BQ Field Quadratics

A bilinear field such as the electromagnetic field B = ∂^TA has the form F = f 1̂ + k·K̂ + s·σ̂. Its time adjoint flips the σ̂ part, F^T = f 1̂ + k·K̂ − s·σ̂. Two quadratic products follow: the field square FF, which holds the Lorentz invariants, and F^TF, which holds an energy density and its flux in a chosen frame. Choose a field from the BQ papers.

Choose the field

Set the norm channel f = 0

The general rule for a product FG of two such fields, with G = g 1̂ + m·K̂ + t·σ̂, follows from the multiplication table:

FG = (fg − k·m + s·t) 1̂ + (k·t + s·m) T̂
+ (fm + gk + k×m − s×t)·K̂ + (ft + gs + k×t + s×m)·σ̂

For FF take G = F; for F^TF take the first factor with s → −s.

The two quadratic products

The multiplication, term by term

Show with general coordinates f, k_i, s_i FF F^TF

Invariant and observer-dependent

FF is covariant. A field transforms as F → UFU⁻¹, so FF → UFFU⁻¹. Its 1̂ and T̂ channels commute with U and are Lorentz invariant. For a pure bivector field (f = 0) the square has only these two channels.
F^TF is observer-dependent. The time adjoint transforms as F^T → U⁻¹F^TU, so F^TF → U⁻¹F^TU²FU⁻¹: not a similarity transformation. Energy density and flux belong to a chosen time direction, and the product carries that dependence.
Moving to the space-time sector. T̂ commutes with every basis element. Multiplying by T̂ sends 1̂ → T̂, T̂ → −1̂, K̂ → −σ̂ and σ̂ → K̂, so T̂F^TF places an energy–flux pair in the minquat channels T̂ and K̂.

Sources and status

The electromagnetic products BB and B^TB, their invariants and the energy–Poynting reading follow de Haas (2026), BQ electromagnetism and fluid dynamics, eq. 92–106 and 133; the fluid products HH and H^TH follow eq. 277–280 of the same paper. The angular-momentum bivector, the interaction bilinear and the fluid bilinear are included because they have the same 1 + 3 + 3 form; their quadratics are computed with the same rule, and their readings are marked as extensions.

All channels are computed in the page from the general rule and checked term by term against the multiplication table. Part of the BQ algebra models; see also bq-galaxy-models.