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Clifford algebras, meson algebras and higher order generalisations

Michel Dubois-Violette, Blas Torrecillas

TL;DR

The paper analyzes the neutral (homogeneous) parts of Clifford and meson algebras, showing that $C_0(E)=\\wedge E$ corresponds to ordinary fermionic statistics (parastatistics of order 1) while $D_0(E)$ corresponds to fermionic parastatistics of order 2. It introduces the universal enveloping algebra $\\mathcal F(E)$ via the relation $[[x,y],z]=0$ and shows how $D_0(E)$ arises as $\\mathcal F(E)/(x^3)$, connecting to parafermi systems through the Green ansatz. The work then generalizes to higher-order parastatistics by defining $\\Phi_n(E)$ and $\\Psi_n(E)$, with $\\Phi_n(E)=\\mathcal F(E)/(x^{n+1})$ and $\\Psi_n(E)$ as a deformation embedded into $n$-fold tensor products of Clifford algebras, providing explicit algebraic relations and representation-theoretic structure under $GL(E)$. Collectively, these constructions extend Clifford and meson algebras to a systematic family parametrized by the parastatistics order $n$, offering a framework for exploring connections to the Jordan spin factor and potential physical interpretations in higher-order fermionic theories.

Abstract

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.

Clifford algebras, meson algebras and higher order generalisations

TL;DR

The paper analyzes the neutral (homogeneous) parts of Clifford and meson algebras, showing that corresponds to ordinary fermionic statistics (parastatistics of order 1) while corresponds to fermionic parastatistics of order 2. It introduces the universal enveloping algebra via the relation and shows how arises as , connecting to parafermi systems through the Green ansatz. The work then generalizes to higher-order parastatistics by defining and , with and as a deformation embedded into -fold tensor products of Clifford algebras, providing explicit algebraic relations and representation-theoretic structure under . Collectively, these constructions extend Clifford and meson algebras to a systematic family parametrized by the parastatistics order , offering a framework for exploring connections to the Jordan spin factor and potential physical interpretations in higher-order fermionic theories.

Abstract

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.
Paper Structure (8 sections, 7 theorems, 60 equations)

This paper contains 8 sections, 7 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. The neutral meson algebra $D_0(E)$
  3. Another approach and its generalization
  4. The key result
  5. The remarkable algebra ${\mathcal{F}}(E)$
  6. Higher order generalizations
  7. Higher order generalizations of Clifford and meson algebras
  8. Conclusion

Key Result

Lemma 1

One has the following identity for $N\geq p\geq q\geq 0$.

Theorems & Definitions (7)

  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Lemma 5
  • Lemma 6
  • Theorem 7