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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