17. Group Theory

1. Introduction to Group Theory

2. Representation of Groups

3. Symmetry & Physics

4. Discrete Groups

5. Direct Products

6. Symmetric Groups

7. Continuous Groups

8. Lorentz Group

9. Lorentz Covariance of Maxwell’s Equations

10. Space Groups

1. Introduction to Group TheorySymmetry :

1.Spatial symmetry of crystals ~ X-ray diffraction patterns.

2.Spatial symmetry of molecules ~ Selection rules in vibrational spectra.

3.Symmetry of periodic systems ~ e-properties: energy bands, conductivity, …

Invariance under transformations :

1.Linear displacement ~ Conservation of (linear) momentum.

2.Rotation ~ Conservation of angular momentum.

3.Between (inertial) frames ~ General (special) relativity.

Theories of elementary particles begin with symmetries & conservation laws.

Group theory was invented to handle symmetries & invariance.

3. Identity

1. Closure

2. Associativity

Definition of a Group

a b G

A group { G, } is a set G with a multiplication such that a, b, c G ,

a b c a b c a b c

unique I G I a a I a

4. Inverse1 1 1a G a a a a I

Group { G, } is usually called simply group G and a b, ab.

Two easily proved theorems :

1.Every a1 is unique.

2. Rearrangement theorema G G

Refs: W.K.Tung, “Group Theory in Physics” (85)M.Tinkham, “Group Theory & QM” (64)

Abelian group : is commutative, i.e.,

More Definitions

Discrete group : 1-1 map between set G & a subset of the natural number.

( label of elements of G is discrete )

Finite group : Group with a finite number n of elements.

n = order of the group.

Continuous group with n-parameter : 1-1 map between set G & subset of Rn .

,ab ba a b G

Cyclic group Cn of order n : 2, , , nnC a a a I Cn is abelian

Group {G, } is homomorphic to group { H , } :

a map f : G H that preserves multiplications, i.e.,

, ,a b c a b c G f a f b f c

If f is 1-1 onto ( f1 exists ), then {G, } and { H , } are isomorphic. Subgroup of group {G, } : Subset of G that is closed under .

Example 17.1.1. D3 Symmetry of an Equilateral Triangle

Dihedral group

Subgroups :

23 3 3, ,C I C C

2 2,C I C

2,I C

2,I C

gi

gj

Table of gi gj for D3

Mathematica

Example 17.1.2. Rotation of a Circular Disk

cos sin

sin cos

x x

y y

R r r

Rotation in x-y plane by angle :

0I R R R R R R

1-D continuous abelian group.

1R R

,G R

Example 17.1.3. An Abstarct Group

Vierergruppe (4-group) :

I A B C

I I A B C

A A I C B

B B C I A

C C B A I

An abstract group is defined by its multiplication table alone.

Example 17.1.4. Isomorphism & Homomorphism: C4

C4 = Group of symmetry operations of a square that can’t be flipped.

2 34 4 4 2 4, , ,C I C C C C

34 2 4

34 2 4

34 4 2 4

32 2 4 43 3

4 4 4 2

I C C C

I I C C C

C C C C I

C C C I C

C C I C C

1, , 1 ,G i i

1 1

1 1 1

1 1

1 1 1

1 1

i i

i i

i i i

i i

i i i

C4 & G are isomorphic.

Subgroup: 2 2,C I C

1, 1

abelian

2. Representation of Groups

A representation of a group is a set of linear transformations on a

vector space that obey the same multiplication table as the group.

Matrix representation : Representation in which the linear

transformations tak the form of invertible matrices

( done by choosing a particular basis for the vector space ).

Unitary representation : Representation by unitary matrices.

Every matrix representation is isomorphic to a unitary reprsentation.

Example 17.2.1. A Unitary Representation

Unitary representations for :

1 0

0 1EU I

3

1 3

2 2

3 1

2 2

EU C

23

1 3

2 2

3 1

2 2

EU C

2

1 0

0 1EU C

2

1 3

2 2

3 1

2 2

EU C

2

1 3

2 2

3 1

2 2

EU C

1 1A

U g g G

2

23 3

2 2 2

1 , ,

1 , ,A g I C C

U gg C C C

23 3 3 2 2 2, , , , ,D I C C C C C

W g

W2

Let U(G) be a representation of G, then is also a representation.

More Definitions & Properties

A representation U(G) is faithful if U(G) is isomorphic to G.

U G U g g G

Every group has a trivial representation with 1U g g G

1W G V U G V

W(G) & U(G) are equivalent representations :

A representation U(G) is reducible if every U(g) is equivalent to the same block

diagonal form, i.e., for some 1 2W g W g W g

A representation U(G) is irreducible if it is not reducible.

1 2U U U We then write :

~W U

~W U

All irreducible representations (IRs) of an abelian group are 1-D.

Commuting matrices can be simultaneously digonalized

W1

Example 17.2.2. A Reducible Representation

A reducible representation for :

1 0 0

0 1 0

0 0 1

U I

3

0 1 0

0 0 1

1 0 0

U C

23

0 0 1

1 0 0

0 1 0

U C

2

0 0 1

0 1 0

1 0 0

U C

2

1 0 0

0 0 1

0 1 0

U C

2

0 1 0

1 0 0

0 0 1

U C

Using & ,

we get the equivalent block diagonal form

1 1 1

3 3 3

1 2 1

36 61 1

02 2

V

1U V U V

1 0 0

0 1 0

0 0 1

U I

3

1 0 0

1 30

2 2

3 10

2 2

U C

23

1 0 0

1 30

2 2

3 10

2 2

U C

2

1 0 0

0 1 0

0 0 1

U C

2

1 0 0

1 30

2 2

3 10

2 2

U C

2

1 0 0

1 30

2 2

3 10

2 2

U C

23 3 3 2 2 2, , , , ,D I C C C C C

Example 17.2.3. Representations of a Continuous Group

Symmetry of a circular disk : cos sin

sin cosR

,G R

Let & 11

12

iV

i

1U V R V

cos sin 0

0 cos sin

iU

i

0

0

i

i

e

e

G is abelian R is reducible.

Independent IRs :

i nnU e 0, 1, 2, 3,n

Only U1 & U1 are faithful.

i.e., is also an eigenfunction with eigenvalue E.

i.e., is the tranformed hamiltonian

& is the transformed wave function

3. Symmetry & Physics

Let R be a tranformation operator such as rotation or translation.

R H R E

H E r r

H T V

1R H R R ER

If H is invariant under R : 1R H R H

1R H R

R

, 0R H

R

possibility of degeneracy.

Actual degeneracy depends on the symmetry group of H & can be calculated,

without solving the Schrodinger eq., by means of the representation theory .

H R ER

i.e., is a representation of G on the space spanned by .

Starting with any function , we can generate a set

1; with , 1,2, ,i i GS R R G R I i n

1

d

i j j ij

R U R

R G

1

d

i j j ij

S R S U R

S G 1 1

d d

k k j j ij k

U S U R

1

d

k k ik

U S R

1

d

k i k j j ij

U SR U S U R

Or, in matrix form : SR S R U U U i ji jR U RU

;U R R G U

Next, we orthonormalize S using, say, the Gram-Schmidt scheme, to get

order of

symm. group of Gn G

G H

; 1,2, ,i Gi d n

= basis that spans an d –D space.

U is in general reducible, i.e.,

where m = number of blocks equivalent to the same IR U ( ).

U m U

Starting with any function , we can generate a basis

for a d-D representation for G.

;i i orthnormalizedR R G ; 1,2, ,i i d

;U R R G U

For arbitary , we can take one state from each U ( ) block to get

a basis to set up a matrix eigen-equation of H to calculate E.

, 0H R R G EH I w.r.t. a basis for an IR of G.( Shur’s lemma )

If is an eigenfunction of H, then U is an IR.

Example 17.3.1. An Even H

H is even in x H x H x

Let be the operator then x x , sG I C

multiplication table

sC I

I I

I

1

2

1 1

1 1

sC I

A

A

IR

Cs is abelian All IRs are 1-D.

For an arbitrary (x) :

,S x x x

I x x

I x x

1 0

0 1I

U

x x

x x

0 1

1 0

U

1 0

0 1I

W

1 0

0 1

W

1 1

1 1

V

1W V U V

1 1

1 1

x x x

x x x

1

2

A

A

Even

Odd

V

= basis for W

Mathematica

where P ( ) = projector onto the space of unitary IR U ( ).

( ) (g) = Character (trace) of U ( ) (g) .

n = dimension of IR.

nG = order of G.

R(g) = operator corresponding to g.

For any f (x),

, if not empty, is the ith basis vector for the IR U ( ).

, if not empty, is a basis vector for the IR U ( ).

; 1, ,i jP f x j n

Generation of IR Basis

Using Schur’s lemma, one can show that (Tung, §4.2)

*

i j i jg GG

nP U g R g

n

g GG

nP g R g

n

; 1, ,P f x j n

2

1 2 3 1 2 3

1

3A

r r r r r r r

Example 17.3.2.QM: Triangular Symmetry

3G D

3 atoms at vertices Ri of an equilaterial triangle :

1

1 2 3

2

3A

r r r r

i ir r R

Starting with atomic s-wave function (r1) at R1 :

*

i j i jg GG

nP U g R g

n

1 0

0 1EU I

3

1 3

2 2

3 1

2 2

EU C

23

1 3

2 2

3 1

2 2

EU C

2

1 0

0 1EU C

2

1 3

2 2

3 1

2 2

EU C

2

1 3

2 2

3 1

2 2

EU C

11 1 2 3 1 3 2

2 1 1 1 1

3 2 2 2 2E r r r r r r r

1 2 3

22

3r r r

12 2 3 3 2

2 3 3 3 3

3 2 2 2 2E r r r r

r 2 3

2

3r r

0

2

23 3

2 2 2

1 , ,

1 , ,A g I C C

U gg C C C

1 1A

U g g G

4. Discrete Groups

For any a G, the set is called a class of G. 1 ;C g a g g G

Rearrangement theorem

A class can be generated by any one of its members.

( a can be any member of C ).

C is usually identified by one of its elements.

Classes :

Example 17.4.1. Classes of D3 23 3 3 2 2 2, , , , ,D I C C C C C

Mathematica

ga

Table of g a g1 for D3

Classes of D3 are :

23 3 2 2 2, , , , ,I C C C C C

Usually denoted as

3 2, 2 , 3I C C

1Tr g a g Tr a g All members of a class have the same character(trace).

Orthogonality relations :

GC

C C n

Dimensionality theorem :

2Gn n

G

CCC

nC C

n

GCC

C

nC C

n

GC

C C n

2Gn n

Normalized full representation table of D3 :

Take each row (column) as vector :

They’re all orthonormalized.

Sum over column (row) then gives the completeness condition.

G

n

n

Example 17.4.2. Orthogonality Relations: D3

D3

Character table of D3

Mathematica

2Gn n

A1 , E :

C GC

n C C n

1 1 2 2 1 1 3 1 0 0 E , E : 1 2 2 2 1 1 3 0 0 6

2 2 21 1 2 6

1 1 1 1 1 0 0

GCC

C

nC C

n

C3 , C2 :

1 1 1 1 1 1 3 C3 , C3 :

row orthogonality

Completeness

6

2

Example 17.4.3. Counting IRs

2 34 4 4 4, , ,C I C C C 2 3~ , , ,I a a a

multiplication tableC4

gb

Table of g b g1 for C4

C4

Character table

Example 17.4.4. Decomposing a Reducible Representation

Other Discrete Groups

5. Direct Products

6. Symmetric Groups

7. Continuous Groups

8. Lorentz Group

9. Lorentz Covariance of Maxwell’s Equations

10. Space Groups