← The BQ algebra at the Dirac level

BQ-Dirac Triple Product Explorer

The product of two Dirac-level four-vectors is even: C = AB = s𝟙 + cα·α + cΣ·Σ, a scalar and two bivectors. Multiplying it from the left by a third vector D = d0β0 + d·β gives the triple product E = D(AB), which is odd: it fills the vector channels βμ and the pseudovector channels β̃μ, and nothing else. Choose D, A and B from the four-vectors used in the BQ papers.

E = (d0s − d·cα) β0 + (ds − d0cα − d×cΣ)·β − (d·cΣ) β̃0 + (d×cα − d0cΣ)·β̃

Choose the three four-vectors

Triple products from the papers

No time adjoint is used at the Dirac level. D stands to the left of C throughout, so when D = ∂ its derivatives act on the channels of C.

Channels of E = D(AB)

The multiplication, term by term

Check with numbers, by explicit 4×4 matrix multiplication

How the channels arise

  • From the multiplication table. β0αi = −βi and β0Σi = −β̃i; βiαi = −β0 and βiΣi = −β̃0; for i ≠ j, βiαj = εijkβ̃k and βiΣj = −εijkβk.
  • Selective coupling. β̃0 sees only cΣ; β0 sees only s and cα; the spatial β and β̃ outputs mix all three.
  • Grades. A vector times an even element is odd: E has only grade 1 (βμ) and grade 3 (β̃μ) parts; 𝟙, α, Σ and χ stay empty.
  • Vector and pseudovector. The βμ part is a space-time four-vector such as a four-force; the β̃μ part is its pseudovector companion. For Maxwell, the vector sector carries the sourced equations, the pseudovector sector the homogeneous ones.

The Pauli triple product inside

With D = [[0, D], [−DT, 0]] and C = diag(−ABT, −ATB), the triple product is block off-diagonal:

D(AB) = [[0, −D(ATB)], [DTABT, 0]]

The upper-right block is minus the BQ-Pauli triple product E = D(ATB). Reading the channels gives [E]β0 = −eT, [E]β = −eK, [E]β̃0 = e1, [E]β̃ = eσ: the four Pauli channel structures, with the Pauli minquat (T̂, K̂) mapped to the vectors and the pauliquat (1̂, σ̂) to the pseudovectors.

Named channels. Where the papers name the channels of C the page uses those names: b1, E/c and B for ∂A; −L, S and T for JA; S, M and L for RP; −𝓛, Tσ and TK for UG; σE, −Π/c and Ω for ∂G. Otherwise s, cα and cΣ.

Maxwell, Lorentz force and fluid products from de Haas (2026), BQ-Weyl electromagnetism, eqs. 90–113 and 397–409. Channels are derived symbolically in the page and checked numerically from the explicit matrices. Part of the BQ algebra at the Dirac level; the Pauli version is the BQ triple product explorer.