BQ-Dirac Multiplication Table in the Weyl Representation

← The BQ algebra at the Dirac level

At the Dirac level the BQ algebra lives in 4×4 complex matrices, in the Weyl representation. Every one of its sixteen basis elements is a 2×2 block matrix whose blocks are BQ-Pauli elements: the Dirac level is the Pauli level one storey up. A product of two Dirac elements is therefore a block product whose four blocks are BQ-Pauli products, and it can be checked a second time as an ordinary 4×4 complex matrix product.

β0 = [[0, T̂], [T̂, 0]],  βi = [[0, K̂i], [−K̂i, 0]],  αi = β0βi,  ΣI = βJβK,  χ = −β0βIβJβK,  β̃μ = χβμ

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

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, σI, σJ, σK, with K̂i = T̂σi. 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. Only β0, βI, βJ, βK are entered by hand, from their Pauli blocks. The other twelve elements are computed on this page as block products: αi = β0βi, ΣI = βJβK (cyclic), χ = −β0βIβJβK, β̃μ = χβμ.
  • Off-diagonal vectors. The vector βK is block off-diagonal, [[0, K̂], [−K̂, 0]]. The block-diagonal matrix diag(K̂, −K̂) is not one of the sixteen basis elements; diagonal blocks belong to the even grades 0, 2 and 4.
  • Even and odd. Even grades (𝟙, α, Σ, χ) are block diagonal, odd grades (β, β̃) block off-diagonal: a product of two odd or two even elements is even, a mixed product is odd.
  • Check. Each product is computed twice, by blocks through the BQ-Pauli table and as a 4×4 matrix product, and compared entry by entry; the result is matched against all sixteen ± basis matrices.

Definitions from de Haas (2026), From Weyl to Dirac, and the BQ-Dirac algebra script of the project (Weyl representation). The same table in the Dirac representation, and the matrix S between the two, is on the page BQ Weyl and Dirac representations. Part of the BQ algebra at the Dirac level; the BQ-Pauli models are at the main index.