Supergeometry and Quantum Field Theory, or: What is a Classical Configuration?

TL;DR

This paper explores the conceptual and mathematical foundations of classical fermion fields in quantum field theory, proposing that their configuration space should be modeled as an infinite-dimensional supermanifold, and discusses approaches to supergeometry.

Contribution

It develops an infinite-dimensional supermanifold framework for classical fermion configurations, clarifies the choice of supermanifold approach, and links superfunctions to classical configuration functionals.

Findings

Superfunctions on the supermanifold correspond to classical configuration functionals.

The Berezin-Kostant-Leites approach is more fundamental than the Rogers approach.

A mathematical program for classical field models with fermions is outlined.

Abstract

We discuss of the conceptual difficulties connected with the anticommutativity of classical fermion fields, and we argue that the "space" of all classical configurations of a model with such fields should be described as an infinite-dimensional supermanifold M. We discuss the two main approaches to supermanifolds, and we examine the reasons why many physicists tend to prefer the Rogers approach although the Berezin-Kostant-Leites approach is the more fundamental one. We develop the infinite-dimensional variant of the latter, and we show that the functionals on classical configurations considered in a previous paper are nothing but superfunctions on M. We present a programme for future mathematical work, which applies to any classical field model with fermion fields. This programme is (partially) implemented in successor papers.