symmetry-analysis

name	symmetry_analysis
description	Use when identifying symmetries of a physical system, applying group theory, classifying representations, or deriving selection rules.

Symmetry Analysis

Apply this skill when the problem benefits from identifying and exploiting symmetries — to simplify equations, classify states, derive selection rules, or understand degeneracies and conservation laws.

Goal

Identify the symmetry group of the system, classify physical quantities under its representations, and use symmetry to constrain or simplify the solution.

Scope

• Discrete symmetries: parity (P), time reversal (T), charge conjugation (C), lattice point groups
• Continuous symmetries: rotation SO(3)/SU(2), translation, Lorentz/Poincare, internal gauge symmetries
• Noether's theorem: continuous symmetry -> conserved current
• Representation theory: irreducible representations, tensor products, Clebsch-Gordan decomposition
• Selection rules: matrix element vanishing by symmetry
• Spontaneous symmetry breaking and Goldstone's theorem

Inputs

• system_description: Hamiltonian, Lagrangian, or equation of motion
• symmetry_group: Known or suspected symmetry group (may need to be identified)
• quantities_of_interest: Operators, states, or matrix elements to classify

Outputs

• symmetry_group_identified: The symmetry group and its generators
• representation_classification: How states/operators transform under the group
• selection_rules: Which matrix elements vanish by symmetry
• conserved_quantities: Conserved charges/currents from continuous symmetries
• degeneracy_structure: Multiplet structure and degeneracy pattern

Workflow

1. Identify the symmetry group.

   • Examine the Hamiltonian/Lagrangian for invariance under transformations.
   • List all generators and verify they close under commutation (Lie algebra).

2. Classify states by irreducible representations.

   • For finite groups: use character tables.
   • For Lie groups: use weight diagrams, Casimir operators, or highest-weight construction.

3. Derive selection rules.

   • A matrix element <f|O|i> vanishes unless the tensor product of representations of |i>, O, and <f| contains the trivial representation.
   • For angular momentum: apply Wigner-Eckart theorem and triangle rule.

4. Identify conserved quantities.

   • For each continuous symmetry generator G: [H, G] = 0 implies G is conserved.
   • Write the explicit conserved current via Noether's theorem if working with a Lagrangian.

5. Analyze symmetry breaking (if applicable).

   • Determine if the ground state breaks a symmetry of the Hamiltonian.
   • Count Goldstone bosons: one for each broken continuous generator (for relativistic systems).

Quality Checks

• The identified generators must satisfy the correct commutation relations of the group.
• Selection rules must be consistent with known experimental results or exact solutions.
• The number of Goldstone bosons must match the number of broken generators (in relativistic theories).
• Conserved quantities should have vanishing Poisson bracket (classical) or commutator (quantum) with the Hamiltonian.

Constraints

• Do not assume a symmetry without verifying invariance of the full Hamiltonian (including interactions).
• Selection rules are necessary conditions for non-vanishing matrix elements, not sufficient; a matrix element allowed by symmetry may still be numerically small.
• Spontaneous symmetry breaking in finite systems (e.g., finite quantum systems) requires careful treatment; true SSB occurs only in the thermodynamic/infinite-volume limit.