The Theory of Symmetry Actions in Quantum Mechanics

Gianni Cassinelli,

Pekka J. Lahti,

Alberto Levrero

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Also available online This is a book about representing symmetry in quantum mechanics. The book is on a graduate and/or researcher...

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This is a book about representing symmetry in quantum mechanics. The book is on a graduate and/or researcher level and it is written with an attempt to be concise, to respect conceptual clarity and mathematical rigor. The basic structures of quantum mechanics are used to identify the automorphism group of quantum mechanics. The main concept of a symmetry action is defined as a group homomorphism from a given group, the group of symmetries, to the automorphism group of quantum mechanics. The structure of symmetry actions is determined under the assumption that the symmetry group is a Lie group. The Galilei invariance is used to illustrate the general theory by giving a systematic presentation of a Galilei invariant elementary particle. A brief description of the Galilei invariant wave equations is also given.
From the reviews: "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 presentation is self-contained and is supplied with necessary background material and the list of relevant references." (Alexandr S. Holevo, Zentralblatt MATH, Vol. 1077, 2006) "The book is devoted to the study of symmetries in quantum mechanics. The book ends with an appendix which collects the necessary mathematical definitions, facts, and theorems needed in the book. In my opinion, the presentation of the material is of the highest quality. Clarity and mathematical rigor make the understanding easy and they make the reading joyful. The book is very much recommended for graduate students and/or researchers of either physics or mathematics interested in the theory of quantum mechanics." (L. Molnár, Publicationes Mathematicae Debrecen, Vol. 68 (1-2), 2006)
Inhalt
A Synopsis of Quantum Mechanics.- The Automorphism Group of Quantum Mechanics.- The Symmetry Actions and Their Representations.- The Galilei Groups.- Galilei Invariant Elementary Particles.- Galilei Invariant Wave Equations.