Classical and Quantum Fermions Linked by an Algebraic Deformation

TL;DR

This paper explores the algebraic deformation linking classical and quantum fermions through their representations, showing how classical fermions relate to Dirac fermions via a parameterized algebraic deformation.

Contribution

It identifies the minimal faithful irreducible representation of the single-fermion algebra and interprets quantization as an algebraic deformation connecting classical and quantum fermions.

Findings

Classical fermions correspond to a minimal faithful irreducible representation.

Quantization is modeled as a deformation of algebraic representations.

Deformation maps classical fermion representations to those of Dirac fermions.

Abstract

We study the regular representation $\rho_\zeta$ of the single-fermion algebra ${\cal A}_\zeta$, i.e., $c^2=c^{+2}=0$, $cc^++c^+c=\zeta~1$, for $\zeta\in [0,1]$. We show that $\rho_0$ is a four-dimensional nonunitary representation of ${\cal A}_0$ which is faithfully irreducible (it does not admit a proper faithful subrepresentation). Moreover, $\rho_0$ is the minimal faithfully irreducible representation of ${\cal A}_0$ in the sense that every faithful representation of ${\cal A}_0$ has a subrepresentation that is equivalent to $\rho_0$. We therefore identify a classical fermion with $\rho_0$ and view its quantization as the deformation: $\zeta:0\to 1$ of $\rho_\zeta$. The latter has the effect of mapping $\rho_0$ into the four-dimensional, unitary, (faithfully) reducible representation $\rho_1$ of ${\cal A}_1$ that is precisely the representation associated with a Dirac fermion.