The theory of symmetry actions in quantum mechanics. With an application to the Galilei group
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.
scientific article; zbMATH DE number 2145806