Canonical transformations for fermions in superanalysis

TL;DR

This paper develops a superanalysis framework for canonical fermionic transformations, constructing a continuous orthogonal group representation on Grassmann modules that extends to Fock space, enabling a unitary implementation of Bogoliubov transformations.

Contribution

It introduces a superanalysis-based approach to fermionic canonical transformations, providing a new continuous group representation on Grassmann modules and its unitary realization on Fock space.

Findings

Constructed a continuous orthogonal group representation on Grassmann modules.

Derived a unitary ray representation implementing Bogoliubov transformations.

Extended the calculus of superanalysis to infinite fermionic degrees of freedom.

Abstract

Canonical transformations (Bogoliubov transformations) for fermions with an infinite number of degrees of freedom are studied within a calculus of superanalysis. A continuous representation of the orthogonal group is constructed on a Grassmann module extension of the Fock space. The pull-back of these operators to the Fock space yields a unitary ray representation of the group that implements the Bogoliubov transformations.