BQ-Dirac Lorentz Covariance of the Triple Product

A triple product E = D(AB) is odd: a vector part e₀β₀ + e·β and a pseudovector part r₀β̃₀ + r·β̃. Boosting the three factors along I boosts the triple by the same similarity as vectors and bilinears, E^L = ΛEΛ⁻¹, and both sectors transform as four-vectors. Equations such as Maxwell’s ∂B = −μ₀J and the Lorentz force JB therefore keep their form in every frame. Choose D, A and B freely, or click an equation from the papers, and set a velocity.

E^L = D^L(A^LB^L) = ΛEΛ⁻¹: e₀′ = γ(e₀ − βe_I), e_I′ = γ(e_I − βe₀), r₀′ = γ(r₀ − βr_I), r_I′ = γ(r_I − βr₀), J, K unchanged

Triple and boost

D A B

Set the scalar channel s of AB to 0

Equations from the papers

Velocity β = v/c along I

Covariance of the triple

Why: one similarity for vectors, bilinears and triples

Check with numbers, by explicit 4×4 matrices

Three kinds of product, one rule

Pauli level. A vector goes as U⁻¹AU⁻¹, a bilinear A^TB as UCU⁻¹, a triple D(A^TB) as U⁻¹EU⁻¹: three rules, set by where the time adjoint stands.
Dirac level. All three go as Λ…Λ⁻¹. What moves is decided by the grade: the vector and pseudovector channels by the lemma β_μΛ_ν^μ = Λβ_νΛ⁻¹, the bivectors in mixing pairs, the scalar and χ not at all.
Two four-vectors in one. β̃_μ = χβ_μ and χ commutes with Λ, so the pseudovector sector is boosted exactly like the vector sector: the Dirac form of the Pauli statement that (e_T, e_K) and (e₁, e_σ) are both four-vectors.

Sources and status

Triple rule, Maxwell and the Lorentz force from de Haas (2026), BQ-Weyl electromagnetism, eqs. 90–113, and the fluid equations from eqs. 397–409 of the same paper; Λ and X^L = ΛXΛ⁻¹ from eqs. 161–196. The Pauli version is the BQ triple-product covariance model.

All identities, including the four-gradient acting on a general field, are checked exactly in bq_dirac_triple_lorentz.py. Part of the BQ algebra at the Dirac level.