BQ-Dirac Lorentz Transformations: Weyl and Dirac

The same boost along I, written in the two representations. In the Weyl representation the boost is Λ_W = diag(U⁻¹, U), the doubled Pauli rotor of the previous three models. The Dirac boost is not new: it is Λ_W carried over by the Weyl-to-Dirac matrix S. Choose a four-vector and a velocity to see the boost of a in both representations side by side. The boosted coordinates come out the same in both; only the 4×4 matrices that hold them differ.

Λ_W = e^{(ψ/2)σ_I^W}, Λ_D = e^{(ψ/2)σ_I^D} = SΛ_WS⁻¹, a^L = ΛaΛ⁻¹ in both, S = ¹⁄_√2[[1, 1], [−1, 1]]

Vector and boost

Velocity β = v/c along I

The two boost matrices

The exponential form: Λ = e^(ψ/2)σ_I in both representations

The boost of a in both representations

From Λ_W to Λ_D through S

How the Dirac boost mixes time and space

What it means for spinors

Check by explicit 4×4 matrix multiplication

What changes and what does not

Changed by S: the matrices Λ, a, a^L and the spinor Ψ. Not changed: the coordinates a^μ and a′^μ, the rotor form c𝟙 − sα_I, the invariant aa = −(a₀² − a²)𝟙, and the rotations diag(r, r), which are the same matrices in both representations. The boost is the one transformation whose matrix changes character: block diagonal in Weyl, inter-block in Dirac.

Sources and status

S and the Dirac basis follow de Haas (2026), From Weyl to Dirac, eqs. 1–24; Λ_W = diag(U⁻¹, U), Ψ^L = ΛΨ and Λ_D² follow de Haas (2025), BQ Dirac Gravity, eqs. 78–82 and 103–107. Reported, not changed: eq. 107 writes Λ_D² = 𝟙 cosh ψ + α_I sinh ψ; with α_I^D = [[0, −σ_I], [−σ_I, 0]] of From Weyl to Dirac eq. 14 the same matrix is cosh ψ 𝟙 − sinh ψ α_I^D. The matrix [[cosh ψ 1, sinh ψ σ_I], [sinh ψ σ_I, cosh ψ 1]] itself agrees. The two agree if the α_I of eq. 107 is the standard Dirac α = [[0, σ_I], [σ_I, 0]], which is σ_I^D = −α_I^D on this page.

Every identity is checked exactly in bq_dirac_lorentz_representations.py. Part of the BQ algebra at the Dirac level; the Weyl-only boost is BQ-Dirac Lorentz transformations, the matrix S is on BQ Weyl and Dirac representations.