Supergeometry and Hermitian conjugation (Q810954)

The starting point of this paper is the contradiction between the rules of \({\mathbb{Z}}_ 2\)-graded algebras and that of the quantum mechanics since Hermitian conjugation, which is the classical limit of taking operator adjoint in quantized theories, is not compatible with the sign rule of \({\mathbb{Z}}_ 2\)-graded algebras. In order to reconcile the Hermitian conjugation with the sign rule of \({\mathbb{Z}}_ 2\)-graded algebras, the author defines the second sign rule. This rule allows him to modify the Berezin approach to the theory of supermanifolds and to construct the concept of Hermitian supermanifold. Since almost all theorems of the traditional theory yield corresponding theorems in the Hermitian setting, the author restricts himself onto a short account of specific features of finite-dimensional Hermitian supergeometry. At the end of the paper some items of the dictionary between the notions of traditional supergeometry and the Hermitian language are listed.