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Supergeometry and Quantum Field Theory, or: What is a Classical Configuration?
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.Comment: 46 pages, LateX2E+AMSLaTe