BQ Triple Product Explorer

The bilinear C = A^TB of two minquats has three channels, c₁1̂ + c_K·K̂ + c_σ·σ̂. Multiplying it from the left by a third minquat D = d₀T̂ + d·K̂, with K̂ = (Î, Ĵ, K̂), gives the triple product E = D(A^TB), which fills all four BQ channels 1̂, T̂, K̂ and σ̂. Choose D, A and B from the four-vectors used in the BQ papers.

Choose the three four-vectors

D A B

Set the norm channel c₁ = 0 Write C’s channels out in full

Triple products from the papers

The superscript T is the BQ time-reversal operation, A^T = −a₀T̂ + a·K̂. D stands to the left of C throughout, so when D = ∂ its derivatives act on the channels of C.

Channels of E = D(A^TB)

E = −(d·c_K) 1̂ + (d₀c₁ + d·c_σ) T̂ + (dc₁ + d₀c_σ + d×c_K)·K̂ + (d×c_σ − d₀c_K)·σ̂

The multiplication, term by term

Check with numbers, by explicit 2×2 matrix multiplication

How the channels arise

From the multiplication table. (d·K̂)(c_K·K̂) = −d·c_K 1̂ + (d×c_K)·K̂ and (d·K̂)(c_σ·σ̂) = (d·c_σ)T̂ + (d×c_σ)·σ̂; the time part gives d₀T̂(c_K·K̂) = −d₀c_K·σ̂ and d₀T̂(c_σ·σ̂) = d₀c_σ·K̂.
Selective coupling. The 1̂ output sees only c_K; the T̂ output sees only c₁ and c_σ; the K̂ and σ̂ outputs mix all three.
Closure. E stays inside the eight-element BQ-Pauli basis, 1 + 3 + 3 + 1: no new algebraic element appears.
Minquat and pauliquat sectors. The T̂ + K̂ part of E is a minquat (a space-time four-vector such as a four-force); the 1̂ + σ̂ part is a pauliquat in the spin-norm sector.

Named channels and the full form

Where the papers name the channels of C, the page uses those names: B and E/c for ∂^TA, L, T and S for J^TA, −S, L and M for R^TP, 𝓛, T_K and T_σ for U^TG, and −σ_E, Ω and −Π/c for ∂^TG. Otherwise it writes c₁, c_K and c_σ and lists their values. “Write C’s channels out in full” replaces every name by its expression in brackets; vector identities such as ∇·(∇×X) = 0 are then left for the reader.