The theory of symmetry actions in quantum mechanics. With an application to the Galilei group
type: Item
instance of: scholarly article
title: The theory of symmetry actions in quantum mechanics. With an application to the Galilei group
zbMATH Open document ID: 1077.81002
author: Gianni Cassinelli
author: Ernesto De Vito
author: Alberto Levrero
author: Pekka Lahti
published in: Lecture Notes in Physics
publication date: 2005-03-15T00:00:00Z
review text: This slim and carefully written book is devoted to projective representations of continuous groups targeted to the fundamental symmetry group of nonrelativistic quantum mechanics -- the Galilei group. The aim is to present the theory needed to construct the universal central extension \(\overline{G}\) from the physical symmetry group \(G\). Chap. 2 is devoted to the general automorphism group of quantum mechanics in Hilbert space and related fundamental results such as the Wigner representation theorem. The construction of \(\overline{G}\) presented in Chap. 3 is quite explicit and can be performed by simple linear algebraic tools. As an application, in Chaps. 4, 5 the case of the Galilei group is treated in full detail both in \(3+1\) and \(2+1\) dimensions, the latter case being related to recently increased interest in the surface phenomena. Chap. 6 is devoted to the study of the Galilei invariant wave equations, showing how these wave equations emerge directly from the symmetry principle without using traditional Lagrangian formalism. In particular it is proven that for a particle of spin \(j\) there exists a linear wave equation such that the particle acquires a gyromagnetic internal moment with the gyromagnetic ratio \(1/j\). The presentation is self-contained and is supplied with necessary background material on quantum mechanics, Hilbert space and representation theory in Chap. 1 and Appendix, and the list of relevant references.
Mathematics Subject Classification ID: 81-02
Mathematics Subject Classification ID: 81R05
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 81R20
zbMATH DE Number: 2145806
zbMATH Keywords: symmetry
zbMATH Keywords: projective representation
zbMATH Keywords: universal central extension
zbMATH Keywords: Galilei group
MaRDI profile type: Publication
full work available at URL: b99455
OpenAlex ID: W1658554953
DOI: 10.1007/B99455
reviewed by: Alexander S. Holevo
description: scientific article; zbMATH DE number 2145806