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

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

1996-07-16

arxiv.org/abs/hep-th/9607132v1

Rev.Math.Phys.9:993-1052,1997

Physics

High Energy Physics

High Energy Physics - Theory

46 pages, LateX2E+AMSLaTeX

Scientific paper

10.1142/S0129055X97000348

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.

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