# The natural language of Symmetry Group Theory

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"The natural language of Symmetry Group Theory"

What is symmetry?
Immunity (of aspects of a system) to a possible change
The natural language of Symmetry - Group Theory
We need a super mathematics in which the operations are as unknown as the quantities
they operate on, and a super-mathematician who does not know what he is doing when
he performs these operations. Such a super-mathematics is the Theory of Groups.
- Sir Arthur Stanley Eddington

•GROUP = set of objects (denoted ‘G’) that can be combined by a binary operation
(called group multiplication - denoted by )
•ELEMENTS = the objects that form the group (generally denoted by ‘g’)
•GENERATORS = Minimal set of elements that can be used to obtain (via group
multiplication) all elements of the group

RULES FOR GROUPS:

•Must be closed under multiplication () - if a,b are in G then ab is also in G
•Must contain identity (the ‘do nothing’ element) - call it ‘E’
•Inverse of each element must also be part of group (gg -1 = E)
•Multiplication must be associative - a  (b  c) = (a  b)  c [not necessarily
commutative]
Groups

Discrete groups                           Continuous groups
Elements can be enumerated              Elements are generated by continuously
Ex. Dihedral groups (esp. D4)              varying one or more parameters.
Ex. Lie Groups

Ex. Of continuous group (also Lie gp.)
Group of all Rotations in 2D space - SO(2) group

 x 2  cos        sin    x1
 y 2   sin      cos   y1
                          

cos       sin  
U ( )  
cos 
Det(U) = 1
 sin             
Lie Groups

•Lie Group: A group whose elements can be parameterized by a finite
number of parameters i.e. continuous group where:
1. If g(ai)  g(bi) = g(ci) then - ci is an analytical fn. of ai and bi .
2. The group manifold is differentiable.
( 1 and 2 are actually equivalent)

•Group Generators: Because of above conditions, any element can be
generated by a Taylor expansion and expressed as :
U ( i )  e i 1  A1  2  A2 
(where we have generalized for N parameters).

Convention: Call A1, A2 ,etc. As the generators (local behavior
determined by these).
Lie Algebras

•Commutation is def as : [A,B] = AB - BA

•If generators (A i) are closed under commutation, i.e. Ai , A j    f ijk Ak
k
then they form a Lie Algebra.

Generators and physical reality

•Hermitian conjugate:        A
take transpose of matrix and complex conjugate of elements
U U = 1               A = A
iA
•U = e ------ if U is unitary , A must be hermitian

Hermitian operators ~ observables with real eigenvalues in QM
Symmetry : restated in terms of Group Theory

State of a system:          |                    [Dirac notation]
Transformation:             U| = |             [Action on state]
Linear Transformation:      U ( | + | ) = U| + U| [distributive]
Composition:                U1U2( | ) = U1(U2 | ) = U1 |
Transformation group:       If U1 , U2 , ... , Un obey the group rules,
they form a group (under composition)
-1
Action on operator:         U U        (symmetry transformation)

Again, What is Symmetry?
Symmetry is the invariance of a system under the action of a group
-1
U U        =
Why use Symmetry in physics?

1.      Conservation Laws (Noether’s Theorem):
For every continuous symmetry of the laws of physics, there must exist
a conservation law.

2.      Dynamics of system:
•Hamiltonian             ~ total energy operator
•Many-body problems:     know Hamiltonian, but full system too
complex to solve
•Low energy modes:       All microscopic interactions not significant
Collective modes more important
•Need effective Hamiltonian
Use symmetry principles to constrain general form of effective
Hamiltonian + strength parameters ~ usually fitted from experiment
High TC Superconductivity

•The Cuprates (ex. Lanthanum + Strontium doping)

CuO4 lattice

•BCS or New mechanism? - d-wave pairing with long-range order.
The procedure - 1
1.   Find relevant degrees of freedom for system
2.   Associate second-quantized operators with them (i.e.
Combinations of creation and annihilation operators)

3.   If these are closed under commutation, they form a Lie Algebra
which is associated with a group ~ symmetry group of system.
    Subgroup:     A subset of the group that satisfies the group
requirements among themselves ~ G  A .
    Direct product & subgroup chain: G = A1 A2  A3 ...
if (1) elements of different subgroups commute
and (2) g = a1 a2 a3 ... (uniquely )
The Procedure - 2
4.   Identify the subgroups and subgroup chains ~ these define the
dynamical symmetries of the system. (next slide.)
5.   Within each subgroup, find products of generators that commute
with all generators ~ these are Casimir operators - Ci.
[Ci ,A] = 0  CiA = ACi  ACiA-1 = Ci

 Ci’s are invariant under the action of the group !!

6.   Since we know that effective Hamiltonian must (to some degree
of approximation) also be invariant ~ use casimirs to construct
Hamiltonian
7.   The most general Hamiltonian is a linear combination of the
Casimir invariants of the subgroup chains -
 =  aiCi
where the coefficients are strength parameters (experimental fit)
Dynamical symmetries and Subgroup Chains

Hamiltonian

Physical implications
•Good experimental
agreement with phase
diagram.
Casimirs and the SU(4) Hamiltonian

Casimir
operators

Model Hamiltonian:

Effect of parameter (p) :
High TC Superconductivity - SU(4) lie algebra

•Physical intuition and experimental clues:
Mechanism: D-wave pairing
Ground states:Antiferromagnetic insulators
•So, relevant operators must create singlet and triplet d-wave pairs
•So, we form a (truncated) space ~ ‘collective subspace’ whose basis
states are various combinations of such pairs -

•We then identify 16 operators that are physically relevant:
16 operators ~ U(4) group [# generators of SU(N) = N2 ]
Noether’s Theorem
•If  is the Hamiltonian for a system and is invariant under the action of
a group  U U -1 = 
•Operating on the right with U, U U -1 U =  U
•i.e. Commutator is zero      U -  U = 0 = [ U ,  ]
•Quantum Mechanical equation of motion : dU           U i
      U , H 
dt    t  
U
•So, if        0 , then U is a constant of the motion
t
•Continuous compact groups can be represented by Unitary matrices.
iA
•U can be expressed as    U e           (i.e. a Taylor expansion)
•Since U is unitary, we can prove that A is Hermitian
•So, A corresponds to an observable and U constant  A constant
•So, eigenvalues of A are constant ‘Quantum numbers’  conserved
2
Uf ( x)  f ( x   )    f ( x)  f ' ( x)          f ' ' ( x)  
2!
 d 
  d 
   n    n                 
d                 i  i 
 
              f ( x)  e
n 
dx
f ( x)  e        dx 
f ( x )  e i  A f ( x )
 n 0 n! dx 

       i 1  A1 1  A1      
Uf ( x )  e                         f ( x)
Nature of U and A

•For any finite or (compact) infinite group, we can find Unitary matrices
that represent the group elements
iA
•U = e         = exp(iA)                (A - generator,  - parameter)
•U = unitary               U U = 1     (U - Hermitian conjugate)
•    exp(-iA) exp(iA) = 1
•              exp ( i(A - A) ) = 1
•                       (A - A) = 0      A = A
•So, A is Hermitian and it therefore corresponds to an observable
•ex. A can be Px - the generator of 1D translations
•ex. A can be Lz - the generator of rotations around one axis
Angular momentum theory

1.   System is in state with angular momentum ~ | ~ state is
invariant under 3D rotations of the system.
2.   So, system obeys lie algebra defined by generators of rotation
group ~ su(2) algebra ~ SU(2) group [simpler to use]
3.   Commutation rule:     [Lx,Ly] = i  Lz , etc.
4.   Maximally commuting subset of generators ~ only one generator

5.   Cartan subalgebra ~ Lz
Stepping operators ~ L+ = Lx + i Ly L- = Lx - i Ly
Casimir operator ~ C = L2 = Lx2 + Ly2 + Lz2
6.   C commutes with all group elements ~ CU = UC ~ UCU-1 = C
C is invariant under the action of the group

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 views: 29 posted: 3/10/2012 language: English pages: 20