BQ Lorentz Covariance of the Triple Product

A triple product E = D(A^TB) multiplies a four-vector, boosted as U⁻¹DU⁻¹, with a bilinear, boosted as UCU⁻¹. The inner U⁻¹U cancels, so E is boosted like a four-vector again: E^L = U⁻¹EU⁻¹. Its minquat sector e_TT̂ + e_K·K̂ and its pauliquat sector e₁1̂ + e_σ·σ̂ each transform as a four-vector, so equations such as ∂B = μ₀J and JB = F keep their form in every inertial frame. Choose a triple and a velocity.

Triple and boost

D A B

Velocity β = v/c along Î

Covariance of the equation

The boost of the triple product in seven steps

The steps use the triple and velocity chosen above. As on the bilinear page, ch = cosh(ψ/2) and sh = sinh(ψ/2) are the C and S of the slides.

Both sectors on their hyperbolas

EE^L = U⁻¹EU⁻¹E^L⁻¹ = UEUinvariant hyperbola

Left: the minquat sector (e_K,I, e_T), like (a₁, a₀) of a four-vector. Right: the pauliquat sector (e_σ,I, e₁). The same rapidity moves both points along their hyperbolas e_T² − e_K,I² and e₁² − e_σ,I²; the transverse components do not change.

Check by explicit matrix multiplication

How the three kinds of product transform

Vector A = a₀T̂ + a·K̂: A^L = U⁻¹AU⁻¹, and its adjoint (A^L)^T = UA^TU.
Bilinear C = A^TB: C^L = (UA^TU)(U⁻¹BU⁻¹) = UCU⁻¹, a similarity; the norm is invariant and the K̂ and σ̂ channels mix.
Triple E = DC: E^L = (U⁻¹DU⁻¹)(UCU⁻¹) = U⁻¹EU⁻¹, vector-like again. Every odd product of four-vectors and adjoints is vector-like, every even product is a similarity, as for the grades of a Clifford algebra.
Two four-vectors in one. U⁻¹1̂U⁻¹ = coshψ 1̂ − sinhψ σ̂_I mirrors U⁻¹T̂U⁻¹ = coshψ T̂ − sinhψ Î, so (e₁, e_σ) transforms exactly like (e_T, e_K). Three invariants follow: e_T² − e_K², e₁² − e_σ² and e_Te₁ − e_K·e_σ.

What covariance means here

If DC = X holds in one frame, then D^LC^L = U⁻¹(DC)U⁻¹ = U⁻¹XU⁻¹ = X^L holds in the boosted frame: the equation keeps its form, and X must itself be a four-vector-like object. A sector that vanishes in one frame vanishes in all, so the homogeneous Maxwell equations (the pauliquat sector of ∂B) and the BQ Bianchi identities hold in every frame.

For a four-gradient on the left the check below is done with real fields: the field is boosted as C′(x′) = UC(x)U⁻¹, its derivatives are taken in the new coordinates x′ = Λx, and ∂′C′ is compared with U⁻¹(∂C)U⁻¹.

Triple rule and the Maxwell and Lorentz-force examples from the BQ-Pauli slides; the fluid and interaction equations from de Haas (2026), BQ electromagnetism and fluid dynamics. Part of the BQ algebra models.