Classical and quantum fermions linked by an algebraic deformation

Abstract

We study the regular representation p(zeta) of the single-fermion algebra A(zeta) i.e., c(2) = C+2 = 0, cc(+)+c(+)c = zeta 1, for is an element of [0, 1]. We show that p(o) is a four-dimensional nonunitary representation of A(o) which is faithfully irreducible (it does not admit a proper faithful subrepresentation). Moreover, p(o) is the minimal faithfully irreducible representation of A(o) in the sense that every faithful representation of A(o) has a subrepresentation that is equivalent to p(o). We therefore identify a classical fermion with po and view its quantization as the deformation: zeta : 0 -> 1 of p(zeta). The latter has the effect of mapping po into the four-dimensional, unitary, (faithfully) reducible representation p(1) of A(l) that is reminiscent of a Dirac fermion.