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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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Immunity (of aspects of a system) to a possible change
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 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
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 =
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
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 )
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 !!
Effect of parameter (p) :
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