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What is symmetry? Immunity (of aspects of a system) to a possible change. The natural language of Symmetry - Group Theory.

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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
  • 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]

Ex. Of continuous group (also Lie gp.)

Group of all Rotations in 2D space - SO(2) group

Det(U) = 1


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 :
  • (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 (Ai) are closed under commutation, i.e. then they form a Lie Algebra.

Generators and physical reality

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

U U = 1

A = A

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)

Action on operator:U U -1 (symmetry transformation)

Again, What is Symmetry?

Symmetry is the invariance of a system under the action of a group

U U -1 = 


Why use Symmetry in physics?

  • 1.Conservation Laws (Noether’s Theorem):
  • 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

For every continuous symmetry of the laws of physics, there must exist a conservation law.

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

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)

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


Dynamical symmetries and Subgroup Chains


Physical implications


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 :
  • So, if , then U is a constant of the motion
  • Continuous compact groups can be represented by Unitary matrices.
  • U can be expressed as (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

Nature of U and A

  • For any finite or (compact) infinite group, we can find Unitary matricesthat represent the group elements
  • U = eiA = 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