BQ Lorentz Transformation of the Bilinear

The bilinear C = A^TB = c₁1̂ + c_K·K̂ + c_σ·σ̂ of two four-vectors is boosted along Î by boosting both factors. The time adjoint A^T transforms as UA^TU and B as U⁻¹BU⁻¹, so the inner pair cancels and C transforms by a similarity, C^L = UCU⁻¹, with the rotor U = e^ψσ̂_I/2. Choose two four-vectors and a velocity.

Vectors and boost

A B

Velocity β = v/c along Î

The transformations of C = A^TB

The boost of the bilinear in seven representations

The steps follow the BQ-Pauli slides on the Lorentz transformation of the bilinear and use the vectors and velocity chosen above. In this section ch = cosh(ψ/2) and sh = sinh(ψ/2), the C and S of the slides, so that C keeps its meaning as the bilinear.

The two transverse planes

CC^L = UCU⁻¹C^L⁻¹ = U⁻¹CUinvariant hyperbola

The boost leaves c₁, c_K,I and c_σ,I alone and turns each transverse pair (c_K,J, c_σ,K) and (c_K,K, c_σ,J) through the rapidity ψ, in opposite senses. The point slides along its hyperbola c_K,J² − c_σ,K² = const (left) or c_K,K² − c_σ,J² = const (right).

Check by explicit matrix multiplication

Why a similarity, and what stays invariant

Vectors and bilinears transform differently. A four-vector is boosted from both sides with the same factor, A^L = U⁻¹AU⁻¹. In the bilinear the adjoint brings the opposite factor, (A^L)^T = UA^TU, so (A^L)^TB^L = UA^T(UU⁻¹)BU⁻¹ = UCU⁻¹.
Invariant channels. 1̂, T̂, Î and σ̂_I commute with σ̂_I, so c₁, the T̂ channel and the longitudinal components c_K,I, c_σ,I are unchanged. Ĵ, K̂, σ̂_J, σ̂_K anticommute with σ̂_I and mix in the pairs (Ĵ, σ̂_K) and (K̂, σ̂_J).
Invariant combinations. The complex three-vector 𝒞 = c_σ + ic_K is rotated by a complex orthogonal matrix, so 𝒞·𝒞 = (c_σ² − c_K²) + 2i c_K·c_σ is invariant. For the EM field these are B² − E²/c² and E·B/c. For two plain four-vectors c_K·c_σ = (a×b)·(a₀b − ab₀) = 0.
The time adjoint. With C^T = c₁1̂ + c_K·K̂ − c_σ·σ̂ (as on the field-quadratics page), (C^L)^T = U⁻¹C^TU = (C^T)^L⁻¹, the same reversal as for vectors.

Direction of the boost

As for the vectors, the rule UCU⁻¹ with a diagonal U holds for boosts along Î. A boost in another direction is first rotated into Î; a rotation R acts on the bilinear as RCR⁻¹ as well, so every bilinear transforms by a similarity under the full Lorentz group.

Definitions as in de Haas, BQ-Pauli slides (Lorentz transformation of the EM field B = ∂^TA, the direct BQ derivation with U = C1̂ + Sσ̂_I, and the connection with Minkowski’s 1908 complex field transformation). Part of the BQ algebra models.