← The BQ algebra at the Dirac level

BQ-Dirac Field Quadratics

A field at the Dirac level is an even element, F = f𝟙 + p·α + q·Σ: a scalar, a boost bivector and a spin bivector, such as the electromagnetic field B = ∂A. Two quadratic products follow. The field square FF holds the Lorentz invariants. The sandwich FaF takes a four-vector a to a four-vector: with a = βμ it gives the energy density, the energy flux and the stress. No time adjoint is needed at this level. Choose a field from the BQ papers.

FF = (f² + p² − q²) 𝟙 + 2fp·α + 2fq·Σ − 2(p·q) χ
FaF = [(f² − p² − q²)a0 + 2a·(p×q)] β0 + [(f² + p² + q²)a − 2(a·p)p − 2(a·q)q − 2a0(p×q)]·β − 2f(a·q) β̃0 + 2f(a×p − a0q)·β̃

Choose the field

Two fields F and G = g𝟙 + m·α + n·Σ multiply by the BQ-Dirac table into the even algebra, including the pseudoscalar χ:

FG = (fg + p·m − q·n) 𝟙 + (fm + gp − p×n − q×m)·α
+ (fn + gq + p×m − q×n)·Σ − (p·n + q·m) χ

For FF take G = F: the cross products cancel. The sandwich is F(aF), a vector times an even element (the triple rule) followed by one more even factor.

The two quadratic products

The four basis insertions FβμF

Column μ holds the channels of FβμF. Because the sandwich is linear in a, these four columns fix it for every four-vector: FaF = aμ FβμF.

The field square, term by term

Check with numbers, by explicit 4×4 matrix multiplication and a Lorentz boost

Invariant, covariant and frame-resolved

  • The square is invariant. A boost along I is X → ΛXΛ−1 with Λ = diag(U−1, U), U = diag(eψ/2, e−ψ/2). Then F2 → ΛF2Λ−1, and 𝟙 and χ commute with Λ: for a pure bivector field the square is a pair of Lorentz scalars.
  • The sandwich is covariant. With a transformed as well, FLaLFL = Λ(FaF)Λ−1: a vector-valued map, whose coefficients form a mixed rank-two tensor.
  • A fixed β0 is frame-resolved. FLβ0FL is the energy and flux for the new observer, not the transformed old ones. The observer dependence that the Pauli product FTF carried is now an explicit vector argument.
  • Grades. For f = 0 the square lies in Λ0 ⊕ Λ4 and the sandwich in Λ1. A scalar channel f adds bivectors to the square and a pseudovector Λ3 part to the sandwich.

The Pauli quadratics inside

With the BQ-Pauli field F = −f1 + q·K̂ + p·σ and its time adjoint FT = −f1 + q·K̂ − p·σ (the T̂ and σ channels change sign), the Dirac field is block diagonal, and so are its products:

F = diag(−FT, −F),   FF = diag(FTFT, FF),   FaF = [[0, FTAF], [−FATFT, 0]]

The lower block of the square is the Pauli field square: [FF]𝟙 = [FF]1, [FF]χ = −[FF]T̂, [FF]α = −[FF]σ, [FF]Σ = −[FF]K̂. With a = β0 (A = T̂) the upper block of the sandwich is T̂FTF, the Pauli energy–flux product moved into the space-time channels: [Fβ0F]β0 = [T̂FTF]T̂, β = K̂, β̃0 = −1, β̃ = −σ.

Electromagnetic products from de Haas (2026), BQ-Weyl electromagnetism, eqs. 118–133 and 145–222; fluid products from eqs. 286–321 of the same paper. The bilinear fields RP, JA and UG have the same 1 + 3 + 3 form; their quadratics follow the same rules and their readings are marked as extensions. The paper writes the boost rotor as U = exp((ψ/2)Î) in eq. 161, which is a phase diag(eiψ/2, e−iψ/2); the matrix Λ of eq. 163 needs U = exp((ψ/2)σI), which is what this page uses. Part of the BQ algebra at the Dirac level; the Pauli version is BQ field quadratics.