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What is symmetry? Immunity (of aspects of a system) to a possible change

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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
- RULES FOR GROUPS:
- Must be closed under multiplication () - if a,b are in G then ab is also in G
- Must contain identity (the ‘do nothing’ element) - call it ‘E’
- Inverse of each element must also be part of group (gg -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:Atake transpose of matrix and complex conjugate of elements
- U = eiA ------ 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 significantCollective 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

Hamiltonian

Physical implications

- Good experimental agreement with phase diagram.

Extra Slides

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 = eiA = exp(iA)(A - generator, - parameter)
- U = unitary UU = 1 (U - Hermitian conjugate)
- exp(-iA) exp(iA) = 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 ~ LzStepping operators ~ L+ = Lx + i Ly L- = Lx - i LyCasimir operator ~ C = L2 = Lx2 + Ly2 + Lz2

6.C commutes with all group elements ~ CU = UC ~ UCU-1 = CC is invariant under the action of the group