← The BQ algebra at the Dirac level

BQ-Dirac Lorentz Transformation of the Bilinear

The bilinear of two Dirac-level four-vectors is even, C = AB = s𝟙 + cα·α + cΣ·Σ. Boosting both factors along I boosts the bilinear by the same similarity as the vectors, CL = ΛCΛ−1. On the channels the scalar and the I components stay, and the transverse boost and spin bivectors mix in pairs, exactly as E and B. Choose A and B freely from the four-vectors of the papers, or click one of the familiar bilinears, and set a velocity.

CL = ΛCΛ−1:   s′ = s,   c′α,I = cα,I,   c′Σ,I = cΣ,I,   c′α,J = γ(cα,J − βcΣ,K),   c′Σ,K = γ(cΣ,K − βcα,J),   c′α,K = γ(cα,K + βcΣ,J),   c′Σ,J = γ(cΣ,J + βcα,K)

Bilinear and boost

Bilinears from the papers

γ = cosh ψ and βγ = sinh ψ with the rapidity ψ = artanh β; for physical fields βc = v.

The boosted bilinear

Why: one similarity, and the channels it mixes

Check with numbers, by explicit 4×4 matrices

Vectors and bilinears

  • One rule for both. At the Pauli level a vector goes as U−1AU−1 and the bilinear as UCU−1, two different rules. At the Dirac level both go by Λ…Λ−1; the difference sits in which blocks are filled.
  • Half and full rapidity. The rotor carries ψ/2. Elements that anticommute with αI pick up Λ², so the transverse channels turn with the full rapidity ψ; elements that commute with αI do not change.
  • No time adjoint. The Pauli bilinear ATB needs the adjoint of A. At the Dirac level the product AB is plain, and ATB appears by itself as the lower block.

Sources and status

The Pauli rule CL = UCU−1 and its channel form follow the BQ-Pauli papers and the Pauli model; Λ = diag(U−1, U) and XL = ΛXΛ−1 follow de Haas (2026), BQ-Weyl electromagnetism, eqs. 161–196, where the boost of the field B and the invariance of B² are given. The fluid and bilinear readings follow by the same rule.

All identities are checked exactly in bq_dirac_bilinear_lorentz.py. Part of the BQ algebra at the Dirac level; see also the Lorentz transformations of vectors.