 John F. Dawson, all rights reserved. 84

CHAPTER 9. SYMMETRIES 9.1. GALILEAN TRANSFORMATIONS

9.1.1 The Galilean group

We need to show that elements of a Galilean transformation form a group. We write the transformationas: ⌃0 = G(⌃), where ⌃ refers to the coordinate system and G = (R,v,a, ⌧) to the elements describing thetransformation. A group of elements is defined by the following four requirements:

Definition 17 (group). A group G is a set of objects, the elements of the group, which we call G, anda multiplication, or combination, rule for combining any two of them to form a product, subject to thefollowing four conditions:

1. The product G1

G2

of any two group elements must be another group element G3

.2. Group multiplication is associative: (G

1

G2

)G3

= G1

(G2

G3

).3. There is a unique group element I, called the identity, such that I G = G for all G in the group.4. For any G there is an inverse, written G�1 such that G G�1 = G�1 G = I.

We first show that one Galilean transformation followed by a second Galilean transformation is also aGalilean transformation. This statement is contained in the following theorem:

Theorem 17 (Composition rule). The multiplication law for the Galilean group is

G00 = G0G = (R0,v0,a0, ⌧ 0) (R,v,a, ⌧) ,

= (R0R,v0 + R0v,a0 + R0a + v0⌧, ⌧ 0 + ⌧) .(9.4)

Proof. We find:

x0 = Rx + vt + a , t0 = t + ⌧ ,

x00 = R0x0 + v0t0 + a0 = R0Rx + (R0v + v0)t + R0a + v0⌧ + a0

⌘ R00x + v00t + a00

t00 = t0 + ⌧ 0 = t + ⌧ + ⌧ 0 ⌘ t + ⌧ 00

where

R00 = R0R , v00 = R0v + v0

a00 = R0a + v0⌧ + a0 ⌧ 00 = ⌧ 0 + ⌧ .

That is, R00 is also an orthogonal matrix with unit determinant, and v00 and a00 are vectors.

Thus the Galilean group G is the set of all elements G = (R,v,a, ⌧), consisting of ten real parameters,three for the rotation matrix R, three each for boosts v and for space translations a, and one for timetranslations ⌧ .

Definition 18. The identity element is 1 = (1, 0, 0, 0), and the inverse element of G is:

G�1 = (R�1,�R�1v,�R�1(a� v⌧),�⌧) , (9.5)

as can be easily checked.

Thus the elements of Galilean transformations form a group.

Example 26 (Matrix representation). It is easy to show that the following 5 ⇥ 5 matrix representation ofthe Galilean group elements:

G =

0

@

R v a0 1 ⌧0 0 1

1

A , (9.6)

forms a group, where group multiplication is defined to be matrix multiplication: G00 = G0G. Here R isunderstood to be a 3⇥ 3 matrix and v and a are 3⇥ 1 column vectors.



Remark 14. An infinitesimal Galilean transformation of the coordinate system is given in vector notationby:

�x = �✓ x⇥ n + �v t + �a ,

�t = �⌧ .(9.7)

The elements of the transformation are given by 1 + �G, where �G = (�✓,�v,�a,�⌧ ).

Example 27. We can find di↵erential representations of the generators of the transformation in classicalphysics. We start by considering complex functions (x, t) which transform “like scalars” under Galileantransformations, that is:

0(x0, t0) = (x, t) . (9.8)

For infinitesimal transformations, this reads:

0(x0, t0) = (x0 ��x, t0 � �t) = (x0, t0)��x ·r0 (x0, t0)��t @t0 (x0, t0) + · · · , (9.9)

and, to first order, the change in functional form of (x, t) is given by:

� (x, t) = ��x ·r + �t @t

(x, t) , (9.10)

Here we have put x0 ! x and t0 ! t. Substituting (9.7) into the above gives:

� (x, t) = ��✓ n · x⇥r + t �v ·r + �a ·r + �⌧ @t

(x, t) . (9.11)

We define the ten di↵erential generator operators (J,K,P, H) of Galilean transformations by

� (x, t) =i

~�

�✓ n · J + �v ·K��a ·P + �⌧ H

(x, t) , (9.12)

Here we have introduced a constant ~ so as to make the units of J, K, P, and H to be the classical units ofangular momentum, impulse, linear momentum, and energy, respectively.1 Comparing (9.11) to (9.12), wefind classical di↵erential representations of the generators:

J =~ix⇥r , K = �~t

ir , P =

~i

r , H = i~@

@t. (9.13)

When acting on complex functions (x, t), these ten generators produce the corresponding changes in thefunctional form of the functions.

Example 28. Using the di↵erential representation (9.13), it is easy to show that the generators obey thealgebra:

[Ji, Jj ] = i~ ✏ijkJk ,

[Ji, Kj ] = i~ ✏ijkKk ,

[Ji, Pj ] = i~ ✏ijkPk ,

[Ki, Kj ] = 0 ,

[Pi, Pj ] = 0 ,

[Ki, Pj ] = 0 ,

[Ji, H] = 0 ,

[Pi, H] = 0 ,

[Ki, H] = i~ Pi .

(9.14)

9.1.2 Group structure

If the generators of a group all commute, then the group is called Abelian. An invariant Abelian subgroupconsists of a subset of generators that commute with each other and whose commutators with any othermember of the group also belong to the subgroup. For the Galilean group, the largest Abelian subgroup isthe six-parameter group U = [L,P] generating boosts and translations. The largest abelian subgroup of thefactor group, G/U , is the group D = [H], generating time translations. This leaves the semi-simple group

1The size of ~ is fixed by the physics.



R = [J], generating rotations. A semi-simple group is one which transform among themselves and cannot bereduced further by removal of an Abelian subgroup. So the Galilean group can be written as the semidirectproduct of a six parameter abelian group U with the semidirect product of a one parameter abelian groupD by a three parameter simple group R,

G = (R⇥D)⇥ U . (9.15)

In contrast, the Poincare group is the simidirect product of a simple group L generating Lorentz transfor-mations by an abelian group C generating space and time translations,

P = L⇥ C . (9.16)

9.2 Galilean transformations in quantum mechanics

Now let | (⌃) i be a vector in V which refers to a specific coordinate system ⌃ and let | (⌃0) i be a vectorwhich refers to the coordinate system ⌃0 = G⌃. Then we know by Wigner’s theorem that:

| (⌃0) i = U(G) | (⌃) i , (9.17)

where U(G) is unitary.2 In non-relativistic quantum mechanics, we want to find unitary transformationsU(G) for the Galilean group. We do this by applying the classical group multiplication properties to unitarytransformations. That is, if (9.17) represents a transformation from ⌃ to ⌃0 by G, and a similar relationholds for a transformation from ⌃0 to ⌃00 by G0, then the combined transformation is given by:

| (⌃00) i = U(G0) | (⌃0) i = U(G0) U(G) | (⌃) i . (9.18)

However the direct transformation from ⌃ to ⌃00 is given classically by G00 = G0G, and quantum mechanicallyby:

| (⌃00) i0 = U(G00) | (⌃) i = U(G0G) | (⌃) i . (9.19)

Now | (⌃00) i and | (⌃00) i0 must belong to the same ray, and therefore can only di↵er by a phase. Thus wecan deduce that:

U(G0)U(G) = ei�(G0,G)/~ U(G0G) , (9.20)

where �(G0, G) is real and depends only on the group elements G and G0. Unitary representations of operatorswhich obey Eq. (9.20) with non-zero phases are called projective representations. If the phase �(G0, G) = 0,they are called faithful representations. The Galilean group generally is projective, not faithful.3 The groupcomposition rule, Eq. (9.20), will be used to find the unitary transformation U(G).

Now we can take the unit element to be: U(1) = 1. So using the group composition rule (9.20), unitarityrequires that:

U†(G)U(G) = U�1(G)U(G) = U(G�1)U(G) = ei�(G�1,G)/~ U(1, 0) = 1 . (9.21)

so that �(G�1, G) = 0. We will use this unitarity requirement in section 9.2.1 below.Infinitesimal transformations are generated from the unity element by the set �G = (�!,�v,�a,�⌧),

where �!ij = ✏ijknk�✓ = ��!ji is an antisymmetric matrix. We write the unitary transformation for thisinfinitesimal transformation as:

U(1 + �G) = 1 +i

~

n

�!ij Jij/2 + �vi Ki ��ai Pi + �⌧ Ho

+ · · ·

= 1 +i

~

n

�✓ n · J + �v ·K��a ·P + �⌧ Ho

+ · · · ,(9.22)

2We will consider anti-unitary symmetry transformations later.3In contrast, the Poincare group is faithful.



where Ji, Ki, Pi and H are operators on V which generater rotations, boosts, and space and time translations,respectively. Here �!ij = ✏ijk nk �✓ is an antisymmetric matrix representing an infinitesimal rotation aboutan axis defined by the unit vector nk by an angle �✓. In a similar way, we write the antisymmetric matrixof operators Jij as Jij = ✏ijkJk, where Jk is a set of three operators.

Remark 15. Again, we have introduced a constant ~ so that the units of the operators J, K, P, and H aregiven by units of angular momentum, impulse, linear momentum, and energy, respectively. The value of ~must be fixed by experiment.4

Remark 16. The sign of the operators Pi and H, relative to Jk in (9.22) is arbitrary — the one we havechosen is conventional.

In the next section, we find the phase factor �(G0;G) in Eq. (9.20) for unitary representations of theGalilean group.

9.2.1 Phase factors for the Galilean group.

The phases �(G0, G) must obey basic properties required by the transformation rules. Since U�1(G)U(G) =U(G�1)U(G) = 1, we find from the unitarity requirement (9.21),

�(G�1, G) = 0 . (9.23)

Also, the associative law for group transformations,

U(G00) (U(G0)U(G)) = (U(G00)U(G0)) U(G) ,

requires that�(G00, G0G) + �(G0, G) = �(G00, G0) + �(G00G0, G) . (9.24)

From (9.23) and (9.24), we easily obtain �(1, 1) = �(1, G) = �(G, 1) = 0. Eqs. (9.23) and (9.24) are thedefining equations for the phase factor �(G0, G), and will be used in Bargmann’s theorem (18) to find thephase factor below.

Note that (9.23) and (9.24) can be satisfied by any �(G0, G) of the form

�(G0, G) = �(G0G)� �(G0)� �(G) . (9.25)

Then the phase can be eliminated by a trivial change of phase of the unitary transformation, U(G) =ei�(G)U(G). Thus two phases �(G0, G) and �0(G0, G) which di↵er from each other by functions of theform (9.25) are equivalent. For Galilean transformations, unlike the case for the Poincare group, the phase�(G0, G) cannot be eliminated by a simple redefinition of the unitary operators. This phase makes the studeof unitary representations of the Galilean group much harder than the Poincare group in relativistic quantummechanics.

It turns out that the phase factors for the Galilean group are not easy to find. The result is stated in atheorem due to Bargmann[2]:

Theorem 18 (Bargmann). The phase factor for the Galilean group is given by:

�(G0, G) =M

2{v0 ·R0(a)� v0 ·R0(v) ⌧ � a0 ·R0(v) } , (9.26)

with M any real number.

4Plank introduced ~ in order to make the classical partition function dimensionless. The value of ~ was fixed by theexperimental black-body radiation law.



Proof. A proper Galilean transformation is given by Eq. (9.3). The group multiplication rules are given inEq. (9.4):

R00 = R0R ,

v00 = v0 + R0(v) ,

a00 = a0 + v0⌧ + R0(a) ,

⌧ 00 = ⌧ 0 + ⌧ .

(9.27)

We first note that v and a transform linearly. Therefore, it is useful to introduce a six-component columnmatrix ⇠ and a 6⇥ 6 matrix ⇥(⌧), which we write as:

⇠ =✓

va

◆

, ⇥(⌧) =✓

1 0⌧ 1

◆

, (9.28)

so that we can write the group multiplication rules for these parameters as:

⇠00 = ⇥(⌧) ⇠0 + R0 ⇠ , (9.29)

which is linear in the ⇠ variables. We label the rest of the parameters by g = (R, ⌧), which obey the groupmultiplication rules:

R00 = R0R , ⌧ 00 = ⌧ 0 + ⌧ . (9.30)

We note here that the unit element of g is g = (1, 0). We also note that the matrices ⇥(⌧) are a faithfulrepresentation of the subgroup of ⌧ transformations. That is, we find:

⇥(⌧ 00) = ⇥(⌧ 0) ⇥(⌧) . (9.31)

We seek now the form of �(G0, G) by solving the defining equation (9.24):

�(G00, G0G) + �(G0, G) = �(G00, G0) + �(G00G0, G) . (9.32)

The only way this can be satisfied is if �(G0, G) is bilinear in ⇠, because the transformation of these variablesis linear. Thus we make the Ansatz:

�(G0, G) = ⇠0T �(g0, g) ⇠ , (9.33)

where �(g0, g) is a 6⇥ 6 matrix, but depends only on the elements g and g0. We now work out all four termsin Eq. (9.32). We find:

�(G00, G0G) = ⇠00T �(g00, g0g)⇥

⇥(⌧) ⇠0 + R0 ⇠⇤

= ⇠00T �(g00, g0g) ⇥(⌧) ⇠0 + ⇠00T �(g00, gg)R0 ⇠ ,

�(G0, G) = ⇠0T �(g0, g) ⇠ ,

�(G00, G0) = ⇠00T �(g00, g0) ⇠0 ,

�(G00G0, G) =⇥

⇠0T R00T + ⇠00T ⇥T (⌧ 0)⇤

�(g00g0, g) ⇠

= ⇠0T R00T �(g00g0, g) ⇠ + ⇠00T ⇥T (⌧ 0) �(g00g0, g) ⇠ .

(9.34)

Substituting these results into (9.32), and equating coe�cients for the three bilinear forms, we find for thethree pairs: (⇠0; ⇠), (⇠00; ⇠0), and (⇠00; ⇠):

�(g0, g) = R00T �(g00g0, g) , (9.35)�(g00, g0g) ⇥(⌧) = �(g00, g0) (9.36)

�(g00, g0g) R0 = ⇥T (⌧ 0) �(g00g0, g) . (9.37)



These relations provide functional equations for the matrix elements. We start by using the orthogonalityof R and writing (9.35) in the form:

�(g00g0, g) = R00 �(g0, g) (9.38)

Since g0 is arbitrary, we can set it equal the unit element: g0 = (1, 0). Then g00g0 = g00, and we find:

�(g00, g) = R00 �(1, g) . (9.39)

When this result is substituted into (9.36) and (9.37), we find:

R00 �(1, g0g) ⇥(⌧) = R00 �(1, g0) (9.40)

R00 �(1, g0g) R0 = ⇥T (⌧ 0) R00R0 �(1, g) . (9.41)

and from (9.40), we find:�(1, g0g) ⇥(⌧) = �(1, g0) . (9.42)

Here g0 is arbitrary, so that we can it to the unit element: g0 = 1, and find:

�(1, g) ⇥(⌧) = �(1, 1) . (9.43)

Now in (9.41), R00 and R0 act only on vectors and commute with the matrices ⇥ and �, so we can write thisas:

�(1, g0g) = ⇥T (⌧ 0) �(1, g) . (9.44)

Again in (9.44), we can set g = 1, from which we find:

�(1, g0) = ⇥T (⌧ 0) �(1, 1) . (9.45)

So combining (9.43) and (9.45), we find that �(1, 1) must satisfy the equation:

�(1, 1) = ⇥T (⌧) �(1, 1) ⇥(⌧) , (9.46)

for all values of ⌧ . Which means that �(1, 1) must be a constant 6 ⇥ 6 matrix, independent of ⌧ . In orderto solve (9.46), we write out �(1, 1) in component form:

�(1, 1) =✓

�11

�12

�21

�22

◆

, (9.47)

so that (9.46) requires:

�11

= �11

+ ⌧ (�12

+ �21

) + ⌧2 �22

, (9.48)�

12

= �12

+ ⌧ �22

, (9.49)�

21

= �21

+ ⌧ �22

, (9.50)�

22

= �22

, (9.51)

which must hold for all values of ⌧ . This is possible only if �22

= 0, and that �21

= ��12

. �11

is thenarbitrary. So let us put �

12

= M/2 and �11

= M 0/2. So the general solution for the phase matrix containstwo constants. We write the result as:

�(1, 1) =M

2Z +

M 0

2Z 0 , where Z =

✓

0 1�1 0

◆

, Z 0 =✓

1 00 0

◆

, (9.52)

From Eqs. (9.33), (9.39), and (9.45), we find:

�(G0, G) = ⇠0T �(g0, g) ⇠ , �(g0, g) = ⇥T (⌧) �(1, 1) R0 . (9.53)



Recall that R0 commutes with ⇥(⌧) and �(1, 1). It turns out that the term involving M 0Z 0 is a trivial phase.For this term, we find:

�Z0(G0, G) =M 0

2⇠0T ⇥T (⌧) Z 0R0(⇠)

=M 0

2v0 ·R0(v) =

M 0

4�

v00 2 � v2 � v0 2

,

(9.54)

So (9.54) is a trivial phase and can be absorbed into the definition of U(g). So then from Eq. (9.52), thephase is given by:

�(G0, G) = +M

2⇠0T ⇥T (⌧) Z R0(⇠) = �M

2[R0(⇠) ]T Z ⇥(⌧) ⇠0 .

=M

2{v0 ·R0(a)� v0 ·R0(v) ⌧ � a0 ·R0(v) } ,

(9.55)

which is what we quoted in the theorem. In the first line, we have used the fact that Z is antisymmetric:ZT = �Z. This phase is non-trival! For example, we might try to do the same tricks we used for the trivalphase in Eq. (9.54), and write:

⇠00T Z ⇠00 =�

[R0(⇠) ]T + ⇠0T ⇥T (⌧)

Z�

⇥(⌧) ⇠0 + R0(⇠)

= ⇠0T Z ⇠0 + ⇠T Z ⇠ + ⇠0T ⇥T (⌧) Z R0(⇠) + [ R0(⇠) ]T Z ⇥(⌧) ⇠0 .(9.56)

But the last two terms cancel rather than add because of the antisymmetry of Z. So we cannot turn (9.55)into a trival phase the way we did for (9.54). This completes the proof.

Remark 17. Bargmann gave this phase in his classic paper on continuous groups[2], and indicated howhe found it in a footnote to that paper. Notice that M appears here as an undetermined multiplicativeparameter. Since we have introduced a constant ~ with the dimensions of action in the definition of thephase, M has units of mass.

We can write the phase as:

�(G0, G) = 1

2

M R0ij{v0iaj � a0ivj � v0ivj⌧ ] (9.57)

Notice that �(G�1, G) = 0.The phase for infinitesimal transformations are given by:

�(G, 1 + �G) = 1

2

M Rij [vi�aj � ai�vj ] + · · · , (9.58)

�(1 + �G, G) = 1

2

M [�vi(ai � vi⌧)��aivi] + · · · ,

Next, we find the transformation properties of the generators.

9.2.2 Unitary transformations of the generators

In this section, we find the unitary transformation U(G) for the generators of the Galilean group. We startby finding the transformation rules for all the generators. This is stated in the following theorem:

Theorem 19. The generators transform according to the rules:

U†(G)JU(G) = R{J + K⇥ v + a⇥ (P + M v)} , (9.59)

U†(G)KU(G) = R{K� (P + M v) ⌧ + M a} , (9.60)

U†(G)PU(G) = R{P + M v} , (9.61)

U†(G) H U(G) = H + v ·P + 1

2

Mv2 . (9.62)

where v = R�1(v) and a = R�1(a).



Proof. We start by considering the transformations:

U†(G) U(1 + �G)U(G) , (9.63)

where G and 1 + �G are two di↵erent transformations. On one hand, using the definition (9.22) forinfinitesimal transformations in terms of the generators, (9.63) is given by:

1 +i

2~�!ij U†(G)Jij U(G) +

i

~�vi U†(G) Ki U(G) (9.64)

� i

~�ai U†(G) Pi U(G) +

i

~�⌧ U†(G) H U(G) + · · ·

On the other, using the composition rule (9.20), Eq. (9.63) can be written as:

ei[�(G�1,(1+�G)G)+�((1+�G),G)]/~ U(G�1(1 + �G)G) (9.65)

= ei�(G,�G)/~ U(1 + �G0) .

where �G0 = G�1 �G G. Working out this transformation, we find the result:

�!0ij = RkiRlj �!kl ,

�v0i = Rji (�!jk vk + �vj) ,

�a0i = Rji (�!jk ak + �vj⌧ + �aj � vj�⌧)�⌧ 0 = �⌧ ,

and the phase �(G,�G) is defined by:

�(G,�G) = �(G�1, (1 + �G)G) + �(1 + �G, G) . (9.66)

We can simplify the calculation of the phase using an identity derived from (9.24):

�(G, G�1(1 + �G)G) + �(G�1, (1 + �G)G)

= �(G, G�1) + �(GG�1, (1 + �G)G) = �(1, (1 + �G)G) = 0 ,

and therefore, since G�1(1 + �G)G = 1 + �G0, we have:

�(G�1, (1 + �G)G) = ��(G, 1 + �G0) .

So the phase �(G,�G) is given by:

�(G,�G) = �(1 + �G, G)� �(G, 1 + �G0) . (9.67)

Now using (9.58), we find to first order:

�(1 + �G, G) = 1

2

M [�vi(ai � vi⌧)��aivi] + · · · ,

�(G, 1 + �G0) = 1

2

M Rij [vi�a0j � ai�v0j ] + · · · ,

= 1

2

M {vi(�!ijaj + �vi⌧ + �ai � v2�⌧)� ai(�vi + �!ijvj)}+ · · · ,

from which we find,

�(G,�G) = 1

2

�!ij M(aivj � ajvi) + �vi M(ai � vi⌧)��ai Mvi (9.68)

+ �⌧ 1

2

Mv2 + · · · .



For the unitary operator U(1 + �G0), we find:

U(1 + �G0) = 1 +i

2~�!0ij Jij +

i

~�v0i Ki � i

~�a0i Pi +

i

~�⌧ 0H + · · · ,

= 1 +i

2~�!ij [RikRjlJkl + 2Ril(vjKl � ajPl)]

+i

~�vi Rij(Kj � ⌧Pj)� i

~�ai RijPj

+i

~�⌧ (H + RijviPj) + · · · , (9.69)

Combining relations (9.68) and (9.69), we find, to first order, the expansion:

ei�(G,�G)/~ U(1 + �G0)

= 1 +i

2~�!ij [RikRjlJkl + 2Ril(vjKl � ajPl) + M(aivj � ajvi)]

+i

~�vi [Rij(Kj � ⌧Pj) + M(ai � vi⌧)]

� i

~�ai [RijPj + Mvi]

+i

~�⌧ [H + RijviPj + 1

2

Mv2] + · · · , (9.70)

Comparing coe�cients of �!ij , �vi, �ai, and �⌧ in (9.64) and (9.70), we get:

U†(G) Jij U(G) = RikRjlJkl + 2Ril(vjKl � ajPl) + M(aivj � ajvi)= RikRjlJkl + (K 0

ivj �K 0jvi)� (P 0iaj � P 0jai) + M(aivj � ajvi)

U†(G) Ki U(G) = Rij(Kj � ⌧Pj) + M(ai � vi⌧)

U†(G) Pi U(G) = RijPj + Mvi

U†(G) H U(G) = H + viP0i + 1

2

Mv2

where, K 0i = RijKj and P 0i = RijPj . In the second line, we have used the antisymmetry of Jij . These

equations simplify if we rewrite them in terms of the components of the angular momentum vector Jk ratherthan the antisymmetric tensor Jij . We have the definitions:

Jij = ✏ijkJk ,

K 0ivj �K 0

jvi = ✏ijk[K0 ⇥ v]k ,

P 0iaj � P 0jai = ✏ijk[P0 ⇥ a]k ,

viaj � vjai = ✏ijk[v ⇥ a]k .

The identity,RikRjl ✏klm = det[R ] ✏ijnRnm , (9.71)

is obtained from the definition of the determinant of R and the orthogonality relations for R. For propertransformations, which is what we consider here, det[ R ] = 1. So the above equations become, in vectornotation,

U†(G)JU(G) = R{J + K⇥ v + a⇥ (P + M v)} ,

U†(G)KU(G) = R{K� (P + M v) ⌧ + M a} ,

U†(G)PU(G) = R{P + M v} ,

U†(G) H U(G) = H + v ·P + 1

2

Mv2 .

where v = R�1(v) and a = R�1(a). This completes the proof of the theorem, as stated.



Exercise 8. Using the indentity (9.71) with det[R ] = +1, show that R(A⇥B) = R(A)⇥R(B).

We next turn to a discussion of the commutation relations for the generators.

9.2.3 Commutation relations of the generators

In this section, we prove a theorem which gives the commutation relations for the generators of the Galileangroup. The set of commutation relations for the group can be thought of as rules for “multiplying” any twooperators, and are called a Lie algebra.

Theorem 20. The ten generators of the Galilean transformation satisfy the commutation relations:




[Ki, Kj ] = 0 ,

[Pi, Pj ] = 0 ,

[Ki, Pj ] = i~ M�ij ,

[Ji, H] = 0 ,

[Pi, H] = 0 ,

[Ki, H] = i~ Pi .

(9.72)

Proof. The proof starts by taking each of the transformations U(G) in theorem 19 to be infinitesimal.These infinitesimal transformations have nothing to do with the infinitesimal transformations in the previoustheorem — they are di↵erent transformations. We start with Eq. (9.59) where we find, to first order:

⇢

1� i

~Jk�✓k � i

~Kk�vk +

i

~Pk�ak � i

~H�⌧ + · · ·

�

⇥ Ji

⇢

1 +i

~Jk�✓k +

i

~Kk�vk � i

~Pk�ak +

i

~H�⌧ + · · ·

�

= Ji + ✏ijkJj�✓k + ✏ijkKj�vk + ✏kji�akPj + · · · .

Comparing coe�cients of �✓k, �vk, �ak, and �⌧ , we find the commutators of Ji with all the other gener-ators:




[Ji, H] = 0 .

From (9.60), we find, to first order:⇢

1� i

~Jk�✓k � i

~Kk�vk +

i

~Pk�ak � i

~H�⌧ + · · ·

�

⇥ Ki

⇢

1 +i

~Jk�✓k +

i

~Kk�vk � i

~Pk�ak +

i

~H�⌧ + · · ·

�

= Ki + ✏ijkKj�✓k + M�ai � Pi�⌧ + · · · ,

from which we find the commutators of Ki will all the generators. In addition to the ones found above, weget:

[Ki, Kj ] = 0 , [Ki, Pj ] = i~ M�ij , [Ki, H] = i~ Pi .

The commutators of Pi with the generators are found from (9.61). We find, to first order:⇢

1� i

~Jk�✓k � i

~Kk�vk +

i

~Pk�ak � i

~H�⌧ + · · ·

�

⇥ Pi

⇢

1 +i

~Jk�✓k +

i

~Kk�vk � i

~Pk�ak +

i

~H�⌧ + · · ·

�

= Pi + ✏ijkPj�✓k + M�vi + · · · ,



from which we find the commutators of Ki with all the generators. In addition to the ones found above, weget:

[Pi, Pj ] = 0 , [Pi, H] = 0 .

The last commmutation relations of H with the generators confirm the previous results. This completes theproof.

The phase parameter M is called a central charge of the Galilean algebra.

9.2.4 Center of mass operator

For M 6= 0, it is useful to define operators which describes the location and velocity of the center of mass:

Definition 19. The center of mass operator X is defined at t = 0 by X = K/M . We also define the velocityof the center of mass as V = P/M .

If no external forces act on the system, the center of mass changes in time according to:

X(t) = X + V t . (9.73)

There can still be internal forces acting on various parts of the system: we only assume here that the centerof mass of the system as a whole moves force free. Using the transformation rules from Theorem 19, X(t)transforms according to:

U†(G)X(t0) U(G) = U†(G) {K + P (t + ⌧) }U(G)/M= R{K� (P + M v) ⌧ + M a + P (t + ⌧) + M v (t + ⌧)}/M= R{K + P t}/M + v t + a= RX(t) + v t + a , where t0 = t + ⌧ .

(9.74)

Di↵erentiating (9.74) with respect to t0, we find:

U†(G) X(t0)U(G) = RX(t) + v ,

U†(G) X(t0)U(G) = RX(t) ,

so the acceleration of the center of mass is an invariant.We can rewrite the transformation rules and commutation relations of the generators of the Galilean

group using X = K/M and V = P/M rather than K and P. From Eqs. (9.59–9.62), we find:

U†(G)JU(G) = R{J + MX⇥ v + M a⇥ (V + v)}= R{J + M (X + a)⇥ v + M a⇥V} ,

U†(G)XU(G) = R{X� (V + v) ⌧ + a} ,

U†(G)V U(G) = R{V + v} ,

U†(G) H U(G) = H + M v ·V + 1

2

Mv2 .

(9.75)

where v = R�1(v) and a = R�1(a). Eqs. (9.72) become:


[Ji, Xj ] = i~ ✏ijkXk ,


[Xi, Xj ] = 0 ,

[Pi, Pj ] = 0 ,

[Xi, Pj ] = i~ �ij ,

[Ji, H] = 0 ,

[Pi, H] = 0 ,

[Xi, H] = i~ Vi .

(9.76)



Remark 18. For a single particle, the center of mass operator is the operator which describes the location ofthe particle. The existence of such an operator means that we can localize a particle with a measurement ofX. The commutation relations between X and the other generators are as we might expect from the canonicalquantization postulates which we study in the next chapter. Here, we have obtained these quantization rulesdirectly from the generators of the Galilean group, and from our point of view, they are consequences ofrequiring Galilean symmetry for the particle system, and are not additional postulates of quantum theory.We shall see in a subsequent chapter how to construct quantum mechanics from classical actions.Remark 19. Since in the Cartesian system of coordinates, X and P are Hermitian operators, we can alwayswrite an eigenvalue equation for them:

X |x i = x |x i , (9.77)P |p i = p |p i , (9.78)

where xi and pi are real continuous numbers in the range �1 < xi < 1 and �1 < pi < 1. In Section 9.4below, we will find a relationship between these two di↵erent basis sets.

9.2.5 Casimir invariants

Casimir operators are operators that are invariant under the transformation group and commute with allthe generators of the group. The Galilean transformation is rank two, so we know from a general theoremin group theory that there are just two Casimir operators. These will turn out to be what we will call theinternal energy W and the magnitude of the spin S, or internal angular momentum. We start with theinternal energy operator.

Definition 20 (Internal energy). For M 6= 0, we define the internal energy operator W by:

W = H � P 2

2M. (9.79)

Theorem 21. The internal energy, defined Eq. (9.79), is invariant under Galilean transformations:

Proof. Using Theorem 19, we have:

U†(G) W U(G) = H + v ·P + 1

2

Mv2 � [R{P + M v}]22M

= H � P 2

2M= W ,

as required.

The internal energy operator W is Hermitian and commutes with all the group generators, its eigenvaluesw can be any real number. So we can write:

H = W +P 2

2M. (9.80)

The orbital and spin angular momentum operators are defined by:

Definition 21 (Orbital angular momentum). For M 6= 0, we define the orbital angular momentum by:

L = X⇥P = (K⇥P)/M . (9.81)

The orbital angular momentum of the system is independent of time:

L(t) = X(t)⇥P(t) = {X + Pt/M}⇥P = X⇥P = L . (9.82)



Definition 22 (Spin). For M 6= 0, we define the spin, or internal angular momentum by:

S = J� L , (9.83)

where L is defined in Eq. (9.81).

The spin is what is left over after subracting the orbital angular momentum from the total angularmomentum. Since the orbital angular momentum is not defined for M = 0, the same is true for the spinoperator. However for M 6= 0, we can write:

J = L + S . (9.84)

The following theorem describes the transformation properties of the orbital and spin operators.

Theorem 22. The orbital and spin operators transform under Galilean transformations according to therule:

U†(G)LU(G) = R{L + X⇥ ( M v ) + ( a� (V + v ) ⌧ )⇥P } , (9.85)

U†(G)SU(G) = R{S } , (9.86)

and obeys the commutation relations:

[Li, Lj ] = i~ ✏ijkLk , [Si, Sj ] = i~ ✏ijkSk , [Li, Sj ] = 0 . (9.87)

Proof. The orbital results are easy to prove using results from Eqs. (9.75). For the spin, using theorem 19,we find:

U†(G)SU(G) = R{J + K⇥ v + a⇥ (P + M v) }�R{K� (P + M v) ⌧ + M a }⇥R{P + M v}/M

= R{J + K⇥ v + a⇥ (P + M v)� [K� (P + M v) ⌧ + M a ]⇥ [P + M v ]/M }

= R{J� (K⇥P)/M } = R{S } ,

as required. The commutator [Li, Jj ] = 0 is easy to establish. For [ Li, Lj ], we note that:

[Li, Lj ] = ✏inm✏jn0m0 [XnPm, Xn0Pm0 ]

= ✏inm✏jn0m0�

Xn0 [Xn, Pm0 ]Pm + Xn [Pm, Xn0 ]Pm0

= i~ ✏inm✏jn0m0�

�n,m0 Xn0 Pm � �n0,m Xn Pm0

= i~�

✏inm✏jn0n Xn0 Pm � ✏inm✏jmm0 Xn Pm0

= i~�

( �mj�in0 � �mn0�ij )Xn0 Pm � ( �im0�nj � �ij�nm0 ) Xn Pm0

= i~�

Xi Pj � �ij ( Xm Pm )�Xj Pi + �ij ( Xn Pn )

= i~�

Xi Pj �Xj Pi

= i~ ✏ijk Lk ,

(9.88)

as required. The last commutator [Si, Sj ] follows directly from the commutator results for Ji and Li.

Remark 20. Additionally, we note that [ Si, Xj ] = [Si, Pj ] = [Si, H ] = 0.Remark 21. So this theorem showns that even under boosts and translations, in addition to rotations, thespin operator is sensitive only to the rotation of the coordinate system, which is not true for either theorbital angular momentum or the total angular momentum operators. However the square of the spin vectoroperator S2, is invariant under general Galilean transformations,

U�1(G) S2 U(G) = S2 , (9.89)

and is the second Casimir invariant. In Section ??, we will find that the possible eigenvalues of S2 are givenby: s = 0, 1/2, 1, 3/2, 2, . . ..



Remark 22. To summerize this section, we have found two hermitian Casimir operators, W and S2, whichare invariant under the group G. We can therefore label the irreducible representations of G by the set ofquantities: [M |w, s], where w and s label the eigenvalues of these operators, and M the central charge.

So we can find common eigenvectors of W , S2, and either X or P. We write these as:

| [M |w, s];x,� i , and | [M |w, s];p,� i . (9.90)

Here � labels the component of spin. The latter eigenvector is also an eigenvector of H, with eigenvalue:

H | [M |w, s];p,� i = Ew,p | [M |w, s];p,� i , Ew,p = w +p2

2M. (9.91)

We discuss the massless case in Section 9.2.8.

9.2.6 Extension of the Galilean group

If we wish, we may extend the Galilean group by considering M to be a generator of the group. This isbecause the phase factor �(G0, G) is linear in M and M commutes with all elements of the group. Thus wecan invent a new group element ⌘ and write:

G = (G, ⌘) = (R,v,a, ⌧, ⌘) , (9.92)

and which transforms according to the rule:

G0G = (G0G, ⌘0 + ⌘ + ⇠(G0, G)) , (9.93)

where ⇠(G0, G) is the coe�cient of M in (9.26)

⇠(G0, G) = �12{v0 ·R0 v ⌧ + a0 ·R0 v � v0 ·R0 a} . (9.94)

The infinitesimal unitary operators in Hilbert space become:

U(1 + �G) = 1 +i

~{J · n ✓ + K · v �P · a + H⌧ + M⌘ }+ · · · , (9.95)

and since M is now regarded as a generator and ⌘ as a group element, the extended eleven parameter Galileangroup can now be represented as a true unitary representation rather than a projective representation: thephase factor has been redefined as a transformation property of the extended group element ⌘, and the phaseM redefined as a operator.

For the extended Galilean group G with M 6= 0, the largest abelian invariant subgroup is now the fivedimensional subgroup C = [P, H, M ] generating space and time translations plus ⌘. The abelian invariantsubgroup of the factor group G/C is then the three parameter subgroup V = [K] generating boosts, leavingthe semi-simple three-dimensional group of rotations R = [R]. So the extended Galilean group has theproduct structure:

G = (R⇥ V)⇥ C . (9.96)

Here the subgroup R⇥ V = [J,K] generates the six dimensional group of rotations and boosts.

9.2.7 Finite dimensional representations

We examine in this section finite dimensional representations of the subgroup R ⇥ V = [J,K] of rotationsand boosts. These generators obey the subalgebra:

[Ji, Jj ] = i~ ✏ijkJk , [Ji, Kj ] = i~ ✏ijkKk , [Ki, Kj ] = 0 . (9.97)



In order to emphasize that what we are doing here is completely classical, let us define:

Ji =~2

⌃i , Ki =~2

�i , (9.98)

in which case ⌃i and �i satisfy the algebra:

[ ⌃i,⌃j ] = 2 i ✏ijk⌃k , [ ⌃i,�j ] = 2 i ✏ijk�k , [ �i,�j ] = 0 . (9.99)

which eliminates ~. It is simple to find a 4 ⇥ 4 matrix representation of ⌃i and �i. We find two suchcomplimentary representations:

⌃i =✓

�i 00 �i

◆

, �(+)

i =✓

0 0�i 0

◆

, �(�)

i =✓

0 �i

0 0

◆

, (9.100)

both of which satisfy the set (9.99):

[ ⌃i,�(±)

j ] = 2 i ✏ijk�(±)

k , �(±)

i �(±)

j = 0 . (9.101)

We also find:[ �(+)

i ,�(�)

j ] = �ij I + i ✏ijk ⌃k . (9.102)

In addition, [ �(�)

i ]† = �(+)

i so �(±)

i is not Hermitian. Nevertheless, we can define finite transformations byexponentiation. Let us define a rotation operator U(R) by:

U(R) = eiˆn·⌃ ✓/2 = I cos ✓/2 + i(n ·⌃) sin ✓/2 , (9.103)

and boost operators V (±)(v) by:

V (+)(v) = ev·�(+)/2 = I + v · �(+)/2 =✓

1 0� · v/2 1

◆

, (9.104)

andV (�)(v) = ev·�(�)/2 = I + v · �(�)/2 =

✓

1 � · v/20 1

◆

. (9.105)

These last two equations follow from the fact that �(±)

i �(±)

j = 0. For this same reason,

V (±)(v0) V (±)(v) = V (±)(v0 + v) . (9.106)

We can easily construct the inverses of V (±)(v). We find:

[V (±)(v) ]�1 = V (±)(�v) = e�v·�(±)/2 = I � v · �(±) . (9.107)

So the inverses of V (±)(v) are not the adjoints. This means that the V (±)(v) operators are not unitary.We now define combined rotation and boost operators by:

⇤(±)(R,v) = V (±)(v) U(R) , [ ⇤(±)(R,v) ]�1 = U†(R) [V (±)(v) ]�1 = U†(R)V (⌥)(v) . (9.108)

We find the results:

U†(R) ⌃i U(R) = Rij ⌃j ,

U†(R) �(±)

i U(R) = Rij �(±)

j ,

[V (±)(v) ]�1 ⌃i V (±)(v) = ⌃i � 2i ✏ijk �(±)

j vk ,

[V (±)(v) ]�1 �(±)

i V (±)(v) = �(±)

i ,

U†(R) V (±)(v) U(R) = V (±)(R�1(v)) .

(9.109)



So for the combined transformation,

[ ⇤(±)(R,v) ]�1 ⌃⇤(±)(R,v) = R(⌃ )� 2i R(�(±) )⇥ v ,

[ ⇤(±)(R,v) ]�1 �(±) ⇤(±)(R,v) = R(�(±) ) .(9.110)

Comparing (9.110) with the transformations of J and K in Theorem 19, we see that ⇤(±)(R,v) are adjointrepresentations of the subgroup rotations and boosts, although not unitary ones. The replacement v ! �ivis a reflection of the fact that V (±)(v) is not unitary. The ⇤(±)(R,v) matrices are faithful representationsof the (R,v) subgroup of the Galilean group:

⇤(±)(R0,v0) ⇤(±)(R,v) = V (±)(v0) U(R0) V (±)(v) U(R)

= V (±)(v0)�

U(R0) V (±)(v) U†(R0)

U(R0) U(R)

= V (±)(v0) V (±)(R0(v)) U(R0R) = V (±)(v0 + R0(v)) U(R0R)

= ⇤(±)(R0R,v0 + R0(v)) .

(9.111)

We can, in fact, display an explicit Galilean transformation for the subgroup consisting of the (R,v) elements.Let us define two 4⇥ 4 matrices X(±)(x, t) by:

Definition 23.

X(+)(x, t) =✓

t 0x · � �t

◆

, X(�)(x, t) =✓�t x · �

0 t

◆

. (9.112)

Then we can prove the following theorem:

Theorem 23. The matrices X(±)(x, t) transform under the subgroup of rotations and boosts according to:

⇤(±)(R,v) X(±)(x, t) [⇤(±)(R,v) ]�1 = X(±)(x0, t0) , (9.113)

where x0 = R(x) + vt and t0 = t.

Proof. This remarkable result is an alternative way of writing Galilean transformations for the subgroup ofrotations and boosts in terms of transformations of 4⇥ 4 matrices in the “adjoint” representation. With theabove definitions, the proof is straightforward and is left for the reader.

Exercise 9. Prove Theorem 23.

In this section, we have found two 4⇥4 dimensional matrix representations of the Galilean group. Theserepresentations turned out not to be unitary. Finite dimensional representations of the Lorentz group inrelativistic theories are also not unitary. Nevertheless, finite representations of the Galilean group will beuseful when discussing wave equations.

9.2.8 The massless case

When M = 0, the phase for unitary representations of the Galilean group vanish, and the representationbecomes a faithful one, which is simpler. For this case, the generators transform according to the equations:

U†(G)JU(G) = R{J + K⇥ v + a⇥P} ,

U†(G)KU(G) = R{K�P ⌧} ,

U†(G)PU(G) = R{P} ,

U†(G) H U(G) = H + v ·P .

(9.114)


CHAPTER 9. SYMMETRIES 9.3. TIME TRANSLATIONS

where v = R�1(v) and a = R�1(a). The generators obey the algebra:




[Ki, Kj ] = 0 ,

[Pi, Pj ] = 0 ,

[Ki, Pj ] = 0 ,

[Ji, H] = 0 ,

[Pi, H] = 0 ,

[Ki, H] = i~ Pi .

(9.115)

We first note that P simply rotates like a vector under the full group, so P 2 is the first Casimir invariant.We also note that if we define W = K⇥P, then

U†(G)W U(G) = R{K�P ⌧}⇥R{P} = R{K⇥P} = R{W} . (9.116)

So W is a second vector which simply rotates like a vector under the full group, so W 2 is also an invariant.We also note that W is perpendicular to both P and K: W ·P = W ·K = 0. Note that W does not satisfyangular momentum commutator relations.

9.3 Time translations

We have only constructed the unitary operator U(1 + �G) for infinitesimal Galilean transformations. Sincethe generators do not commute, we cannot construct the unitary operator U(G) for a finite Galilean transfor-mation by application of a series of infinitesimal ones. However we can easily construct the unitary operatorU(G) for restricted Galilean transformations, like time, space, and boost transformations alone. We do thisin the next two sections.

The unitary operator for pure time translations is given by:

UH(⌧) = limN!1

1 +i

~H ⌧

N

�N

= eiH ⌧/~ . (9.117)

It time-translates the operator X(t) by an amount ⌧ :

U†H(⌧)X(t0) UH(⌧) = X(t) , t0 = t + ⌧ , (9.118)

and leaves P unchanged:U†

H(⌧)PUH(⌧) = P . (9.119)

Invariance of the laws of nature under time translation is a statement of the fact that an experiment withparticles done today will give the same results as an experiment done yesterday — there is no way ofmeasuring absolute time.

We first consider transformations to a frame where we have set the clocks to zero. That is, we put t0 = 0so that ⌧ = �t. Then (9.118) becomes:

X(t) = UH(t)XU†H(t) = X + V t . (9.120)

where X = K/M and V = P/M . From Eq. (9.120), we find:

X(t) {UH(t) |x i} = UH(t)X |x i = x {UH(t) |x i} . (9.121)

So if we define the ket |x, t i by:|x, t i = UH(t) |x i = eiHt/~ |x i , (9.122)

then (9.121) becomes an eigenvalue equation for the operator X(t) at time t:

X(t) |x, t i = x |x, t i , x 2 R3 . (9.123)

Note that the eigenvalue x of this equation is not a function of t. It is just a real vector.


9.4. SPACE TRANSLATIONS AND BOOSTS CHAPTER 9. SYMMETRIES

From Eq. (9.122), we see that the base vector |x, t i satisfies a first order di↵erential equation:

� i~ddt

|x, t i = H |x, t i , (9.124)

and from (9.120), we obtain Heisenberg’s di↵erential equation of motion for X(t):

ddt

X(t) = [X(t), H]/i~ = P/M . (9.125)

The general transformation of the base vectors |x, t i between two frames, which di↵er by clock time ⌧ only,is given by:

|x, t0 i = UH(t0) |x i = UH(t0)U†H(t) |x, t i = UH(⌧) |x, t i , (9.126)

where ⌧ = t0 � t.The inner product of |x, t i with an arbitrary vector | i is given by:

(x, t) = hx, t | i = hx |U†H(t)| i = hx | (t) i , (9.127)

where the time-dependent “state vector” | (t) i is defined by:

| (t) i = U†H(t) | i = e�iHt/~ | i . (9.128)

This state vector satisfies a di↵erential equation given by:

i~ddt| (t) i = H | (t) i , (9.129)

which is called Schrodinger’s equation. This equation gives the trajectory of the state vector in Hilbertspace. Thus, we can consider two pictures: base vectors moving (the Heisenberg picture) or state vectormoving (the Schrodinger picture). They are di↵erent views of the same physics. From our point of view,and remarkably, Schrodinger’s equation is a result of requiring Galilean symmetry, and is not a fundamentalpostulate of the theory.

The state vector in the primed frame is related to that in the unprimed frame by:

| (t0) i = U†H(t0) | i = U†

H(t0)UH(t) | (t) i = U†H(⌧) | (t) i , (9.130)

We next turn to space translations and boosts.

9.4 Space translations and boosts

The unitary operators for pure space translations and pure boosts are built up of infinitesimal transformationsalong any path:

UP

(a) = limN!1

1� i

~P · aN

�N

= e�iP·a/~ , (9.131)

UX

(v) = limN!1

1 +i

~K · v

N

�N

= eiK·v/~ = eiMv·X/~ , (9.132)

The space translation operator UP

(a) is diagonal in momentum eigenvectors, and the boost operator UX

(v)is diagonal in position eigenvectors. From the transformation rules, we have:

U†P

(a)XUP

(a) = X + a , (9.133)

U†X

(v)PUX

(v) = P + Mv . (9.134)


9.4. SPACE TRANSLATIONS AND BOOSTS CHAPTER 9. SYMMETRIES

In a similar way, for pure boosts, p0 = p+Mv, wave functions in momentum space transforms accordingto:

0(p0) = hp0 | 0 i = hp |U†X

(v)UX

(v) | i = hp | i = (p) , (9.147)

and we find:

h p |X | i = �~i

rp

(x) . (9.148)

For the combined unitary operator for space translations and boosts, we note that the combined trans-formations give: (1,v, 0, 0)(1, 0,a, 0) = (1,v,a, 0). So, using Bargmann’s theorem, Eq. (9.26), for the phase,and Eq. (B.16) in Appendix ??, we find the results:

UX,P(v,a) = ei(Mv·X�P·a)/~ = e+i 1

2

Mv·a/~ UP

(a) UX

(v) , (9.149)

= e�i 1

2

Mv·a/~ UX

(v) UP

(a) ,

So for combined space translations and boosts we find:

UX,P(v,a) |x i = e+i 1

2

Mv·a/~ UP

(a) UX

(v) |x i= e+i(Mv·x+

1

2

Mv·a)/~ UP

(a) |x i= e+i(Mv·x+

1

2

Mv·a)/~ |x + a iU

X,P(v,a) |p i = e�i 1

2

Mv·a/~ UX

(v) UP

(a) |p i= e�i(p·a+

1

2

Mv·a)/~ UX

(v) |p i= e�i(p·a+

1

2

Mv·a)/~ |p + Mv i .

Writing x0 = x + a and p0 = p + Mv, and inverting these expressions, we find

|x0 i = e�i(Mv·x+

1

2

Mv·a)/~ UX,P(v,a) |x i , (9.150)

|p0 i = e+i(p·a+

1

2

Mv·a)/~ UX,P(v,a) |p i . (9.151)

For combined transformations, wave functions in coordinate and momentum space transform according tothe rule:

0(x0) = hx0 | 0 i = hx0 |UX,P(v,a) | i = e+i(Mv·x+

1

2

Mv·a)/~ (x) , (9.152)

0(p0) = hp0 | 0 i = hp0 |UX,P(v,a) | i = e�i(p·a+

1

2

Mv·a)/~ (p) . (9.153)

These functions transform like scalars, but with an essential coordinate or momenutm dependent phase,characteristic of Gailiean transformations.

Example 29. It is easy to show that Eq. (9.152), is the Fourier transform of (9.153),

0(x0) =Z

d3p0

(2⇡~)3eip0·x0/~ 0(p0)

= e+i(Mv·x+

1

2

Mv·a)/~Z

d3p

(2⇡~)3eip·x/~ (p) = e+i(Mv·x+

1

2

Mv·a)/~ (x) .

as required by Eq. (9.143).

We discuss the case of combined space and time translations with boosts, but without rotations, inAppendix ??. We turn next to rotations.


CHAPTER 9. SYMMETRIES 9.5. ROTATIONS

9.5 Rotations

In this section, we discuss pure rotations. Because of the importance of rotations and angular momentumin quantum mechanics, this topic is discussed in great detail in Chapter ??. We will therefore restrict ourdiscussion here to general properties of pure rotations and angular momentum algebra.

9.5.1 The rotation operator

The total angular momentum is the sum of orbital plus spin: J = L + S, with [ Li, Sj ] = 0. Commoneigenvectors of these two operators are then the direct product of these two states:

| `, m`; s, ms i = | `, m` i | s, ms i . (9.154)

The rotation operator is given by the combined rotation of orbital and spin operators:

UJ

(R) = eiˆn·J ✓/~ = eiˆn·L ✓/~ eiˆn·S ✓/~ = UL

(R) US

(R) . (9.155)

The orbital rotation operator acts only on eigenstates of the position operator X, or momentum operator P,For pure rotations, the rotation operator can be found by N sequential infinitesimal transformations

�✓ = ✓/N about a fixed axis n:

UJ

(n, ✓) = limN!1

1 +i

~n · J ✓

N

�N

= eiˆn·J ✓/~ . (9.156)

For pure rotations, the Galilean phase factor is zero so that we have:

UJ

(R0)UJ

(R) = UJ

(R0R) . (9.157)

From Theorem 19 and Eq. (9.59), for pure rotations, we have:

U†J

(n, ✓) Ji UJ

(n, ✓) = Rij(n, ✓) Jj ⌘ Ji(n, ✓) . (9.158)

We discuss parameterizations of the rotation matrices R(n, ✓) in Appendix ??. Here Ji(n, ✓) is the ith

component of the operator J evaluated in the rotated system. Setting i = z, we find for the z-component:

Jz(n, ✓) U†J

(n, ✓) = U†J

(n, ✓)Jz (9.159)

We also know that J2 = J2

x + J2

y + J2

z is an invariant:

U†J

(n, ✓) J2 UJ

(n, ✓) = J2 . (9.160)

So from Eq. (9.159), we find that:

Jz(n, ✓)�

U†J

(n, ✓) | j,m i = ~ m�

U†J

(n, ✓) | j,m i , (9.161)

from which we conclude that the quantity in brackets is an eigenvector of Jz(n, ✓) with eigenvalue ~m. Thatis, we can write:

| j,m(n, ✓) i = U†J

(n, ✓) | j,m i . (9.162)

It is also an eigenvector of J2 with eigenvalue ~2 j(j +1). It is useful to define a rotation matrix D(j)m0,m(n, ✓)

by:D(j)

m,m0(n, ✓) = h jm |UJ

(n, ✓) | jm0 i . (9.163)


9.5. ROTATIONS CHAPTER 9. SYMMETRIES

Matrix elements of the rotation operator are diagonal in j. The rotation matrices have the properties:

jX

m0=�j

D(j)m,m0(R) D(j) ⇤

m00,m0(R) = �m,m00 , (9.164)

D(j) ⇤m,m0(R) = D(j)

m0,m(R�1) = (�)m0�m D(j)�m,�m0(R) . (9.165)

We can express | n, ✓; j,m i in terms of the rotation matrices. We write:

| j,m(n, ✓) i =jX

m0=�j

D(j) ⇤m,m0(n, ✓) | j,m0 i . (9.166)

In the coordinate representation of orbital angular momenta, spherical harmonics are defined by: Y`,m(⌦) =h⌦ | `, m i. Using Eq. (9.166), we find:

Y`,m(⌦0) = h⌦0 | `, m i = h⌦ |U†J

(n, ✓) | `, m i

=X

m0=�`

D(`) ⇤m,m0(n, ✓) h⌦ | `, m0 i =

X

m0=�`

D(`) ⇤m,m0(n, ✓) Y`,m0(⌦) ,

(9.167)

where ⌦ and ⌦0 are spherical angles of the same point measured in two di↵erent coordinate systems, rotatedrelative to each other.

9.5.2 Rotations of the basis sets

Now L and therefore J does not commute with either X or P. Therefore they cannot have commoneigenvectors. However S does commute with with both X or P. Supressing the dependence on w and M ,the common eigenvectors are:

|x, sm i , and |p, sm i . (9.168)

A general rotation of the ket |x, sm i can be obtained by first translating to the state where x = 0, then rotat-ing, and then translating back to a rotated state x0 = R(x). That is, (R, 0, 0, 0) = (1,x0, 0, 0)(R, 0, 0, 0)(1,�x, 0, 0).The trick is that the orbital angular momentum operator L acting on a state with x = 0 gives zero, so onthis state J = S. The phases all work out to be zero in this case, so we find:

UJ

(R) |x, sm i = UP

(x0) UJ

(R) UP

(�x) |x, sm i= U

P

(x0) UJ

(R) |0, sm i= U

P

(x0) US

(R) |0, sm i=X

m0

UP

(x0) |0, sm0 iD(s)m0,m(R)

=X

m0

|x0, sm0 iD(s)m0,m(R) . (9.169)

Inverting this expression, we find:

U†J

(R) |x0, sm0 i =X

m

D(s)⇤m0,m(R) |x, sm i , (9.170)

which gives: 0sm0(x0) =

X

m

D(s)m0,m(R) sm(x) , (9.171)

where hx, sm | i = sm(x) with | 0 i = U(R) | i.


CHAPTER 9. SYMMETRIES 9.6. GENERAL GALILEAN TRANSFORMATIONS

9.6 General Galilean transformations

The general Galilean transformation for space and time translations and rotations is given by:

x0 = R(x) + vt + a ,

t0 = t + ⌧ . (9.172)

Starting from the state | sm;x, t i, we generate a full Galilean transformation G = (R,v,a, ⌧) by first doinga time translation back to t = 0, a space translation back to the origin x = 0, then a rotation (which nowcan be done with the spin operator alone), then a space translation to the new value x0, then a boost to thev frame, and finally a time translation forward to t0. This is given by the set:

G = (1, 0, 0, t0)(1,v, 0, 0)(1, 0,x0, 0)(R, 0, 0, 0)(1, 0,�x, 0)(1, 0, 0,�t) ,

= (1, 0, 0, t0)(1,v, 0, 0)(1, 0,x0, 0)(R, 0, 0, 0)(1, 0,�x,�t) ,

= (1, 0, 0, t0)(1,v, 0, 0)(1, 0,x0, 0)(R, 0,�R(x),�t) ,

= (1, 0, 0, t0)(1,v, 0, 0)(R, 0,x0 �R(x),�t) ,

= (1, 0, 0, t0)(R,v,x0 �R(x)� vt,�t) ,

= (R,v,a, ⌧) , (9.173)

as required. The combined unitary transformation for the full Galilean group is then given by:

UH(t0)UX

(v)UP

(x0)UJ

(R)UP

(�x)UH(�t) = eig(x,t)/~ U(G) . (9.174)

The only contribution to the phase comes from between step four and step five in the above. UsingBargmann’s theorem, we find:

g(x, t) =12

M v · (x0 �R(x)) =12

Mv2 t +12

M v · a . (9.175)

So

U(G) |x, t; sm i = e�ig(x,t)/~ UH(t0)UX

(v)UP

(x0)UJ

(R)UP

(�x)UH(�t) |x, t; sm i= e�ig(x,t)/~ UH(t0)U

X

(v)UP

(x0)UJ

(R)UP

(�x) |x, 0; sm i= e�ig(x,t)/~ UH(t0)U

X

(v)UP

(x0)UJ

(R) |0, 0; sm i= e�ig(x,t)/~ UH(t0)U

X

(v)UP

(x0)US

(R) |0, 0; sm i= e�ig(x,t)/~ UH(t0)U

X

(v)UP

(x0)X

m0

|0, 0; sm0 iD(s)m0,m(R)

= e�ig(x,t)/~ UH(t0)UX

(v)X

m0

|x0, 0; sm0 iD(s)m0,m(R)

= e�ig(x,t)/~ UH(t0)X

m0

eiMv·x0 |x0, 0; sm0 iD(s)m0,m(R)

= eif(x,t)/~ X

m0

|x0, t0; sm0 iD(s)m0,m(R) (9.176)

Where we have defined the phase factor �(G) by:

f(x, t) = Mv · x0 � �(G) = Mv · x0 � 12

Mv2 t� 12

M v · a

= Mv ·R(x) +12

Mv2 t +12

M v · a .(9.177)


9.7. IMPROPER TRANSFORMATIONS CHAPTER 9. SYMMETRIES

Inverting Eq. (9.176), we find:

U†(G) |x0, t0; sm0 i = e�if(x,t)/~ X

m

|x, t; sm iD(s) ⇤m0,m(R) . (9.178)

So that: 0sm0(x0, t0) = eif(x,t)/~ X

m

D(s)m0,m(R) sm(x, t) . (9.179)

where sm(x, t) = hx, t; sm | i, and we have put: | 0 i = U(G) | i. It is important to note here that thephase factor f(x, t) depends on x and t, as well as the parameters of the Galilean transformation.

Exercise 10. Find the general Galilean transformation of momentum eigenvectors: |p, sm i. Show that thetransformed functions sm(p) give the same result as as the Fourier transform of Eq. (9.179).

9.7 Improper transformations

In this section we follow Weinberg[?, p. 77]. We first extend the kinds of Galilean transformations we considerto include parity, time reversal, and charge conjugation. The full Galilean transformations are now describedby:

x0 = rR(x) + vt + a , t0 = t + ⌧ . (9.180)

Here r = det[R ] and can have values of ±1. We still require that lengths are preserved so that R isstill orthogonal, and that the rate of passage of time does not dilate or shrink, only the direction of timecan be reversed. So the full group, including improper transformations, is now represented by the twelveparameters:

G = (R,v,a, ⌧, r, ) . (9.181)

The full group properties are now stated in the next theorem.

Theorem 24. The composition rule for the full Galilean group is given by:

G00 = G0G = (R0,v0,a0, ⌧ 0, r0,0 ) (R,v,a, ⌧, r, )= (R0R,v0 + r0R0(v),a0 + v0⌧ + r0R(a),⌧ 0 + ⌧, r0r,0, )

(9.182)

Proof. The proof follows directly from the complete transformation equations (9.180) and left as an exercise.

9.7.1 Parity

In this section we consider parity transformations (space reversals) of the coordinate system. This is repre-sented by the group elements:

GP = (1, 0, 0, 0,�1,+1) . (9.183)

We note that G�1

P = GP . So using the rules given in Theorem 24, we find for the combined transformation:

G0 = G�1

P G GP = (1, 0, 0, 0,�1,+1) (R,v,a, ⌧, r, ) (1, 0, 0, 0,�1,+1)= (R,�v,�a, ⌧, r, ) .

(9.184)

The phase factors are zero in this case. So we have:

P�1 U(G)P = U(G�1

P G GP ) = U(G0) . (9.185)


CHAPTER 9. SYMMETRIES 9.7. IMPROPER TRANSFORMATIONS

Now if we take r = 1 and = 1, both G and G0 are proper. This means that we can take G = 1 + �G,where �G = ( �!,�v,�a,�⌧, 1, 1 ). Then G0 = 1 + �G0, where �G0 = ( �!,��v,��a,�⌧, 1, 1 ). Sothen U(1 + �G) can be represented by:5

U(1 + �G) = 1 +i

~

n

�✓ n · J + �v ·K��a ·P + �⌧ Ho

+ · · · . (9.186)

Using this in Eq. (9.185), we find:

P�1 JP = J ,

P�1 KP = �K ,

P�1 PP = �P ,

P�1 H P = H .

(9.187)

We note that P is linear and unitary, with eigenvalues of unit magnitude. We also have: P�1 = P† = P.We assume that the Casimir invariants M and W remain unchanged by a parity transformation.

Exercise 11. Show that under parity,

P�1 X(t)P = �X(t) , (9.188)

where X(t) = X + V t, where X = K/M and V = P/M .

We discuss the action of parity on eigenvectors of angular momentum in Section 21.1.4.

9.7.2 Time reversal

Time reversal is represented by the group elements:

GT = (1, 0, 0, 0,+1,�1) , (9.189)

with G�1

T = GT . So again using the rules given in Theorem 24, we find for the combined transformation:

G0 = G�1

T G GT = (1, 0, 0, 0,+1,�1) (R,v,a, ⌧, r, ) (1, 0, 0, 0,+1,�1)= (R,�v,a,�⌧, r, ) .

(9.190)

So we have:T �1 U(G) T = U(G�1

T G GT ) = U(G0) . (9.191)Again, we take r = +1 and = +1, so that G = 1 + �G and G0 = 1 + �G0, where

�G =�

�!,�v,�a,�⌧, 1, 1�

,

�G0 =�

�!,��v,�a,��⌧, 1, 1�

,(9.192)

Both of these transformations are proper. So we can take U(G) and G(G0) to be represented by the infinites-imal form of Eq. (9.186). Since we will require T to be anti-linear and anti-unitary, T �1i T = �i, and, using(9.191), we find:

T �1 J T = �J ,

T �1 K T = K ,

T �1 P T = �P ,

T �1 H T = H .

(9.193)

We also assume that M and W are unchanged by a time-reversal transformation. The eigenvalues of T arealso of unit magnitude. We also have: T �1 = T † = T . We discuss time reversal of angular momentumeigenvectors in Section 21.1.4.

5We do not use the extended group in this discussion.


9.7. IMPROPER TRANSFORMATIONS CHAPTER 9. SYMMETRIES

Exercise 12. Show that under time reversal,

T �1 X(t) T = X(�t) , (9.194)

where X(t) = X + V t, where X = K/M and V = P/M .

For combined parity and time-reversal transformations, we find:

(PT )�1 J (PT ) = �J ,

(PT )�1 K (PT ) = �K ,

(PT )�1 P (PT ) = P ,

(PT )�1 H (PT ) = H .

(9.195)

9.7.3 Charge conjugation

The charge conjugation operator C changes particles into antiparticles. This is not a space-time symmetry,but one that reverses the sign of the mass and spin. That is, we assume that:

C�1 M C = �M , C�1 S C = �S . (9.196)

In addition, we take C to be linear and unitary, and:

C�1 J C = �J ,

C�1 K C = �K ,

C�1 P C = P ,

C�1 H C = H .

(9.197)

The eigenvalues of C are again of unit magnitude. If we define X = K/M , and V = P/M , then this meansthat

C�1 X C = X ,

C�1 V C = �V ,(9.198)

So we have the following theorem:

Theorem 25 (PT C). From Eqs. (9.195) and (9.197), the combined (PT C) operation when acting on thegenerators of the Galilean transformation, leaves the generators unchanged:

(PT C)�1 J (PT C) = J ,

(PT C)�1 K (PT C) = K ,

(PT C)�1 P (PT C) = P ,

(PT C)�1 H (PT C) = H .

(9.199)

That is, the generators are invariant under (PT C).

Exercise 13. Show that under charge conjugation,

C�1 X(t) C = X(�t) , (9.200)

where X(t) = X + V t, with X = K/M and V = P/M . So when acting on the equation of motion ofX(t), charge conjugation has the same e↵ect as time reversal. We can interpret this as meaning that innon-relativistic physics, we can think of an antiparticle as a negative mass particle moving backwards intime.


CHAPTER 9. SYMMETRIES 9.8. SCALE AND CONFORMAL TRANSFORMATIONS

Let us be precise. If | i represents a single particle state, then | c i = C | i is the charge conjugate state.Ignoring spin for the moment, if |m

0

, w0

, E0

;x, t i are eigenstates of X(t) and M with positive eigenvaluesm = m

0

> 0, w = w0

> 0 and E = E0

> 0, then

C |m0

, w0

, E0

;x, t i = | �m0

,�w0

,�E0

;x, t i , (9.201)

is an eigenvector X(t), M , W , and H with negative eigenvalues m = �m0

< 0, w = �m0

< 0, andE = �E

0

< 0. So the charge conjugate wave function with m, w, and E all positive:

c(m0

, w0

, E0

;x, t) = hm0

, w0

, E0

;x, t | c i = hm0

, w0

, E0

;x, t | C | i= h�m

0

,�w0

,�E0

;x, t | i = (�m0

,�w0

,�E0

;x, t) , (9.202)

is the same as the wave function with m0

, w0

, and E0

negative. We will study single particle wave functionsin the next chapter. Charge conjugate symmetry says that, in priciple, we cannot tell the di↵erence betweena world consisting of particles or a world consisting of antiparticles.

9.8 Scale and conformal transformations

Scale transformations are changes in the measures of length and time. An interesting question is if there areways to determine a length or time scale in absolute terms, or are these just arbitrary measures. If there areno physical systems that can set these scales, we say that the fundamental forces in Nature must be scaleinvariant. Conformal invariance is a combined space-time expansion of the measures of length and time, andgeneralizes scale changes. We discuss these additional space-time symmetries in the next two sections.

9.8.1 Scale transformations

Scale transformations are of the form:

x0i = ↵xi , t0 = � t . (9.203)

We require, in particular, that if (x, t) satisfies Schrodinger’s equation with w = 0 for a spinless free particlein ⌃, then 0(x0, t0) satisfies Schrodinger’s equation in ⌃0. Probability must remain the same, so we requirethat

| 0(x0, t0) |2 d3x0 = | (x, t) |2 d3x . (9.204)

With this observation, it is easy to prove the following theorem.

Theorem 26. Under scale transformations x0 = ↵x and t0 = �t, spinless scalar solutions of Schrodinger’sequation transform according to:

0(x0, t0) = ↵�3/2 eig(x,t)/~ (x, t) . (9.205)

with � = ↵2 and g(x, t) = C, a constant phase.

Exercise 14. Prove Theorem 26.

We put ↵ = es and then � = e2s, so that infinitesimal scale transformations become:

�x = �sx , �t = 2�s t . (9.206)

We now follow our work in example 27 to find a di↵erential representation of the scale generator D. UsingEq. (9.205), infinitesimal scale changes of scalar functions are given by:

0(x0, t0) = e�3�s/2 (x0 ��x, t0 ��t)

=�

1� 3�s/2 + · · · � 1��sx ·r��s 2 t @t + · · · (x0, t0)

=n

1��s�

3/2 + x ·r + 2 t @t

+ · · ·o

(x0, t0)

(9.207)


9.8. SCALE AND CONFORMAL TRANSFORMATIONS CHAPTER 9. SYMMETRIES

The dilation generator D is defined by:

� (x, t) = 0(x, t)� (x, t) = �i�s D (x, t) , (9.208)

from which we find:D = �3

2i +

1ix ·r� 2i t @t = �3

2i + x ·P� 2 t H . (9.209)

We can drop the factor of �3i/2 since this produces only a constant phase. Using the di↵erential represen-tations in Eqs. (9.13), we find the commutation relations for D:

[D,Pi ] = iPi , [D,H ] = 2iH , [D,Ki ] = �iKi , (9.210)

and commutes with Ji. D also commutes with M , but we note that the first Casimir operator W = H =P 2/2M does not commute with D. In fact, we find:

[D,W ] = 2iW . (9.211)

So the internal energy W breaks scale symmetry.

9.8.2 Conformal transformations

Conformal transformations are of the form:

x0i =xi

1� ct, t0 =

t

1� ct, (9.212)

where c has units of reciprocal time (not velocity!) and can be positive or negative. Note that 1/t0 = 1/t� c.For a scalar spin zero free particle satisfying Schrodinger’s equation, probability is again conserved ac-

cording to (9.204), and we find the following result for conformal transformations:

Theorem 27. Under scale transformations x0 = ↵x and t0 = �t, spinless scalar solutions of Schrodinger’sequation transform according to:

0(x0, t0) = (1� ct)3/2 eig(x,t)/~ (x, t) . (9.213)

whereg(x, t) =

12

mc x2

1� ct. (9.214)

Exercise 15. Prove Theorem 27. For this, it is useful to note that:

r0 = (1� ct) r , @0t = (1� ct)2 @t � c(1� ct)x ·r . (9.215)

and that:~irh

eig(x,t)/~ (x, t)i

= eig(x,t)/~h ~

ir + (rg(x, t))

i

(x, t) . (9.216)

Infinitesimal conformal transformations are given by:

�x = �c tx , �t = �c t2 . (9.217)

So from Eq. (9.213), infinitesimal conformal transformations of scalar functions are given by:

0(x0, t0) = (1� t �c)3/2 ei�g(x

0,t0)/~ (x0 ��x, t0 ��t) , (9.218)

where�g(x0, t0) =

12

mx2 �c . (9.219)


CHAPTER 9. SYMMETRIES 9.9. THE SCHRODINGER GROUP

So

0(x0, t0) =n

1� 32

t �c + · · ·on

1 +~2i

mx2 �c + · · ·o

⇥n

1��c tx ·r��c t2 @t + · · ·o

(x0, t0)

=n

1 + �cn

�32

t +~2i

mx2 � tx ·r� t2 @t

o

+ · · ·o

(x0, t0) ,

(9.220)

The conformal generator C is defined by:

� (x, t) = 0(x, t)� (x, t) = i�c C (x, t) , (9.221)

from which we find:

C =3i

2t� ~

2mx2 +

t

ix ·r� i t2 @t

=3i

2t� ~

2mx2 + tx ·P� t2 H

=3i

2t� ~

2mx2 + t D + t2 H .

(9.222)

We find the following commutation relations for C:

[C, H ] = �iD , [C, D ] = �2i C , (9.223)

and commutes with all other operators. Note that scale and conformal transformations do not commute. Soif we put:

G1

=12

(H + C) , G2

=12

(H � C) , G3

=12

D , (9.224)

we find that G satisfies a O(2, 1) algebra:

[G1

, G2

] = �iG3

, [G1

, G3

] = iG2

, [G2

, G3

] = iG1

. (9.225)

Since [ Gi, Jj ] = 0, the group structure of the extended group has O(3)⇥O(2, 1) symmetry.

9.9 The Schrodinger group

The extension of the Galilean group to include scale and conformal transformations is called the Schrodingeror non-relativistic conformal group, which we write as S. We consider combined scale and conformal trans-formations of the following form:

x0 =R(x) + vt + a

�t + �, t0 =

↵t + �

�t + �, ↵� � �� = 1 . (9.226)

Here ↵, �, �, and � are real parameters, only three of which are independent. This transformation containsboth scale and conformal transformations as special interrelated cases. The group elements now consistof twelve independent parameters, but it is useful to write them in terms of thirteen parameters with oneconstraint: S = (R,v,a,↵,�, �, �). The extended transformation is a group. The group multiplicationproperties are contained in the next theorem:

Theorem 28. The multiplication law for the Schrodinger group is given by:

S00 = S0S = (R0,v0,a0,↵0,�0, �0, �0) (R,v,a,↵,�, �, �)= (R0R,R0(v) + ↵v0 + �a0, R0(a) + �v0 + �a0,

↵0↵+ �0�,↵0� + �0�, �0↵+ �0�, �0� + �0� ) .

(9.227)


REFERENCES REFERENCES

A faithful five-dimensional matrix representation is given by:

S =

0

@

R v a0 ↵ �0 � �

1

A , S00 = S0S , (9.228)

which preserves the determinant relation: det[ S ] = ↵� � �� = 1. The unit element is 1 = (1, 0, 0, 1, 0, 0, 1)and the inverse element is:

S�1 = (R�1,��R�1(v) + �R�1(a),�↵R�1(a) + �R�1(v), �,��,��,↵ ) . (9.229)

For infinitesimal transformations, it is useful to write:

↵ = 1 + �s + · · · ,

� = �⌧ + · · · ,

� = ��c + · · · ,

� = 1��s + · · · ,

(9.230)

so that

↵� � �� = ( 1 + �s + · · · ) ( 1��s + · · · )� ( �⌧ + · · · ) (��c + · · · ) = 1 + O(�2) , (9.231)

as required. �⌧ , �s, and �c are now independent variations. So the unitary transformation transformationfor infinitesimal transformations is now written as:

U(1 + �S) = 1 +i

~

n

�✓ n · J + �v ·K��a ·P + �⌧ H + �s D ��c Co

+ · · · , (9.232)

in terms of the twelve generators J, K, P, H, D, and C.

References

[1] E. P. Wigner, Gruppentheorie und ihre Anwendung auf dei Quantenmechanic der Atomspektren (Braun-schweig, Berlin, 1931). English translation: Academic Press, Inc, New York, 1959.

[2] V. Bargmann, “On unitary ray representations of continuous groups,” Ann. Math. 59, 1 (1954).

[3] J.-M. Levy-Leblond, “Galilei group and nonrelativistic quantum mechanics,” J. Math. Phys. 4, 776(1963).

[4] J.-M. Levy-Leblond, “Galilean quantum field theories and a ghostless Lee model,” Commun. Math. Phys.4, 157 (1967).

[5] J.-M. Levy-Leblond, “Nonrelativistic particles and wave equations,” Commun. Math. Phys. 6, 286(1967).

[6] J.-M. Levy-Leblond, “Galilei group and galilean invariance,” in E. M. Loebl (editor), “Group theory andits applications,” volume II, pages 222–296 (Academic Press, New York, NY, 1971).


Galilei Group and Nonrelativistic Quantum MechanicsJeanMarc LevyLeblond Citation: J. Math. Phys. 4, 776 (1963); doi: 10.1063/1.1724319 View online: http://dx.doi.org/10.1063/1.1724319 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v4/i6 Published by the American Institute of Physics. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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JOURNAL OF MATHEMATICAL PHYSICS VOLUME 4, NUMBER 6 JUNE 1963

Galilei Group and Nonrelativistic Quantum Mechanics

JEAN-MARC LEVy-LEBLOND

Laboratoire de Physique Theorique et Hautes Energies, Orsay, France (Received 16 January 1963)

This paper is devoted to the study of the GaJilei group and its representations, The Galilei group presents a certain number of essential differences with respect to the Poincare group, As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones, Consequently, in nonrelativistic quantum mechanics the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the' existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter' this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence bet~een physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular how the number of polarization states reduces to two for the zero-mass case (though in fact there 'are no physical zeromass systems in nonrelativistic mechanics). Finally, we study the two-particle system where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.

INTRODUCTION

SINCE the work by Wignerl came out, the Poincare group (inhomogeneous Lorentz group)

and its unitary representations have become well known. In particular, each relativistic wavefunction corresponds to some unitary representation of the Poincare group, and in a certain sense, one usually says that an elementary particle is associated to a unitary irreducible representation of the group. Within such a definition, an elementary particle is characterized by its mass and spin.

It was much later that such a work was undertaken for the Galilei group, the invariance group of nonrelativistic mechanics. The Galilei group has, in fact, a rather more intricate structure than has the Poincare group and this has important repercussions in the study of the group representations. Indeed, in quantum mechanics, we deal with the unitary projective (i.e. up-to-a-factor) representations of the group concerned. But in most of the physically interesting cases, as Bargmann showed,2

the study of unitary projective representations of the group can be reduced to the study of true unitary representations of its universal covering group. Such is the case of the rotation, Lorentz, and Poincare groups.

On the other hand, as Bargmann also showed,2 the Galilei group owns a (one-dimensional) infinity of projective representations classes, nonequivalent to true representations, and, what is more troublesome, the physically meaningful representations are precisely these nontrivial projective representations.

1 E. P. Wigner, Ann. Math. 40, 149 (1939). 2 V. Bargmann, Ann. Math. 59, 1 (1954).

Inonu and Wigner3 have indeed shown that under no condition can the basis functions of the Galileigroup true representations be interpreted as wavefunctions of physical particles. With these functions, one can construct neither localized states, nor even states with definite velocity. Conversely, Hamermesh4 studying the infinitesimal group operations, has shown that one can construct a position operator only in the case of nontrivial projective representations. One easily sees that the solutions of the Schrodinger equation for a free particle transform precisely according to such representations.

In the following pages, these unitary, irreducible, nontrivial projective representations of the Galilei group will be called, for short, "physical representations."

In the first section, we recall some generalities about the Galilei group and its structure. In the second one, we explicitly construct the physical representations of the Galilei group, with the help of the "little group" technique.6 The third section is devoted to a physical discussion of these representations where we exhibit their connection with the free-particle Schrodinger equation, and obtain a group-theoretical characterization of a nonrelativistic elementary system by its spin, mass and internal energy-this last parameter in fact revealing itself to be arbitrary. We next study the zero-mass case. In this fourth section, we rediscover some of the true representations already studied by Inonu

3 E. Inonu and E. P. Wigner, Nuovo Cimento 9,705 (1952). • M. Ham~rmesh, Ann. Phys. 9, 518 (1960). 6 See, for mstll:nc~, M. Hamermesh, Group Theory (Addi

son-Wesley PublIshmg Company, Inc., Reading, Massachusetts, 1962), Sec. 12-7.

776


GAL I LEI G R 0 U PAN D NON R E LA T I V 1ST Ie QUA N TUM ME C HAN I C S 777

and Wigner. a One can then give to these representations some vague physical meaning, and we observe, as in the zero-mass case of the Poincare group, the uncoupling of the different helicity states. The Lie algebra of the Galilei group and its representations are investigated in the fifth section where we emphasize the difference between the parts played by the mass in relativistic and nonrelativistic quantum mechanics (Bargmann's superselection rule). In the sixth and last section, we decompose the tensor product of two physical representations of the Galilei group into a direct sum (integral) of such physical (irreducible) representations.

I. THE GALILEI GROUP

The proper Galilei group to which we restrict ourselves (we exclude the inversions) contains the translations in space and time, the rotations and the pure Galilei transformations, i.e., transitions to a uniformly moving coordinate system. Let us note the general element of the group by

G = (b, a, v, R), (1.1)

where b is a time translation, a a space translation, v a pure Galilei transformation, and R a rotation. The group acts on the coordinates (x, t) of an event in space-time according to

x' = Rx + vt + a, t' = t + b. (1.2)

We thus get the multiplication law for the group:

G'G = (b', a', v', R')(b, a, v, R)

= (b' + b, a' + R'a + bv', v' + R'v, R'R). (1.3)

The identity for the group is

1 = (0, 0, 0, 1), (1.4)

and the inverse element of G = (b, a, v, R) is given by

G- 1 = (-b, -R-1(a - bv), -R-1v, R-1). (1.5)

One notices at once the complexity of the Galilei group structure.

The Poincare group <P admits a maximal abelian invariant subgroup e (the space and time translations) and the factor group <pIe is a simple group £,

the Lorentz group. Here the situation is more complicated. The

maximal abelian invariant subgroup of the Galilei group g, is a six-parameter group 'U (space translations plus pure Galilei transformations). The factor group g/'U itself admits a one-parameter invariant subgroup 5:>, the time translations, and it is only the factor group (g/'U)/5:> which is a simple group ffi, the rotation group.

In other words, the Poincare group can be written

<P = £ X e, (1.6)

that is, the semidirect product of a simple group £ by an abelian group e. But the Galilei group is

9 = (ffi X 5:» X 'U, (1.7)

the semidirect product by an abelian group 'U of the semidirect product by an abelian group 5:> of a simple group ffi.

We will now use the following result, due to Bargmann,2 which we reproduce without proof:

"The physical representations of the Galilei group are obtained from the projective unitary representations of its universal covering group characterized by the system of factors:

w(G', G) = exp [i(tm) (a' ·R'v - v' ·Ra

+ bv' ·R'v)], (1.8)

where G = (b, a, v, R), G' = (b', a', v', R') and m is any real number.

This means that to each element G of the universal covering group (which one obtains merely by replacing the rotations R by the elements of the unitary unimodular group), corresponds a unitary operator UCG) such that the multiplication law

U(G') U(G) = w(G', G) U(G'G) (1.9)

holds, where G', G, w(G', G) have been defined above. The ensuing transition from the covering group

to the original Galilei group only adds a possible sign ambiguity.

In a more elaborate language, we are looking for true unitary representations of some nontrivial central extension of the Galilei group universal covering group by a one-dimensional abelian group. 6

Let us note

G=(O,G), Oreal, (1.10)

which are the elements of this extension. We have then the multiplication law

G'G = (0' + 0 + HG', G), G'G),

where HG', G), given by

w(G', G) = exp [i~(G', G)],

is an exponent of the group.2

(1.11)

(1.12)

The eleven-parameter group g, with which we

6 F. Lurcat and L. Michel (unpublished). L. Michel, Lectures at the Istanbul Summer School (1962) (to be published).


778 JEAN-MARC LEVY-LEBLOND

deal now, has a structure rather different from the one of the original Galilei group. In fact, the maximal abelian invariant subgroup of 9 is e, the space-time translations plus the one-parameter central subgroup. The factor group 9/ e in turn admits a maximal abelian invariant subgroup 'U, made up of the pure Galilei transformations (3 parameters). Finally, the factor group C§/e)/'U is a simple group m, the rotation group. We can write

9 = (m X 'U) X e, (1.13)

where the products are semidirect products. This is the structure we shall be concerned with. Let us finally notice that setting m = ° in (1.8) brings us back to the study of true representations of the Galilei group:

m = ° ==} w(G', G) = 1 ==} U(G') U(G)

= U(G'G). (1.14)

The central extension 9 becomes a trivial one (direct product).

Unless otherwise specified, we will deal exclusively from now on with the case m ~ 0.

II. PHYSICAL REPRESENTIONS OF THE GALILEI GROUP

We now proceed to construct the physical (i.e., irreducible, unitary, nontrivial projective) representations of the Galilei group, making use of its structure as studied above and following the "little group" technique.s

Let us suppose we have found some physical representations of the Galilei group. If we restrict ourselves to the abelian subgroup e of space-time translations, this representation will decompose into a direct integral of unitary irreducible representations of the subgroup e. These representations are well-known, they are designated by a real vector p and a real number E. We can then choose as a set of basis functions, square-integrable functions 1/;(p, E, n, where r is an additional set of variables which may be needed to distinguish the basis functions belonging to the same irreducible representation of the translation group e. We now know the representation of this subgroup:

U(b, a, 0, l)1/;(p, E, r)

= exp (-ibE + ia·p)lf(p, E, t). (11.1)

Using the mUltiplication law of the group representation (1.9), a factor system, i.e., a real number m (1.8) having been chosen, we look for the representation of the factor group g/ e. Let us note

first the following equalities, inferred from (1.3), (1.8), and (1.9):

U(b, a, v, R) = exp (-i(!m)a·v)

X U(b, a, 0, l)U(O, 0, v, R),

U(b, a, v, R) = exp [i(!m)(a·v - bv.v)]

X U(O, 0, v, R)U(b, R-1(a - by), 0,1),

whence

(II.2)

(II.2')

U(b, a, 0, l)U(O, 0, v, R) = exp [im(a·v - !bv·v)]

X U(O, 0, v, R)U(b, R-1(a - by), 0,1). (II.3)

Letting each member of this equality between operators act upon some basis function 1/;(p, E, r), and taking (11.1) into account, we get

U(b, a, 0, l)U(O, 0, v, R)1/;(p, E, r = exp [im(a·v - !bv·v)]

X exp [-ibE + i(R-1a·bR-1v).p]

X U(O, 0, v, R)1/;(p, E, r), (II .4)

U(b, a, 0, l)U(O, 0, v, R)1/;(p, E, r)

= exp [-ib(E + v·Rp + !mv2) + ia·(Rp + my)]

X U(O, 0, v, R)1/;(p, E, r). (II.5)

That is to say, the function U(O, 0, v, R)1/;(p, E, n transforms according to the representation (p', E') of the space-time translations group, where

p' = Rp + mY, (II.6)

E' = E + v·Rp + !mv2•

Thus, if (p', E') and (p, E) are connected by the relation

E' - (p'2/2m) = E _ (p' /2m) , (11.7)

it is always possible to find an element (0, 0, v, R) of the Galilei group (more precisely of the factor group g/e) such that (II.6) holds.

In other words, if a physical representation of the Galilei group contains an irreducible representation (p, E) of the translation group, it contains all the representations (p', E') given by (11.7). Therefore, there is a one-to-one correspondence between the points of the paraboloid:

E - (p2/2m) = 'V = ct., (II. 7')

and the irreducible representations of the translation group contained in a physical representation of the Galilei group.

This enables us to construct the Hilbert space X


GAL I LEI G R 0 UP AND NON R E L A T I V 1ST I C QUA N TUM M E C HAN I C S 779

of the representation as a direct integral of the Hilbert spaces JCp,E where acts the irreducible representation (p, E) of e:

Je = J df..LCp, E) Jep,E, (11.8)

where J.I.(p, E) is an invariant measure on the paraboloid (II.8). As a matter of fact,

df..LCp, E) = dp dE orE - Cp2/2m) - 'U]. (II.9)

The Little Group

We now search for the "little group",6 i,e., the subgroup ~(p, E) of the Galilei group constituted by those elements (0, 0, v, R) of the factor group s/e such that the function U(O, 0, v, R)if;(p, E, r) still belongs to the irreducible representation (p, E) of e. ~(p, E) is what mathematicians call the "stabilisator" of (p, E). After (II.6), we get for (0, 0, v, R) the conditions

p = Rp + mY,

E = E + V· Rp + !mv2,

which can also be written as

p = Rp + mY,

p2 = (Rp + mv)2.

(II.lO)

(II.10')

These two conditions then are not independent, the second being implied by the first one.

Thus an element of the little group ~(p, E) is uniquely defined by the choice of a rotation R, since in that case, the condition (IL10') determines v.

This correspondence between the little group l)(p, E) and the rotation group (R is an isomorphism. Indeed,

p = Rp + mY, p = R'p + my'

implies

p = R'(Rp + my) + mv' = R'Rp + m(R'v + v').

The product of (0, 0, v', R') and (0, 0, v, R) corresponds to the product of Rand R'.

The little group representations are then well known: they are the rotation group representations. The irreducible ones are labeled by an integer of half-integer number 8. We denote them by D'. They are (28 + I)-dimensional.

We show now that choosing a paraboloid (11.7') (which fixes the possible representations of the translation group) and a representation D' of the little group completely determines a representation of the whole Galilei group.

First we set up a point (Po, Eo) on the paraboloid. Then for each point (p, E) of that same paraboloid, one can select one element

(11.11)

of the Galilei group such that VpE acting on (Po, Eo) according to (11.6) transforms it into Cp, E), which we denote by

(II. 11')

N ow let V be any element of the factor group:

V = (0, 0, v, R), (11.12)

and let (p', E') be the result of the action of V (p, E) according to (II.6):

V(p, E) = (p', E'). (I 1. 13)

This can also be written as

VVpE(Po, Eo) = Vp'E'(PO' Eo), or

Thus, (II.14)

is an element of the little group ~(po, Eo), which, of course, depends on (p, E) and V.

Conversely, every element V of the factor group Sl e can be written in the form

(11.15)

with the definitions (11.11'), (II.I2), and (II.I3). Now, after (II.5), (II.6), and (II.ll'), U(Vp,B)'

if; (Po, Eo) is proportional to if;(p, E). The simplest choice is then to define U(Vp,E) by

(II.I6)

Finally, we choose a representation D' of the little group. The variable r is merely an index, running from -8 to +8, on which act the (28 + I)-dimensional matrices of the representation D':

Vo E f)(po, Eo) :==} U(Vo)if;(po, Eo, S) H

= L if;(Po, Eo, ~)[D'(VO)]~i' (II.17) ~--.

Therefore, for any V = (0, 0, v, R) and any if;(p, E, r), we have, after (IU5), (II.16), and (II.17),

U(V)if;(P, E, r)

U(VP'E,)U(V~PE),V)U-l(VPE)if;Cp, E, r)

U(VP'E,)U(V~PE),V)if;(po, Eo, r)

L U(Vp.E,)if;Cpo, Eo, mD·(V~DB).V)]H ~



U(V)If(p, E, t)

= 1: If(p',E',mD'(Vcip,E),V)],r' (II.18) , Since we also know the operators which represent the translation, the decomposition (II.2) of any element of the whole group, gives us at once the complete solution to our problem:

U(b, a, v, R) If(p , E, t)

= exp [-ictm)a·v + ia·p' - ibE']

X L If(p', E', ~)[D'(V6P.E)(V,R)]H' , where

p' = Rp + mv,

E' = E + v·Rp + !mv2,

and

VciPE),(V,R) = V;?E'(O, 0, v, R) VpE .

(II.19)

These representations are clearly irreducible. They are unitary with the scalar product:

Ccp, If) = f dp dE 0[ E - :~ - '0 ]

x L <I>(p, E, a)lf(p, E, a). (II.20)

Finally, one can verify directly that they are indeed projective representations of the universal covering group of the Galilei group with the system of factors (L8) which we started from.

The transition to the Galilei group itself merely introduces the usual sign ambiguity in the case of half-integers. We have thus obtained the following result:

The physical representations of the Galilei group are characterized by two real numbers m and '0 and an integer or half-integer number s. We designate them by [m I '0, s] and they are given explicitly by (II.19).

III. PHYSICAL DISCUSSION

The interpretation of the preceding results is straightforward. Eq. (II.20) defines the functional space of the representation as the space of squareintegrable functions on the paraboloid:

E - (p2/2m) = '0. (II.7')

This, and the rotation properties of these basis functions, impel us to establish a one-to-one correspondence between a (free) particle of mass m, internal energy '0, spin s, and the physical rep-

resentation [m I '0, s] of the Galilei group. We will now study this correspondence.

Schrodinger Equation and Galilei Group

For the time being, we shall disregard the spin and consider the Schrodinger equation for a spinless particle:

i(alf/at) + (1/2m),11f = '0 If. (III. I)

We are only concerned with the free-particle case, so that in (III.I), '0 is a constant. We wish to study the invariance properties of the Schrodinger equation with respect to the Galilei group transformation. (Let us note that most of quantum mechanics textbooks thoroughly investigate the Lorentz invariance of the Dirac equation but completely overlook the Galilean invariance of the Schrodinger equation.)

We follow the passive point of view. That is, we look at the same state described by (IILI), III a transformed frame of reference defined by

x' = Rx + vt + a (1.2)

t' = t + b.

G = (b, a, v, R) is the Galilean transformation we consider. In the new frame of reference, the state must be described by some wavefunction If'. The physical predictions we get from the two descriptions will be identical if and only if the transformed wavefunction at any point differs from the original wavefunction at the transformed point by at most a phase factor (the density of particles being a scalar) :

If'(x, t) = e-;[(X' ,t ') If(x', t'), (III.2)

where (x', t') depend on (x, t) according to (1.2). Obviously, the new wavefunction has to satisfy the Schrodinger equation

i(alf' /at) + (l/2m},1If' = '01f'. (III.3)

We may now determine the unknown function f. Using (1.2), we find

a/at = a/at' + v·V', (III.4) V = RV'.

The Schrodingerequation (III.3) for the new wavefunction can be rewritten as an equation in f and If:

rica/at') + iv· V' + (1/2m},1' - '0]

X e-if(zo"O)If(x', t') = 0. (III.5)


GA LILEI GROUP AND NONRELA TIVISTIC QU ANTUM M ECHA NICS 781

Dropping the primes and expanding,

(IIL6)

ift(x, t) satisfies the Schrodinger equation (III. I), so that we are left with the two conditions

Vf - mv = D,

(of/at) - (1/2m)t.! + mv2 = D.

(III.7)

We easily integrate these equations, and get

f(x, t) = mv·x - !mv2 t + C, (III.8)

where C is a constant. We see that, unlike the relativistic case, the phase

factor cannot be eliminated. The transformation properties of the Schrodinger wavefunctions are then

if/ex, t) = exp [-imv·x'

+ !imv2 t' - iC]ift(x', t'),

x' = Rx + vt + a, (III.9)

t' t + b.

In the momentum space, we deal with wavefunctions:

</J(p, E) = J e-iP'x+iEt ift(x, t) dx dt.

The Schrodinger equation reads

[E - (p2/2m)]</J(p,E) = 'Ucp(p,E),

so that the support of cp is

E - (p2/2m) = '0,

(IlL 10)

(IILlI)

(IILlI')

and the wavefunctions have to be square-integrable:

J (cp(p, :: + '0)(2 dp = J Icp(p, EW

X o( E - :: - '0) dp dE < ex>. (III.12)

Let us study the Galilean transformation in the momentum space:

cp'(p, E) = J e-;P'JC+,Etift'(x, t) dx dt

= J exp (-ip·R-\x' - vt' + vb - a)

+ iE(t' - b)] X exp [ -imv·x'

+ timv2t' - iC ]ift(X" t') dx' dt'. (III. 13)

We drop the primes and re-order the sum in the exponential:

cp'(p, E) = exp [iRp·a - ibRp·v - iEb - iC]

X J exp [-i(Rp + mv).x + iCE + v·Rp

+ tmv2)t]ift(x, t) dx dt.

We write

p' = Rp + mY, (III.14) E' = E + v·Rp + tmv2

•

Expressing (p, E) in terms of (p', E'), we obtain

cp'(p, E) = exp [-imv·a + timbv2 - iC

+ ia·p' - ibE']cp(p', E').

We now choose

C = -tma·v + tmbv2•

Then

cp'(p, E) exp [-itmv·a + ia·p'

- ibE']cp(p', E').

(IILI5)

(III.16)

(III.17)

One sees at once that (III.17), (III.14), and (III.l2) are identical with (II.19) and (II.20); (III.lI') with (II.7'). We conclude that the solution of the Schrodinger equation for a free spinless particle of mass m and internal energy '0 belongs to the physical representation [m I '0 ,D] of the Galilei group.

Clearly, any choice other than (III.16) for the constant C, would have led us to an irreducible, projective representation of the Galilei group different from but equivalent to the one we consider here. 7

The Spin

We now want to interpret 8 as the spin of the particle described by the physical representation [m I '0, 8]. This, however, is not a Galilei invariant concept. Indeed, in order to understand the meaning

7 The concept of equivalence in the case of projective representations is somewhat distinct from the case of true representations. In fact if {V. I and {V r' I are two projective representations of the group G, they are said to be equivalent if U r' = VU. V-I holds between operator rays2 (U. is the operator ray generated by Vr, i.e. the set of all operators TV., TEe, H = 1). For the operators themselves, we have Vr' = .p(r)VVrV-1, where .p(r) is some complex function of modulus 1 on the group. We see that Vr' is indeed a projective representation of the group with a factor system w'(r, s) = [.p(r).p(s)/.p(rs)]w(r, s) equivalent to the factor system wCr, s) of {V.I.


782 JEAN -MARC LEVY -LEBLOND

of 8, we have to define the little group f)(Po, Eo), that is to pick up some particular point (Po, Eo) of the paraboloid. Let us recall however that all the representations [m I '0, 8] obtained from different points (Po, Eo) (of the same paraboloid!) are equivalent. a

We now choose (Po, Eo) = (0, '0), i.e. the paraboloid top. We then have a most natural choice for Vp •E :

Vp • E = (0,0, m-1p, 1).

Starting from (1.3), we now easily obtain

V6P.E) (v.R)

(0,0,0, R),

with any V = (0, 0, v, R) and (p', E') Consequently, we rewrite (Il.I9) as

U(b, a, v, R)f(p, E, r)

= exp [-iima·v + ia·p' - ibE']

X L f(P', E', ~)[D'(R)]<r , ~

where

p' = Rp + mY,

E' = E + v·Rp + imv2•

(III.I8)

(III.19)

Yep, E).

(III.20)

8 characterizes now exclusively the behavior of our particle with respect to rotations; it is really its intrinsic angular momentum.

Let us notice that the choice (IlLI8) which led us to this result amounts to bringing back the particle at rest by accelerating the initial coordinate system, without rotating it around the direction of the movement (pure Galilei transformation).

Internal Energy

Physically, we are used to saying that, in nonrelativistic mechanics, we can freely choose the origin from which we count the energies. In the particular case of one free particle, this amounts to saying that the internal energy is completely arbitrary. We would like to rediscover this feature from Galilean invariance. This is done quite easily.

Let f(p, E, r) be some basis function of the [m I '0, 8J physical representation of the Galilei group. Let us now define an operator U by:

f(p, E, r) = (Uf)(p, E, r)

= f(P, E + '0, r), (III.21)

We callX'U the Hilbert spaces of functions 1/I(P, E, r) with the scalar product:

(cp, 1/1) = J t: <P(p, E, r)f(P, E, r)

X o( E - f~ - '0) dp dE. (II.20)

It is clear that V realizes a mapping, in fact an homeomorphism of Xv on :Jeo. Since

(4), {;) = J t: 4>(p, E, r)f(p, E, r)

X o(E - ::) dp)E

= J t: ¢(P, E + '0, r)1/I(p, E + 'O'~'r)

X o( E - ::) dp dE

= J t: ¢(P, E', r)f(P, E', r)

X o( E' - :: - '0) dp dE' ,

we have

(4), {;) = (Ucp, Uf) = (c/>, f), (II I. 22)

that is to say, U is an isometric operator and :Je'U and :Jeo are isomorphic Hilbert spaces.

If U(G) is the operator corresponding to G = (b, a, v, R) in the [m I '0, 8] representation, we define

O(G) = UU(G)U- 1• (III.23)

Letting now O(G) act upon some function f(P, E) of X o, using (IIL2I) and (II.19), we get

O(G) = eib'O Uo(G) , (III.24)

where Uo(G) is the operator corresponding to G, in the representation [m I 0, 8] according to the definition (II.I9), or else

U'U(G) = eib'UU-1 Uo(G)U, (III.25)

where we add a subscript '0 to U(G) in order to emphasize the fact that it belongs to the representation [m I '0, 8J. Obviously then, the representations [m I '0, 8] and [m I 0, 8] are equivalent, in the sense of projective representations equivalence. 7 In other words, for an isolated particle, the internal energy '0 has no physical significance.

Antiparticles

Until now, when looking for a physical interpretation of our results, we implicitly assumed tha('m, which we interpreted as the mass, was positive. Actually, the construction of the first section is


GALl LEI GROUP AND NONRELATIVISTIC QU ANTUM MECHANICS 783

valid for any nonzero value of m. What can be said of the negative m case?

Casting a glance at Eq. (II.19), we see at once that if f(p, E, t) transforms according to the [m I '0,8] representation, then

1//(p, E, t) = f( -p, -E, t) (III.26)

transforms according to [-m ,- '0, sl, where by s we mean that fC transforms under rotations by jj' (R), the complex conjugate representation of D' (R). But this does not alter the physical interpretation of 8 as the spin of the particle.

We can immediately verify that the operation f -t fC is an antiunitary one and then, that the representations [rn I '0, 81 and [-m I -'0, 8J are antiunitarily equivalent.

If we consider now charged particles, by the replacement

E' = E + v·Rp.

The basis functions of the representation are now defined onto the Hcylinder":

p2 = p2 = ct. (E arbitrary). (IV.2)

But these representations are no longer irreducible. In fact, let Po (lying on the cylinder IV.2) be the vector around which the rotations, in the D' representation chosen, are diagonal matrices. We call ~(po) the group of rotations around Po. Ro being such a rotation, with an angle IPo, one has

(IV.3)

We once more apply the little group technique.Starting from any vector p of the cylinder (IV.2), we can choose one rotation r p such that

(IVA)

p-tp - QA, E -tE - Qq;, (III.27) Now, R being some rotation which takes p into p':

we see that the above-mentioned antiunitary transformation takes a particle of mass m, internal energy '0, and charge Q into another particle characterized respectively by (-rn, -'0, -Q).

Let us remark that a Fourier transformation, or else direct dealing with the Schrodinger equation in space-time, leads us to the same result with the transformation

1/!C(x, t) = iii(x, t). (III.28)

In other words, if the representation [m I '0, 8] and the charge Q describe some particle, we may describe its antiparticle either by the same representation [m I '0, 8J and the charge -Q, or else by the representation [-m I -'0, 8J and the same charge Q.

This gives some meaning to the negative m case. We notice, however, that, as it is well-known,

if the same Dirac equation describes particle and antiparticle, their nonrelativistic description needs two different Schrodinger equations.

IV. THE ZERO-MASS CASE

We now plainly make m = 0 in the realization (III.20) we obtained for the physical representations [m I '0, 8J of the GaliIei group. We thus get some tru.e representation of the group:

U(O, a, v, R)f(p, E, t) = exp (ia-p' - ibE')

X E f(p', E', mD'(R)hr, (IV. I) t

where

p' = Rp,

p' = Rp, (IV.5)

we get

po = r;,lRrppo, i.e., r;,lRrp = R~·R £ ~(Po).

That is, any rotation can be written in the form

(IV.6)

We next define new functions cb(P, E, p) on our cylinder:

(IV.7)

They transform according to

U(b, a, v, R)¢(p, E, p) = E exp (iap' - ibE') •

X E f(P', E', mD'(R)]n[D'Crp)JiP' •

Inverting the summations, and using (IV.7),

U(b, a, v, R)¢(P, E, p) = exp (iap' - ibE')

X E E 1/!(p', E', mD·(rp'»)tT[D·(R~·R)J.~. < T

Lastly,

UCb, a, v, R)cb(p, E, p)

= exp [iap' - ibE' + ifX{>o(R, p)]cb(P', E' p). (IV.S)

We calculate explicitly the function <Po(R, p) in the Appendix. The subspaces of the functions ¢CP, E, p), with a given p, are thus invariant. We have reduced the primitive representation (IV.I) into (28 + 1) representations which are now irreducible. These are the true representations named "class II" by


784 JEAN-MAR.C LEVY-LEBLOND

Inonu and Wigner.3 They are characterized by an integer p and a positive number p2. We denote them by I ° I P, p I. The very construction of these representations and as we will see later, their Lie algebra and her invariants, designate p as the component of the angular momentum along the direction of the linear momentum, i.e., helicity.

Added to the fact that we deal here with nullmass states, this suggests a close analogy between the representations just found and the irreducible representations of the Poincare group in the zeromass case. In fact, at least in the case of vanishing P, we may interpret the representations 10 I P, p} (P ~ 0) as describing zero-mass and infinite-speed particles, which are indeed the nonrelativistic limit of the zero-mass particles. Naturally, there is no completely consistent interpretation of the representations just found. Nevertheless, we have been able to give some vague meaning to them. And mainly, they display certain features we usually think to be characteristic of the relativistic case (uncoupling of different helicity states) and which, in fact, are latent in the nonrelativistic case.8

•9

V. THE LIE ALGEBRA OF THE GALILEI GROUP AND THE ROLE OF THE MASS IN NON

RELATIVISTIC QUANTUM MECHANICS

Taking the infinitesimal elements of the oneparameter subgroups of the Galilei group (considered as a Lie group) and using the group law (1.3), we calculate their commutators and thus obtain the Lie brackets for the Lie algebra of the group.

We make the most natural choice for the basis elements of the algebra:

r for the time translations, k.(i = 1, 2, 3) for the space translations, ui(i = 1, 2, 3) for the pure Galilei transformations,

Mi(i = 1,2,3) for the rotations.

We then have

[Mi' M j ] = Eijkllfk, lUi' u j ] = [ki' k j ] = 0,

[Mi' Ui] fiikUk, [ki , r] 0, (V.1)

[M i , k j ] Eijkkk, lUi' k j ] 0,

[Mi , r] 0, lUi, rJ k i •

Let us now compute explicitly the infinitesimal

8 It was Wigner 9 who emphasized (in the relativistic case but the same remark is valid here) that a zero-mass system possesses two polarization states but only if we consider the space reflections, since otherwise they would not be connected to each other. On the other hand, for the nonzero mass systems, proper rotational invariance is sufficient for deducing the (28 + 1) polarization states from anyone among them.

9 E. P. Wigner, Rev. Mod. Phys. 29, 255 (1957).

elements of the physical representation (III.20):

M = -px(ajap) - is,

u = m(ajap) + p(ajaE), (V.2)

k = p, r = E,

with obvious notations. Let us notice, as usual, the splitting of the total angular momentum in orbital and intrinsic (spin) parts.

It is now easy to see that the realization (V.2) fulfills all equations (V.1), except that translations and pure Galilei transformations no longer commute. Instead,

(V.3)

This is quite natural. We know that we deal in fact with a projective representation. This means that in (V.2) we obtain a representation of the Lie algebra of a central extension of the Galilei group, and no longer of the Galilei group itself. The Lie algebra element of the one-parameter subgroup by which the extension is made can be called p.. Here, it is represented by p. = m. The extension is central, so that p. commutes with all other Lie algebra elements. But it is nontrivial, so that p. appears in some Lie bracket [see (V.3)].

The enveloping algebra admits the following invariants: 4

2p.r - k 2 = 2mB - p2 = 2m'O,

-(f.J.M + k XU)2 = m2S2 = m2s(s + 1), (V.4)

/J. = m.

We recover, of course, the characterization of physical representations by [m I '0, s]. There is however a rather subtle point we have yet to make clear. We have seen that physical knowledge as well as mathematical considerations on the Galilei group physical representations allow us to conclude that, in fact, the internal energy of an isolated particle is an arbitrary parameter. Precisely we showed that all representations [m I '0, s] and [m I '0', s] are physically equivalent. But we now find '0 as an element of the center of the group algebra. How can any equivalence transformation modify this center? The answer is to be found in the fact that we deal here with an extension of the Galilei group, and such an extension as we consider here has not a uniquely defined Lie algebra. There is a whole class of algebras, in one-to-one correspondence with the unlike but equivalent systems of factors of the projective representation associated with the extension. Here, going from some algebra to another equivalent one, we modify precisely the center element '0, and that one only.


GAL I LEI G R 0 U PAN D NON R E LA T I V 1ST I C QUA N TUM ME C HAN I C S 785

Now the existence within the center of the enveloping algebra of the group, of some basis element of this algebra (here J1., the mass operator), involves a most important physical consequence: it leads to a superselection rule. 10

Let us for instance consider a state vector which results from the superposition of two state vectors having different masses:

(V.5)

where 1/11 and 1/12 respectively belong to the physical representations [m1 I '0, s] and [m2 I '0, s] of the Galilei group. We now consider the behavior of this composed state under the following series of transformations: a translation a; a pure Galilei transformation v; the inverse translation, and the inverse Galilei transformation. Within the group we know that these all commute, whence,

(0,0, -v, 1)(0, -a, 0, 1)(0,0, v, 1)(0, a, 0, 1)

= (0,0, 0, 1), (V.6)

i.e., the identical transformation. With respect to some physical representation,

that series is obviously represented by some phase factor at most. In fact, using (111.20), we find at once

U(O, 0, -v, I)U(O, -a, 0, I)U(O, 0, v, 1)

becomes trivially a direct product and has no more physical consequences.

Starting from the true representation (IV.9) of the Galilei group, and using the explicit form 'Po(R, p) derived in the Appendix, we obtain the following representation:

a a. P2 Ml = -P2 -;- + P3 -;- + tp -+

upa UP2 P pa

a a. P2 M2 = -P3 -;- + PI -;- + t p-+

UPI upa P pa (V.9)

M a + a + . a = -PI -;- P2 -;- tp UP2 UPI

u = p(ajaE) , k = p, T = E.

We may now verify that this representation leads us to the rules (V.l). We also notice the close analogy between these expressions and those obtained for the Poincare-group Lie algebra in the zero-mass case. 12

The enveloping algebra invariants and their values for the {O I P, pI representation are

k2 = p2 = p2, (V.lO)

M·k = ipp = ipP.

This confirms our interpretation of p as the helicity.

x U(O, a, 0, 1) = e- ima'v

•

Thus, our compound state becomes

(V.7) VI. DECOMPOSITION OF THE TENSOR PRODUCT OF TWO PHYSICAL REPRESENTATIONS

1/1 = 1/11 + 1/12---" 1/1 = e-im,a,vl/'l + e- im,a'V2' The tensor product of two physical representations

(V.8) of the Galilei group,

The superposition principle cannot have any meaning for 1/11 and 1/12 if m 1 ~ m2 , since that would mean that an identical transformation could affect the norm of any of their compound states. The relative phase of two states having different masses is completely arbitrary. This is known as the "Bragmann superselection rule."ll It prevents the existence, in nonrelativistic quantum mechanics, of states with a mass spectrum, and therefore of unstable particles.

We see here how the mass plays different parts in relativistic and nonrelativistic quantum theories.

The Lie Algebra in the Zero-Mass Case

We will deal now really with the Lie algebra of the Galilei group itself: the central extension

10 G. C. Wick, A. S. Wightman, and E. P. Wigner, Phys. Rev. 88, 101 (1952).

11 A. S. Wightman, "Lectures on Relativistic Invariance," in Les Bouches 1960 Summer School Proceedings (Hermann et Cie., Paris, 1960), pp. 159-226.

(VI. 1)

is still a (projective) representation whose operators act onto the square-integrable basis functions

1/I(Pl' P2, E I , E2, tl, t2)

according to

U(b, a, v, R)1/I(Pl, P2, E I , E 2 , SI, S2)

= exp [-!i(ml + m2)a·v

where

+ ia(pi + pD - ib(Ei + E~)] X L 1/I(pi, pL Ei, EL ~1' ~2)

hE~

(VI.2)

p; = Rpi + m;v, (i = 1,2)

E~ = E; + V·Rpi + !miv2.

12 J. S. Lomont and H. E. Moses, J. Math. Phys. 3, 405 (1962).



These representations are unitary, the scalar product being defined as

(q" I/t) = J dpl dEl dp2 dE2

X O(EI - 2~1 - '01 ) o(E2 - 2~2 - '0 2)

X L 4i(Pl' P2, E l , E 2 , rl, r2) ',f. (VI.3)

We are going to reduce the Hilbert space, which we just determined, into a direct sum (possibly a direct integral) of invariant Hilbert spaces. We proceed quite similarly to Wightman.ll

We first define new variables:

and call

P = PI + P2, E = El + E2 ,

E =

mlm2 p.=

ml + m2 '

where we have assumed m1 + m2 ~ o.

(VIA)

These variables are precisely those corresponding to the usual separation of the center-of-mass and relative motions for our two-particle system.

We also introduce the internal energy of the compound system, i.e., the difference between its total energy and the center-of-mass kinetic energy:

(VI. 5)

We have now the following expression for the scalar product:

(q" I/t) = 1'" d'O J dP dE o(E - ~ - '0) "U,+"U. 2M

We see, exactly as in the case of the Poincare group,11 the most natural appearance of the orbital angular momenta.

This provides also the profound reason why, when studying the Schrodinger equation, one keeps only integer (and not half-integer) relative angular momenta. But in contradistinction with the relativistic case, the mass is now conserved (and even

X L4i(p,E,q,E,rl, r2)I/t(p,E,q,E,rl, r2), (VI. 6) rd"3

which has been directly derived from (VI.3). The Hilbert space X of the representation is thus reduced to a direct integral of Hilbert spaces X"U, each associated to a paraboloid:

(VI. 7)

Since (P, E) precisely characterizes the representation of the translation subgroup in the tensor product (see VI.I), the condition (VI.7) is then a necessary one for the Hilbert spaces X"U to be invariant. In fact, we can reduce them no further with respect to the translation subgroup representation, and it suffices now to look in each X"U for the subspaces invariant with respect to the little group, which.may be chosen simply as the rotation group.

Equation (VI.6) shows now that, if some point (P, E) is fixed, the basis functions of our representation only depend on some vector q whose length is fixed. Since these are uniform functions onto the sphere q2 = C", we expand them in spherical harmonics, i.e., basis functions of irreducible representations of the rotation group. We then obtain, for the little group (rotation group), the representation

D" ® D" ® (ffi D I).

1-0 (VI.8)

The decomposition of this tensor product into irreducible representations is immediate. Since we have seen at the beginning that a paraboloid '() and an irreducible representation of the little group uniquely define a physical representation of the Galilei group, we finally obtain the complete solution to our problem, which we symbolically write as:

superconserved as we have seen). The kinetic energy of the relative motion of the components is to be found now in the internal energy of the compound system. The weakened concept of equivalence which we introduced earlier and which led us to the arbitrariness of the internal energy of an isolated particle, has now also, in the two-particle case, a quite interesting application. It enables us to change


GAL I LEI G R 0 U PAN D NON R E LA T I V 1ST Ie QUA N TUM M E C HAN I C S 787

[ml I '01, 8tl and [m2 I '02, 82] into [ml I 0, 81] and [m2 I 0, 82], but not to change simultaneously all the [m l + m, / '0, j] of their tensor product decomposition into [m l + ma I 0, jJ. All we can do is to "renormalize" the internal energy of the possible compound states by an amount '0 1 + '02 , Of course this agrees entirely with our previous physical knowledge: once we have fixed up the internal energy of two isolated particles, their compound state has an internal energy which is no longer arbitrary.

In the case where m l + ma = 0, with the help of techniques quite similar to those just used, we obtain the following result, which we quote here for completeness and without proof:

[m 1'01 ,811 @ [-m / '02 , 821

= J$: dP l~m P.(jj .. P,~:. (0 I P, PI + P2 + l},

(VI. 10) or else

[m / '01 , sd @ [-m / '02 , S21

interest they have taken in this work, as well as for their many suggestions and critical remarks.

APPENDIX

We here derive an explicit formula for the angle CPo of the rotation:

(A.I)

where p' = Rp, and the r/s are well defined rotations which take some fixed Po into p:

rppo = p.

We use the spinorial representation of the rotation group. A rotation by angle around an axis n(/n/ = 1) may be written as

R(n, cp) = cos (cp/2) - itt sin (cp/2) , (A.2)

where

n = ~·n, (A.3)

and the To'S are the usual Pauli matrices.

= 1'" dP EB (0 I P, pj0(2,,+I) (2 •• +1). $0 p __ CXl

We choose as rp , the rotation in the (Po, p) plane (VI. 10') which brings Po into p. Writing it as the product

of two plane symmetries, we have Such a result is valid also if we replace one (or two) of the representations [m / '0, s] by [m / '0, sl (see Sec. III); particularly,

[m / '0, sl @ [-m / '0', sl

= 1" dP EB (O I P, p}0(2Hl)', eo p_-m

(VI. 11)

and this justifies our choosing the representations [m / '0, s] and [-m / '0', s] in order to represent, respectively, a particle and its antiparticle.

All the results derived here are of course well known. It is however stimulating to obtain them from Galilean invariance only, and this provides an agreeable and unifying piont of view.

ACKNOWLEDGMENTS

The author is very grateful to Professor Louis

(A A)

where k and ko are the unit vectors lying on p and PoCk = pi/pi, etc .... ), and with the same notations as in (A.3).

Now, from the definition of k' and (A.2), we have

k = (cos ~ - in sin ~)k( cos ~ + itt sin ~) . (A.5)

Bringing (A.S), (AA), and (A.2) into (A.I), using also, repeatedly, the well-known identity

db = a·b + i~·(a x b), (A.6)

and

Michel and Professor Franc;ois Lur<;at for the we get

(2 + feko + feofe) cos ~ - i(tt + kottko + ttkko + feottfeo) sin i Ro = 2(1 + ko ·k')I(I + ko .k)!

Finally, R _ (1 + k·ko) cos (cp/2) + (ko, n, k) sin (cp/2) - iko(n·ko + n·k) sin (cp/2) . (A.7)

o - (1 + k.ko)t[I + cos cpko·k + (1 - cos cp)(n·ko)(n·k) + sin cp(ko, n, k)]l



The expression (A.7) for Ro shows clearly the axis ko of the rotation R o, whose angle CPo is immediately obtained by identifying (A.7), and

Choosing ko as our z axis, and taking successively infinitesimal rotations around the x, y, and z axes, we obtain

Ro = cos (CPo/2) - ifco sin (CPo/2). (A.7')

In particular, for an infinitesimal rotation around n, with an angle of cP « 1,

CPo ~ (n·ko + n·k/1 + k·ko)cp. (A.8

JOURNAL OF MATHEMATICAL PHYSICS

(cpo)x = [kz/O + k,]cp = [Px/(p + p,)]cp,

(CPo). = [k yl(l + k,)]cp = [P./(p + pz)]cp, (A.9)

(CPo), = [1 + k,/(l + k,)Jcp = cpo

We can then immediately derive the expressions (V.9) for the Lie algebra in the zero-mass case.

VOLUME 4. NUMBER 6 JUNE 1963

Principle of General Q Covariance

D. FINKELSTEIN,*

Yeshiva University, New York, New York J. M. JAUCH,

University of Geneva and CERN, Geneva, Switzerland S. SCHIMINOVICH, t

Yeshiva University, New York, New York AND

D. SPEISER,t

University of Geneva, Geneva, Switzerland (Received 10 December 1962)

In this paper the physical implications of quaternion quantum mechanics are further explored. In a quanternionic Hilbert space Xo, the lattice of subspaces has a symmetry group which is isomorphic to the group of all co-unitary transformations in Xo. In contrast to the complex space Xc (ordinary Hilbert space), this group is connected, while for Xc it consists of two disconnected pieces.

The subgroup of transformations in Xo which associates with every quaternion q of magnitude 1, the correspondence if/ ..... qif/q-l for all if/ E X Q (called Q conjugations), is isomorphic to the threedimensional rotation group. We postulate the principle of Q covariance: The physical laws are invariant under Q conjugations. The full significance of this postulate is brought to light in localizable systems where it can be generalized to the principle of general Q covariance: Physical laws are invariant under general Q conjugations. Under the latter we understand conjugation transformations which vary continuously from point to point.

The implementation of this principle forces us to construct a theory of parallel transport of quaternions. The notions of Q-covariant derivative and Q curvature are natural consequences thereof.

There is a further new structure built into the quaternionic frame through the equations of motion. These equations single out a purely imaginary quaternion "I(x) which may be a continuous function of the space-time coordinates. It corresponds to the i in the Schriidinger equation of ordinary quantum mechanics. We consider "I(x) as a fundamental field, much like the tensor gp.. in the general theory of relativity. We give here a classical theory of this field by assuming the simplest invariant Lagrangian which can be constructed out of "I and the covariant Q connection. It is shown that this theory describes three vector fields, two of them with mass and charge, and one massless and neutral. The latter is identifiable with the classical electromagnetic field.

1. INTRODUCTION

I N the development from Galilean to special to general relativity, it was shown by Einstein that

the concepts of Euclidean geometry have only an

* Supported by the National Science Foundation. t Now at the University of Buenos Aires. t Supported by the Swiss Commision for Atomic Research.

approximate validity and that the true laws of geometry are subject to disturbances from place to place. Still more fundamental than the laws of geometry are those of classical logic as expressed in the propositional calculus. In the development from classical to quantum physics it was shown by Bohr that the concepts of classical logic have


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