Classical and Quantum Fermions Linked by an Algebraic Deformation

Abstract

We study the regular representation ${\rho }_{\zeta }$ of the single-fermion algebra ${\mathcal{A}}_{\zeta }$, i.e., $c^2=c^{+2}=0$, $c{c}^{+}+c^{+}c=\zeta 1$, for $\zeta \in [0,1]$. We show that ${\rho }_0$ is a four-dimensional nonunitary representation of ${\mathcal{A}}_0$ which is faithfully irreducible (it does not admit a proper faithful subrepresentation). Moreover, ${\rho }_0$ is the minimal faithfully irreducible representation of ${\mathcal{A}}_0$ in the sense that every faithful representation of ${\mathcal{A}}_0$ 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 ${\mathcal{A}}_1$ that is precisely the representation associated with a Dirac fermion.

7 pages