Classical and Quantum Fermions Linked by an Algebraic Deformation

We study the regular representation ρζ of the single-fermion algebra Aζ , i.e., c2 = c+2 = 0, cc+ + c+c = ζ 1, for ζ ∈ [0, 1]. We show that ρ0 is a four-dimensional nonunitary representation of A0 which is faithfully irreducible (it does not admit a proper faithful subrepresentation). Moreover, ρ0 is the minimal faithfully irreducible representation of A0 in the sense that every faithful representation of A0 has a subrepresentation that is equivalent to ρ0. We therefore identify a classical fermion with ρ0 and view its quantization as the deformation: ζ : 0 → 1 of ρζ . The latter has the effect of mapping ρ0 into the four-dimensional, unitary, (faithfully) reducible representation ρ1 of A1 that is precisely the representation associated with a Dirac fermion.