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
Ex. Of continuous group (also Lie gp.)
Group of all Rotations in 2D space - SO(2) group
Det(U) = 1
Generators and physical reality
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?
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 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 -
where the coefficients are strength parameters (experimental fit)
Ci’s are invariant under the action of the group !!
Dynamical symmetries and Subgroup Chains
Casimirs and the SU(4) Hamiltonian
Effect of parameter (p) :
High TC Superconductivity - SU(4) lie algebra
Nature of U and A
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