Clifford algebras, meson algebras and higher order generalisations

We analyse the homogeneous parts of Clifford and meson algebras and point out that for the Clifford algebra it is related to fermionic statistics, that is, to fermionic parastatistics of order 1 while for the meson algebra it is related to fermionic parastatistics of order 2. We extend these homogeneous algebras into corresponding algebras related to fermionic parastatistics of all orders. We then define correspondingly higher order generalizations of Clifford and meson algebras.

💡 Research Summary

The paper investigates the homogeneous (or “neutral”) parts of two well‑known associative algebras attached to a finite‑dimensional pseudo‑Euclidean space E: the Clifford algebra C(E) and the meson (or Duffin‑Kemmer‑Petiau) algebra D(E). By setting the scalar product to zero, the authors obtain C₀(E)=∧E, the exterior algebra, and D₀(E), a cubic algebra defined by the relation xyx=0 for all x,y∈E. They show that C₀(E) corresponds to fermionic parastatistics of order 1 (ordinary Fermi statistics), while D₀(E) corresponds to fermionic parastatistics of order 2.

A detailed representation‑theoretic analysis reveals that each homogeneous component D⁰ₙ(E) decomposes, as a GL(E)‑module, into a direct sum of irreducible representations labelled by Young diagrams with exactly two columns whose total number of boxes equals n. The dimension formula for a two‑column diagram λ_{p,q} (p≥q) is derived as
dim λ_{p,q}= (p−q+1)/(p+1)·C(p,N)·C(q,N+1),
where N=dim E. The authors prove that the total dimension of D⁰ₙ(E) equals the sum of these dimensions over all p+q=n, establishing a precise combinatorial identity.

The key algebraic insight is the equivalence
xyx=0 ⇔