Boson-Fermion unification implemented by Wick calculus

Abstract

We construct a transformation between Bose Fock space and Fermi Fock space that is super-symmetric in the sense that it converts Boson fields into Fermi fields over a fixed one-particle space. The transformation involves the spectral splitting of the one-particle space into a continuous direct integral of internal spaces. We present a theory of integration on the Fock spaces over the square intergable functions taking values in these internal spaces which we refer to as a Wick calculus: this is a natural generalization of the theory of quantum stochastic calculus which used a single fixed internal space.