BQ Bilinear Product Explorer

A physical four-vector is four real coordinates contracted with the minquat basis, A = AμK̂μ = a0T̂ + a·K̂, with K̂ = (Î, Ĵ, K̂) the spatial basis vector. For two such vectors the physically relevant product is ATB, and the BQ algebra sorts it into three channels on its own: a norm, a rotation bivector and a boost bivector. Choose A and B from the four-vectors used in the BQ papers.

Choose the two four-vectors

Products from the papers

The superscript T is the BQ time-reversal operation, AT = −a0T̂ + a·K̂. It flips the sign of the time coordinate; it is not the matrix transpose, not conjugation and not multiplication by T̂.

Channels of ATB

ATB = (a0b0 − a·b) 1̂ + (a×b)·K̂ + (a0b − ab0)·σ̂

The multiplication, term by term

Check with numbers, by explicit 2×2 matrix multiplication

How the channels arise

  • Norm, 1̂. From T̂T̂ = −1̂ and K̂iK̂i = −1̂ together with the sign flip of AT: a0b0 − a·b = ημνAμBν. The Lorentz signature enters through the basis, not through index lowering.
  • Rotation bivector, K̂. From K̂iK̂j = εijkK̂k for i ≠ j: the cross product a×b.
  • Boost bivector, σ̂. From T̂K̂i = K̂iT̂ = −σ̂i: the time–space mixing a0b − ab0. The factor order is kept as in the product, A on the left and B on the right, so that a derivative in A acts on B: with A = ∂ the term ab0 becomes ∇b0, not b0∇.
  • No T̂ channel. A bilinear of two minquats never has a T̂ part; the numerical check below shows it as exactly zero.
  • ATA. For identical factors a×a = 0 and a0a − aa0 = 0, so only the invariant norm survives.

Reading the derivative ∂

∂ = −(1/c)∂t T̂ + ∇·K̂ is an operator. In ATB each component of A stands to the left of each component of B, so when ∂ is the left factor its derivatives act on B (∂TA gives the fields). When ∂ is the right factor nothing follows it, and the result is an operator written with ∂t and ∇ on the right. The coordinates t and r are independent, so ∂tt = 1, ∇·r = 3, ∇×r = 0, ∇t = 0 and ∂tr = 0.

Definitions as in the BQ-Pauli slides on the bilinear product and in de Haas (2026), BQ electromagnetism and fluid dynamics: P = (E/c)T̂ + p·K̂, ∂ = −(1/c)∂tT̂ + ∇·K̂, A = (φ/c)T̂ + A·K̂, J = cρT̂ + J·K̂, U = γcT̂ + γv·K̂, G = (ε/c)T̂ + g·K̂. Channels are derived symbolically in the page and checked numerically from the explicit matrices. Part of the BQ algebra models; see also bq-galaxy-models.