Dirac reduction algebra

TL;DR

This paper introduces the Dirac reduction algebra, a new algebraic structure derived from the orthosymplectic Lie superalgebra and Weyl-Clifford superalgebra, which helps generate polynomial solutions to the Dirac equation in flat spacetime.

Contribution

It provides a complete presentation of the Dirac reduction algebra and demonstrates its use in generating all polynomial solutions to the Dirac equation.

Findings

Complete presentation of the Dirac reduction algebra

Generation of all polynomial solutions to the Dirac equation

Connection between algebraic structures and solutions to PDEs

Abstract

There is a homomorphism of associative superalgebras from the enveloping algebra of the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ to the Weyl-Clifford superalgebra $W(2n|n)$ with $2n$ even Weyl algebra generators and $n$ odd Clifford algebra generators. Under this homomorphism, the positive odd root vector $x\in\mathfrak{osp}(1|2)$ is sent to the Dirac operator $\gamma^\mu\partial_\mu\in W(2n|n)$ and generates a left ideal $I$. The corresponding reduction (super)algebra, denoted $Z_n$, is the normalizer of $I$ in $W(2n|n)$ modulo $I$. By construction, $Z_n$ acts on the space of all Clifford algebra-valued polynomial solutions to the (massless) Dirac equation. In this paper, we find a complete presentation of (a localization of) this so-termed Dirac reduction algebra. Furthermore, we use the Dirac reduction algebra to generate all polynomial solutions to the Dirac equation in $n$-dimensional flat spacetime.