Fermionic fields in the functional approach to classical field theory

TL;DR

This paper develops a classical framework for Fermionic fields within the algebraic approach, integrating functional-analytic and topological aspects without extra Grassman variables, inspired by quantum field theory developments.

Contribution

It introduces a novel formalism for classical Fermionic fields that naturally incorporates anticommutativity and functional analysis, extending existing algebraic frameworks without additional Grassman degrees of freedom.

Findings

Framework successfully models Fermionic fields classically

Incorporates functional-analytic and topological features

Applicable to interacting models in quantum field theory

Abstract

In this paper, we present a formulation of the classical theory of Fermionic (anticommuting) fields, which fits into the general framework proposed by K.Fredenhagen, M.Duetsch and R.Brunetti. It was inspired by the recent developments in perturbative algebraic quantum field theory and allows for a deeper structural understanding also on the classical level. We propose a modification of this formalism that allows to treat also Fermionic fields. In contrast to other formulations of classical theory of anticommuting variables, we don't introduce additional Grassman degrees of freedom. Instead the anticommutativity is introduced in a natural way on the level of functionals. Moreover our construction incorporates the functional-analytic and topological aspects, which is usually neglected in the treatments of anticommuting fields. We also give an example of an interacting model where our framework can be applied.