Symmetry groups in physics

Part 1

George Jorjadze

Free University of Tbilisi

Zielona Gora - 24.01.2017

G.J. — Symmetry groups in physics Lecture 2 1/21

Contents

• Introduction

• Group axioms

• Examples of groups

• Subgroup

• Group homomorphism

• Group representation

• Matrix exponent

• GL(n) and SL(n) groups

• U(n) and SU(n) groups

• SU(2) group

• O(n) and SO(n) groups

• SO(3) group

G.J. — Symmetry groups in physics Contents 2/21

Introduction

Group theory is a mathematical tool that investigates symmetries ofphysical systems and provides effective description of physical phenomena.The methods of group theory allow to handle the consequences ofsymmetries even without the dynamical content of a physical problem.

Physical systems have discrete or/and continuous symmetries.

Examples of discrete symmetries:

• Space inversion ~x 7→ −~x,• Time inversion t 7→ −t,• Charge conjugation e 7→ −e,• Permutations of identical particles.

Examples of continuous symmetries are:

• Space-time translations,• Space rotations,• Lorentz transformations,• Rescaling of space-time coordinates.

G.J. — Symmetry groups in physics Introduction 3/21

Group axioms

A group G is a set of elements g, h, ... with a binary operation g · h ∈ G,which is called product and it satisfies the following axioms:

• Asscociativity. For any three elements (g, h, k) from G one has

g · (h · k) = (g · h) · k .

• Unit element. There is an element e ∈ G, such that for any g ∈ G

e · g = g · e = g .

• Inverse element. Any g ∈ G has an inverse element, denoted by g−1,with the property

g · g−1 = g−1 · g = e .

Exercise 1. Show that G contains only one unit element; and g ∈ G hasonly one inverse element.

In general, the group product is not commutative (see examples).

G.J. — Symmetry groups in physics Group axioms 4/21

Examples of groups

1. Real numbers R form a group if the group product is defined asthe sum of two numbers. Then, the unit element is 0and the inverse element of x ∈ R is −x.

2. Positive real numbers R+ with standard multiplication form a group.Its unit element is 1 and the inverse of a ∈ R+ is 1/a.

3. Non-degenerated n× n matrices form a group under the matrixmultiplications. The group unit element here is the unit matrix.

4. Unitary transformations of a Hilbert space H form a group.These transformations preserve the transition amplitudes, i.e.the scalar product in H

〈U ψ |U φ 〉 = 〈ψ |φ 〉.

The product of two transformations is defined as their composition

(U1 · U2) |ψ 〉 = U1 (U2 |ψ 〉) .

G.J. — Symmetry groups in physics Examples of groups 5/21

5. Galilei transformations relate space-time coordinates of inertial systemsin Newtonian mechanics

t 7→ t̃ = t+ a , xi 7→ x̃i = Rij xj + ai + vi t , (i, j = 1, 2, 3).

Here, a is a time-translation parameter, Rij is a rotation matrix, ai is avector for space translations and vi is a velocity between the inertialsystems. Galilei transformations form a group, where the group product isdefined as a composition of two transformations, similarly to the previousexample. This group is called Galilei group.

6. Poincare transformations, xµ 7→ x̃µ = Λµ ν xν + aµ, relate space-time

coordinates of inertial systems in relativistic mechanics and they formPoincare group. We will discuss this group later in more detail.

Note that the groups in the first two examples are commutative(g · h = h · g for any pair (g, h)), while others are non-commutative.

In the last three examples groups are given as symmetry transformations ofphysical systems. Thus, groups naturally arise in physics.

G.J. — Symmetry groups in physics Examples of groups 6/21

Subgroup

Definition. A subset H of a group G is called subgroup, if H is a groupunder the product given in G.

Examples:

1. Non-zero complex numbers z 6= 0 form a group under the standardmultiplications. Complex numbers with unit norm |z| = 1 form itssubgroup, which is denoted by U(1).

2. Space-time translations and space rotations form subgroups of Galileigroup. The boosts also form its subgroup.

Exercise 2. Check that the matrices

M(θ) =

(cos θ − sin θsin θ cos θ

), θ ∈ [0, 2π) ,

form a subgroup of the group of 2× 2 non-degenerated matrices.Find the matrix M−1(θ). These group is denoted by SO(2).

G.J. — Symmetry groups in physics Subgroup 7/21

Group homomorphism

Definition. A map F : G 7→ G̃ from a group G to a group G̃ is calledhomomorphism, if

F (g · h) = F (g) · F (h) .

An invertible homomorphism is called isomorphism and the isomorphism ofa group to itself is called automorphism.

Examples:

Let us consider the group of real numbers x ∈ R (see Example 1, p.5) andintroduce two maps a) x 7→ ex, b) x 7→ eix.

Exercise 3. Check that: a) provides isomorphisms of R to R+ ,b) is a homomorphism of R into U(1), which is not an isomorphism.

Exercise 4. Check that the map Fh(g) = h · g · h−1, defined by a groupelement h ∈ G, is an automorphism of G. It is called inner automorphism.

Exercise 5. Check that the map θ 7→ −θ is an automorphism of SO(2)(see Exercise 2), but it is not inner an automorphism.

G.J. — Symmetry groups in physics Group homomorphism 8/21

Group representation

Let us consider a linear space L and its one-to-one linear maps f : L 7→ L.The set of all one-to-one linear maps form a group, denoted by GL.The group product in GL is defined as a composition of two maps:if f1 ∈ GL and f2 ∈ GL, then f1 · f2 (x) = f1(f2(x)), for any x ∈ L.

Definition. A homomorphism of a group G in some GL is a representationof G.

Thus, a representation of G is a set of non-degenerated linear operatorsfg : L 7→ L, labeled by the group elements g ∈ G, which satisfy theoperator relation fg · fh = fg·h .

Definition. Representation of G is called irreducible if L has no subspace,apart from trivial, which is invariant under the action of all operators fg.

Definition. Let fg and f̃g be two representations of a group G in linearspaces L and L̃, respectively. These representations are called equivalent,if there exists an invertible map R : L 7→ L̃ such that f̃g = Rfg R

−1, forall g ∈ G.

G.J. — Symmetry groups in physics Group representation 9/21

Definition. Representation of G is called unitary if it preserves the scalarproduct in L, i.e. if 〈 fg(x) | fg(y) 〉 = 〈x | y 〉, for all g ∈ G.

Examples:

1. R2 as a linear space of 2-vectors ~x = (x1, x2) with scalar product〈 ~x | ~y 〉 = x1y1 + x2y2. The SO(2) matrices M(θ), given in Exercise 2,define linear maps of R2, which can be treated as a representation.

Exercise 6. Check that it is an unitary and irreducible representation (UIR).

2. C1 is a 1-dimensional linear complex space of complex numbers z ∈ Cwith scalar product 〈 z |w 〉 = z∗w. Let us consider a linear representationof U(1) in C1 given by eiθz, where eiθ ∈ U(1) and z ∈ C1.

Exercise 7. Check that this defines UIR of U(1).

3. The groups U(1) and SO(2) are obviously isomorphic. Hence, arepresentation of U(1) is a representation of SO(2) and vice versa.Exercise 6 then provides an example of UIR for U(1).

Exercise 8. Find an invertible map from R2 to C1 that makes the UIR’s ofU(1) given in Exercises 6 and 7 equivalent to each other.

G.J. — Symmetry groups in physics Group representation 10/21

Matrix exponent

Let us consider a n× n matrix A. Its exponent is defined by the series

exp[A] = I +A+A2

2!+ · · ·+ An

n!+ · · ·

Exercise 9. Prove that this series converges for any matrix A.

Exercise 10. Calculate the matrix exponents and check the identities

exp

[(0 a0 0

)]=

(1 a0 1

), exp

[(0 bb 0

)]=

(cosh b sinh bsinh b cosh b

),

exp

[(0 −θθ 0

)]=

(cos θ − sin θsin θ cos θ

).

Exercise 11. Prove the Jacobi’s formula for the determinant of matrixexponent

det (exp[A]) = eTr[A] .

G.J. — Symmetry groups in physics Matrix exponent 11/21

GL(n) group

GL(n) group is given by n× n non-degenerated matrices. This group wasalready introduced on page 5 as an example of groups.

If the matrix elements of M ∈ GL(n) are only real numbers the group isdenoted by GL(n,R), and it is denoted by GL(n,C), if the matrix elementsare complex.

The condition detM 6= 0 restricts the domain of possible values of thematrix elements of M but not their number. Therefore, GL(n,R) andGL(n,C) are n2 and 2n2 parametric groups, respectively.

For example, GL(1,R) = R+ ∪ R− and GL(1,C) is a complex planewithout the origin.

Note that the notation GL(n) is related to the fact that this group isassociated with one-to-one general linear transformations of n-dimensionalvector space.

G.J. — Symmetry groups in physics GL(n) group 12/21

SL(n) group

SL(n) is a subgroup of GL(n). Its elements are n× n matrices with unitdeterminant and this set is a subgroup because of

det[M ·N ] = det[M ] · det[N ] .

Similarly to GL(n), one can consider the groups SL(n,R) and SL(n,C).

Obviously, they are n2 − 1 and 2n2 − 2 parametric groups, respectively.

Due to Jacobi’s formula, exp[A] ∈ SL(n), when A is a traceless matrix.

Exercise 12. If A =

(a bc −a

), with a2 + bc > 0, check that

exp

[(a bc −a

)]= coshβ I +

sinhβ

βA ,

where β =√a2 + bc and I is the unit matrix.

Generalize this formula for a2 + bc ≤ 0.

G.J. — Symmetry groups in physics SL(n) group 13/21

U(N) group

This group is a subgroup of GL(n,C) given by the unitary matrices.

A matrix U is called unitary if

U † U = U U † = I ,

where U †, defined by U †kl = U∗lk, is called Hermitian conjugated to U .

Exercise 13. Check that U(n) is n2 parametric group.

A matrix h is called Hermitian if h† = h.

Exercise 14. Check that if h is Hermitian, then exp[ih] ∈ U(n).

Exercise 15. Check that n× n Hermitian matrices form n2 dimension reallinear space.

Exercise 16. Check that the unitary matrices preserve the scalar product inCn, i.e. 〈U(z) |U(w) 〉 = 〈 z |w 〉, where

〈 z |w 〉 =

n∑k=1

z∗k wk .

G.J. — Symmetry groups in physics U(N) group 14/21

SU(n) group

This group is given as the intersection of SL(n,C) and U(n), whichmeans that M ∈ SU(n), when M is unitary and det[M ] = 1.

Exercise 17. Check that SU(n) is n2 − 1 parametric group.

It is clear that if h is Hermitian and traceless, then exp[ih] ∈ SU(n).

Exercise 18. Check that the number of n× n traceless Hermitian matricesis n2 − 1.

For example, for n = 2 one can choose the matrices

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

),

which are called Pauli matrices.

Exercise 19. Check that Pauli matrices satisfy the identities

σk σl = δklI + iεklm σm ,

where I is the unit matrix, δkl and εklm are Kroneker and Levi-Chivitasymbols, respectively.G.J. — Symmetry groups in physics SU(N) group 15/21

SU(2) group

Exercise 20. Using the result of Exercise 19, check that

exp [iαkσk] = cosα I +sinα

α[iαkσk] ,

with α =√α21 + α2

2 + α23 .

Let us represent M ∈ SU(2) in the form M =

(a bc d

).

One then gets M−1 =

(d −b−c a

), since det[M ] = 1.

Becouse of M−1 = M †, one obtains d = a∗, c = −b∗ and M takes theform

M =

(a b−b∗ a∗

), with |a|2 + |b|2 = 1 .

The last equation defines 3d sphere S3 embedded in R4.Thus, geometrically, SU(2) group is identified with S3.

G.J. — Symmetry groups in physics SU(2) group 16/21

O(N) group

O(n) is the group of n× n real orthogonal matrices.

M matrix is called orthogonal if MkiMkj = δij .

Thus, O(n) is a subgroup of GL(n,R) defined by the matrix relations

MT ·M = M ·MT = I ,

where MT is the transposed matrix to M , given by MTkl = Mlk.

Exercise 21. Check that O(n) is n(n−1)2 parametric group.

Exercise 22. Check that rows (and columns) of an orthogonal matrix Mare orthonormal.

Exercise 23. Check that the orthogonal matrices preserve the scalarproduct in Rn, i.e. 〈M(x) |M(y) 〉 = 〈x | y 〉, where

〈x | y 〉 =

n∑k=1

xk yk .

G.J. — Symmetry groups in physics O(n) group 17/21

SO(n) group

From the definition of O(n) follows that det[M ] = ±1, for M ∈ O(n).Since determinant is a continuous function of the matrix elements, O(n)group contains two non-connected parts:one with unit determinant and another with determinant equal to -1.

Orthogonal matrices with unit determinant form SO(n) group.The corresponding matrices are called rotations.

Exercise 24. Check that SO(2) group is completely described by thematrices M(θ) of Exercise 2.

The set of orthogonal matrices with determinant equal to −1 can bewritten as P ·SO(n), where P corresponds to the reflection of the first axe.For example, in O(2) one gets P = diag(−1, 1), and

P · SO(2) =

(− cos θ sin θsin θ cos θ

).

Exercise 25. Check that exp[A] ∈ SO(n) if AT = −A.

G.J. — Symmetry groups in physics SO(n) group 18/21

SO(3) group

Note that SO(1) group contains only one element, which is number 1.SO(2) group elements are given by the matrices M(θ) (see Exercise 2)and they can be written as exponents presented in Exercise 10.

Now we consider SO(3) matrices. They correspond to rotations in R3.The rotation matrices around x, y and z axes, respectively, are

R1(α) =

1 0 00 cosα − sinα0 sinα cosα

, R2(β) =

cosβ 0 sinβ0 1 0

− sinβ 0 cosβ

,

R3(γ) =

cos γ − sin γ 0sin γ cos γ 0

0 0 1

.

An arbitrary SO(3) group element can be written as a product of thesethree matrices. Therefore, the rotation angles (α, β, γ) can be treated asgroup parameters.

G.J. — Symmetry groups in physics SO(3) group 19/21

A more common parameterization of the SO(3) rotations is given by theEuler angels. These angels describe the orientation of a rotated systemwith respect to the initial system, and this orientation is obtained by thefollowing consecutive rotations

R(ϕ, φ, θ) = R3(ϕ) ·R1(φ) ·R3(θ) .

To get the unique parameterization of the orientation of the rotatedsystem one has to choose the angles within the intervals:

0 ≤ ϕ < 2π, 0 ≤ φ ≤ π, 0 ≤ θ < 2π.Using the form of the matrices R1 and R3, one finds

R(ϕ, φ, θ) =(cosϕ cos θ − sinϕ cosφ sin θ sinϕ cosφ cos θ + cosϕ sin θ sinϕ sinφ− sinϕ cos θ − cosϕ cosφ sin θ cosϕ cosφ cos θ − sinϕ sin θ cosϕ sinφ

sinφ sin θ − sinφ cos θ cosφ

).

Exercise 26. Check that the matrix R(ϕ, φ, θ) indeed has this form.

Exercise 27. Check the orthonormality of the columns and rows inR(ϕ, φ, θ) and calculate its determinant.

G.J. — Symmetry groups in physics SO(3) group 20/21

Future plan

In the next lecture we introduce Lorentz and Poincare groups as symmetrytransformations of spacetime in relativistic theories.

We also consider the symplectic group as the group of phase space linearcanonical transformations in Hamiltonian mechanics.

We then discuss general structures of Lie groups and Lie algebras.

We introduce sl(2,R) algebra as a simple and remarkable example andanalyzed its structure in more detail.

Finally, we construct finite dimensional UIRs of spin group and present aUIR of sl(2,R) algebra.

G.J. — Symmetry groups in physics Future plan 21/21