Classical and Quantum Fermions Linked by an Algebraic Deformation

Abstract

We study the regular representation $ρ_ζ$ of the single-fermion algebra ${\cal A}_ζ$, i.e., $c^2=c^{+2}=0$, $cc^++c^+c=ζ~1$, for $ζ\in [0,1]$. We show that $ρ_0$ is a four-dimensional nonunitary representation of ${\cal A}_0$ which is faithfully irreducible (it does not admit a proper faithful subrepresentation). Moreover, $ρ_0$ 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 $ρ_0$. We therefore identify a classical fermion with $ρ_0$ and view its quantization as the deformation: $ζ:0\to 1$ of $ρ_ζ$. The latter has the effect of mapping $ρ_0$ into the four-dimensional, unitary, (faithfully) reducible representation $ρ_1$ of ${\cal A}_1$ that is precisely the representation associated with a Dirac fermion.