Supergeometry and Hermitian conjugation
type: Item
instance of: scholarly article
title: Supergeometry and Hermitian conjugation
zbMATH Open document ID: 0734.58009
DOI: 10.1016/0393-0440(90)90009-R
author: Thomas Schmitt
published in: Journal of Geometry and Physics
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58A50
zbMATH DE Number: 4214970
zbMATH Keywords: \({\mathbb{Z}}_ 2\)-graded algebras
zbMATH Keywords: supermanifolds
zbMATH Keywords: Hermitian supergeometry
reviewed by: Vyacheslav Abramov
MaRDI profile type: Publication
cites work: Particle spin dynamics as the Grassmann variant of classical mechanics
cites work: Q3344780
cites work: Q4130629
cites work: Q3713209
cites work: Q3702979
cites work: Q3811298
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full work available at URL: 0393-0440(90)90009-r
OpenAlex ID: W1973735989
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description: scientific article; zbMATH DE number 4214970