← The BQ algebra at the Dirac level

BQ-Dirac Bilinear Product Explorer

At the Dirac level a physical four-vector is four real coordinates contracted with the Weyl vector basis, A = Aμβμ = a0β0 + a·β, with β = (βI, βJ, βK). The product of two such vectors needs no time adjoint: β0² = −𝟙 and βi² = +𝟙 carry the Lorentz signature themselves. The product lands in three channels, a scalar, a boost bivector and a spin bivector, and nowhere else. Choose A and B from the four-vectors used in the BQ papers.

AB = −(a0b0 − a·b) 𝟙 + (a0b − ab0)·α + (a×b)·Σ

Choose the two four-vectors

Products from the papers

The slash marks a Dirac-level vector, A = Aμβμ, with the same real coordinates as the Pauli minquat A = AμK̂μ. In block form A = [[0, A], [−AT, 0]], with AT the BQ time adjoint of the Pauli level.

Channels of AB

The multiplication, term by term

Check with numbers, by explicit 4×4 matrix multiplication

How the channels arise

  • Scalar, 𝟙. From β0β0 = −𝟙 and βiβi = +𝟙: −a0b0 + a·b, the Pauli norm a0b0 − a·b with the opposite sign. The signature sits in the squares of the basis vectors, so no time adjoint is needed.
  • Boost bivector, α. From β0βi = αi = −βiβ0: the time–space mixing a0b − ab0. The factor order is kept, A on the left and B on the right, so that a derivative in A acts on B.
  • Spin bivector, Σ. From βiβj = εijkΣk for i ≠ j: the cross product a×b.
  • No odd grades. A product of two vectors is even: no β, β̃ or χ channel. The numerical check shows these nine channels as exactly zero.
  • Squares. AA = −(a0² − a²)𝟙: every Dirac-level vector squares to a multiple of the unit.

The Pauli bilinear inside

With A = [[0, A], [−AT, 0]] the product is block diagonal:

AB = [[−ABT, 0], [0, −ATB]]

The lower block is minus the BQ-Pauli bilinear ATB. Reading the blocks of 𝟙 = diag(1̂, 1̂), αi = diag(σ̂i, −σ̂i) and Σi = diag(−K̂i, −K̂i) gives the channel correspondence 1̂ → −𝟙, K̂ → Σ, σ̂ → α: the same three coefficients as on the Pauli level, now on the scalar, spin and boost elements of the Dirac basis.

Reading ∂̸. ∂ = −(1/c)∂tβ0 + ∇·β is an operator. As the left factor its derivatives act on B; as the right factor nothing follows it, and the result is an operator.

Definitions as in de Haas (2026), BQ-Weyl electromagnetism, with A = (φ/c)β0 + A·β, J = cρβ0 + J·β, U = γcβ0 + γv·β, G = (ε/c)β0 + g·β. Channels are derived symbolically in the page and checked numerically from the explicit matrices. Part of the BQ algebra at the Dirac level; the Pauli version is the BQ bilinear product explorer.