Reference Data Module

Reference data for validation of GENERIC decomposition computations.

This module provides known-correct values for structure constants and other quantities used in validation tests.

References

SU(2) structure constants:

Completely antisymmetric epsilon tensor ε_ijk [σ_i, σ_j] = 2i ε_ijk σ_k where σ are Pauli matrices

SU(3) structure constants:

Standard structure constants for Gell-Mann matrices Values from: Gell-Mann, M. (1962). “Symmetries of Baryons and Mesons” Also verified against: Particle Data Group reviews

qig.reference_data.get_su2_structure_constants()

Get structure constants for SU(2) algebra (Pauli matrices).

The Pauli matrices satisfy: [σ_i, σ_j] = 2i ε_ijk σ_k where ε_ijk is the Levi-Civita symbol.

Returns:

f_abc – Structure constants with normalization [F_a, F_b] = 2i Σ_c f_abc F_c

Return type:

np.ndarray, shape (3, 3, 3)

Notes

The non-zero components are:

f_123 = +1 (cyclic permutations) f_231 = +1 f_312 = +1 f_213 = -1 (anti-cyclic permutations) f_132 = -1 f_321 = -1

qig.reference_data.get_su3_structure_constants()

Get structure constants for SU(3) algebra (Gell-Mann matrices).

The Gell-Mann matrices λ_a satisfy: [λ_a, λ_b] = 2i Σ_c f_abc λ_c

Returns:

f_abc – Structure constants for SU(3)

Return type:

np.ndarray, shape (8, 8, 8)

Notes

The structure constants have the following properties: - Totally antisymmetric: f_abc = -f_bac - Real valued - Satisfy Jacobi identity

Non-zero structure constants (up to permutations and sign): - f_123 = 1 - f_147 = f_246 = f_257 = f_345 = 1/2 - f_156 = f_367 = -1/2 - f_458 = f_678 = sqrt(3)/2

Reference: Gell-Mann (1962), Particle Data Group

qig.reference_data.verify_structure_constant_properties(f_abc, algebra_name='unknown')

Verify mathematical properties of structure constants.

Parameters:

• f_abc (np.ndarray, shape (n, n, n)) – Structure constants

• algebra_name (str) – Name of algebra for reporting

Returns:

results – Dictionary containing verification results: - antisymmetry_error: Max |f_abc + f_bac| - jacobi_error: Max Jacobi identity violation - is_real: Whether all entries are real - passed: Whether all checks passed

Return type:

Dict[str, Any]

qig.reference_data.get_reference_structure_constants(algebra_type)

Get reference structure constants for a given algebra.

Parameters:

algebra_type (str) – Type of algebra: “su2” or “su3”

Returns:

f_abc – Structure constants

Return type:

np.ndarray

Raises:

ValueError – If algebra_type is not recognized

qig.reference_data.print_reference_verification()

Print verification results for reference structure constants.

This module provides reference structure constants for SU(2) and SU(3) algebras, verified against literature values.

See Validation Framework for verification results.