BQ Lorentz Transformations

A Lorentz boost along Î acts on a four-vector A = a₀T̂ + a·K̂ from both sides with the same rotor, A^L = U⁻¹AU⁻¹, where U = e^ψσ̂_I/2 = diag(e^ψ/2, e^−ψ/2) and ψ is the rapidity. The inverse boost is UAU, and the time adjoint A^T transforms the opposite way. Choose a four-vector and a velocity.

Vector and boost

Velocity β = v/c along Î

The rotor

Because σ̂_I² = 1̂, the exponential series splits into even and odd powers: e^ψσ̂_I/2 = cosh(ψ/2) 1̂ + sinh(ψ/2) σ̂_I. With γ = coshψ and βγ = sinhψ, the rapidity is ψ = artanh β and e^±ψ = γ(1 ± β), the relativistic Doppler factor.

The four transformations

From the textbook boost to the BQ rotor, in six steps

The steps follow the BQ-Pauli slides on the Lorentz transformation and use the vector and velocity chosen above.

In the (a₁, a₀) plane

AA^L = U⁻¹AU⁻¹A^L⁻¹ = UAUa₀² − a₁² = constlight cone

The boost slides the point along its hyperbola: the invariant a₀² − a₁² does not change, and the light-cone coordinates a₀ ± a₁ are multiplied by e^∓ψ. The coordinates a₂, a₃ across the boost are unchanged.

Check by explicit matrix multiplication

Why both sides, and why the adjoint reverses

Basis transformation. U⁻¹T̂U⁻¹ = coshψ T̂ − sinhψ Î and U⁻¹ÎU⁻¹ = coshψ Î − sinhψ T̂, while Ĵ and K̂ are unchanged. The coordinate rule a₀′ = γ(a₀ − βa₁), a₁′ = γ(a₁ − βa₀) is the same transformation read on the coordinates instead of the basis: K̂_μΛ_ν^μ = U⁻¹K̂_νU⁻¹.
In the matrix. A = [[i(a₀ + a₁), a₂ + ia₃], [−a₂ + ia₃, i(a₀ − a₁)]]: the boost multiplies the diagonal entries by e^−ψ and e^ψ and leaves the off-diagonal entries alone.
The time adjoint. Since U is real and diagonal, (A^T)^L = U⁻¹A^TU⁻¹ = (UAU)^T = (A^L⁻¹)^T and (A^L)^T = UA^TU. The adjoint of a boosted vector is boosted the other way.
Invariance and covariance. Hence (A^L)^TB^L = UA^TU U⁻¹BU⁻¹ = U(A^TB)U⁻¹: every bilinear transforms by a similarity, and its 1̂ and T̂ channels are invariant, so A^TA is the same in every frame.

Direction of the boost

The rule U⁻¹AU⁻¹ with a diagonal U holds for boosts along Î. Frames moving in another direction are first rotated so that their relative velocity lies along Î (Synge’s communal photon); a boost in another direction can always be replaced by one rotation and one Î-boost, which is where Thomas–Wigner rotation enters.

Definitions as in de Haas, BQ-Pauli papers (U, rapidity, Lemma on basis–component compatibility, and (A^T)^L = (A^L⁻¹)^T). Part of the BQ algebra models.