Abstract
Many methods of modern quantum field theory rely heavily on functionals of classical fields; this notion is however problematic whenever anticommuting fields are present. We propose a calculus for such functionals which avoids the use of auxiliary Grassmann algebras, and which relies on an infinite-dimensional version of Berezin-Leites supermanifold theory. We begin by studying “functional power series expansions” without growth conditions; this already allows to make e.g. the Yang-Mills action functional with fermionic, anticommuting matter fields a well-defined mathematical object. We introduce analytical conditions on power series which enable us to substitute them into each other, and we globalize them to superfunctionals. Also, infinitesimal transformation laws of the fields are discussed.
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