Noncommutative algebra API

Expressions over noncommutative symbols (pool.symbol(name, commutative=False)) read as noncommutative polynomials, with word-rewriting normal forms. See the Noncommutative algebra chapter of the user guide.

alkahest.nc_expand(expr) → Expr

Distribute products keeping factor order, unroll non-negative integer powers of noncommutative factors, collect equal words.

alkahest.nc_terms(expr) → list[tuple[Expr, list[Expr]]]

The (coefficient, word) terms of nc_expand().

alkahest.nc_rewrite(expr, rules, *, max_steps=200000) → Expr

Rewrite to a normal form with literal word rules from make_rule(), matching inside longer words. Raises NcError E-NC-001 when no normal form is reached within max_steps, E-NC-003 for a left side that is not a single word.

alkahest.commutator(a, b, rules=None) → Expr

alkahest.anticommutator(a, b, rules=None) → Expr

a*b ∓ b*a expanded; with rules, rewritten by nc_rewrite().

alkahest.normal_order(expr, bosons=None, fermions=None) → Expr

Normal-order with [a, a†] = 1 (bosons) and {c, c†} = 1, c² = 0 (fermions). bosons / fermions are (annihilator, creator) pairs or lists of them. Raises E-NC-004 for a malformed mode list.

alkahest.vacuum_expectation(expr, bosons=None, fermions=None) → Expr

⟨0|expr|0⟩, the scalar part of normal_order(). Raises E-NC-007 for a factor that is not a listed ladder operator.

alkahest.gamma_trace(slots, pool=None) → Expr

Four-dimensional Dirac trace, metric diag(+1,−1,−1,−1), γ⁵ = iγ⁰γ¹γ²γ³. Slots are 0–3, 5 or a commutative Expr (slashed vector). The result is a polynomial in dot(p, q) and levi_civita(a, b, c, d) (ε^{0123} = +1). Raises E-NC-005 for a bad slot and E-NC-006 past 14 slots.

simplify_pauli reduces any word over sx, sy, sz with the full Pauli algebra; make_rule / simplify_with match product patterns inside longer products (contiguous runs for noncommutative products, sub-multisets for commutative ones).

Refusals raise NcError (E-NC-001 … E-NC-007); see Error handling API.