BQ Weyl and Dirac Representations

← The BQ algebra at the Dirac level

The BQ-Dirac algebra has one set of sixteen basis elements and one multiplication table, but it can be written in more than one set of 4×4 matrices. The BQ construction is done in the Weyl representation; the Dirac representation is needed for massive particles and for the comparison with the standard Dirac formalism. One real orthogonal matrix S carries the one into the other. This page starts with S and why both representations are used, and then shows the sixteen basis matrices and all 256 products in either representation, with one switch.

The transformation matrix S, from Weyl to Dirac

Why we start with Weyl, and when Dirac is needed

Weyl: the construction representation

The Pauli level stays visible. A Weyl vector is a = [[0, A], [−A^T, 0]]: its two blocks are the BQ-Pauli minquat A = a₀T̂ + a·K̂ and its time adjoint. A product of two vectors is then ab = [[−AB^T, 0], [0, −A^TB]]: the Pauli bilinear A^TB sits in it as a block.
Grade and block go together. The even grades (𝟙, α_i, Σ_i, χ) are block diagonal and the odd grades (β_μ, β̃_μ) block off-diagonal.
The Pauli rotor is the boost. Λ = diag(U⁻¹, U), so the Dirac Lorentz transformation of a vector follows in one line from the Pauli one, A^L = U⁻¹AU⁻¹. The same holds for the gravitational rotor: the PG metric, the flows and the CL invariants are computed here.
The chirality is diagonal. χ = diag(T̂, −T̂): the two blocks are the two Weyl sectors, and the massless equations decouple into them.

This is why every BQ-Dirac model on this site is built in the Weyl representation: every Dirac result contains the Pauli result as a block (de Haas 2026, Weyl-level EM, the construction-oriented representation).

Dirac: the application representation

Mass. In the Weyl representation PP = EE factorizes as (P − E)(P + E) = 0, and taken on their own the factors give nontrivial solutions only for m₀ = 0. Massive fermions need spinors and a representation that mixes the parity and time-reversal duplexes: the Dirac representation (de Haas 2025, BQ Dirac Gravity, eqs. 68–76).
Time on the diagonal. β₀ = diag(T̂, −T̂), so a vector is [[a₀T̂, a·K̂], [−a·K̂, −a₀T̂]]. At rest the spinors separate into an upper and a lower block, u(0) ∝ [φ; 0] and v(0) ∝ [0; η]: positive- and negative-energy adapted sectors.
Comparison. With β_μ = iγ_μ, the BQ-Dirac β_μ are i times the standard Dirac-representation γ matrices (axis order I, J, K = z, y, x), so the massive Dirac equation, its non-relativistic limit and the literature are read in this form.
The spinor boost is constructed. The Lorentz operator of Dirac spinors is SΛS⁻¹, the Weyl boost carried over by S, not a postulated matrix.

The working order is therefore Dirac → Weyl → Dirac: take the massive spinor to the Weyl representation with S⁻¹, do the geometric calculation there, and return with S (de Haas 2025, M₂(ℂ) unification, Fig. 6).

	Weyl	Dirac
vector a	[[0, A], [−A^T, 0]]	[[a₀T̂, a·K̂], [−a·K̂, −a₀T̂]]
diagonal	the even grades, χ	β₀, Σ_i, β̃_i, 𝟙
rotation (Σ_i)	separate blocks, equal	separate blocks, equal
boost e^{−(ψ/2)α_I}	diag(U⁻¹, U): separate, opposite	[[cosh, sinh σ_I], [sinh σ_I, cosh]]: hyperbolic mixing
phase e^φχ	diag(e^φT̂, e^−φT̂): separate, opposite	[[cos φ 1, −sin φ T̂], [−sin φ T̂, cos φ 1]]: circular mixing
suited to	construction, rotors, massless fields	massive spinors, rest frame, standard comparison

The boost and the phase are the opposite-sign pair: separate in Weyl, inter-block in Dirac, hyperbolic because α_I² = +𝟙 and circular because χ² = −𝟙 (de Haas 2026, From Weyl to Dirac, eqs. 806–817).

The basis and the table in either representation

Representation

All 256 products land on exactly one basis element, with sign ±1 and no factor i. Pick a cell to see both factors in their Pauli block content, the block product term by term, and the 4×4 product.

The sixteen basis matrices and their Pauli content: Weyl representation

Grades: grade 0 grade 1, vector grade 2, boost bivector grade 2, spin bivector grade 3, pseudovector grade 4, pseudoscalar. Blocks use the BQ-Pauli basis 1, T̂ = i1, Î, Ĵ, K̂, σ_I, σ_J, σ_K, with K̂_i = T̂σ_i. 𝟙 is the 4×4 unit.

The 16 × 16 table (row · column)

Each cell is the product of its row element (left) times its column element (right), coloured by the grade of the result. The selected cell is outlined.

The selected product, with the BQ-Pauli entries

The same product as 4×4 complex matrices

Structure read off the table

Notes

Definitions from de Haas (2026), From Weyl to Dirac, eqs. 1–24, and de Haas (2025), BQ Dirac Gravity, §3.3; checked exactly in bq_dirac_representations.py. The Weyl-only table is the BQ-Dirac multiplication table in the Weyl representation. Part of the BQ algebra at the Dirac level; the BQ-Pauli models are at the main index.