From Berezin Integral to Batalin-vilkovisky Formalism: a Mathematical Physicist’s Point of View

Within the path integral approach quantum field theory can be formulated as an integral over the superspace of fields. The Berezin integral is the crucial element in this construction. Replacing an infinite-dimensional supermanifold of fields by an ordinary supermanifold we get a nice toy model of quantum field theory. In this article I motivate the origin of the Berezin integral and trace some of its applications, up to the Batalin-Vilkovisky formalism. The latter was developed as a method for determining the ghost structure in such theories as gravity and supergravity, whose Hamiltonian formalism has constraints not related to a Lie algebra action.