The Theory of Symmetry Actions in Quantum Mechanics – Book Review
Quantum mechanics has revolutionized our understanding of the fundamental laws of nature. One aspect of the theory that has gained most attention, by both Physicists and lay public, is the irreducible probabilistic nature of physical laws. However, quantum mechanics has also brought to the forefront the essential role that symmetries play in our description and understanding of the world around us. Symmetries are important even in classical mechanics, but in quantum mechanics they become almost indispensable, and they have a particularly nice way of manifesting themselves. Anyone interested in gaining a deeper understanding of quantum mechanics will have to learn a great deal about symmetries, group theory, representation theory and similar subjects.
“The Theory of Symmetry Actions in Quantum Mechanics” is a very technical advanced collection of lectures on the subject. This material is rarely, if ever, covered at such high level of mathematical abstraction in US graduate schools. The approach to the subject is more appropriate for an advanced mathematics course, and reader ought to be very much at ease with advanced concepts from group theory, condensed mathematical notation, theorems, proofs, and similar matters. The topics covered in this book include:
1. A Synopsis of Quantum Mechanics
2. The Automorphism Group of Quantum Mechanics
3. The Symmetry Actions and Their Representations
4. The Galilei Group
5. Galilei Invariant Elementary Particles
6. Galilei Invariant Wave Equation
This book is a comprehensive resource for deeper understanding of symmetries of quantum mechanics on a very abstract functional/group theoretic level, and will be of interest for anyone who is already familiar with the quantum mechanical symmetries and wants to deepen their understanding. There are several subtle points that are emphasized which are usually overlooked in more elementary treatments. However, this is certainly not the best book to be used as a first exposure to this subject. The symmetries of quantum mechanics are covered in most advanced quantum mechanics textbooks. A more systematic, yet relatively elementary, approach can be found in Quantum Mechanics. Symmetries.
Overall, this is a very technically demanding book on symmetries in quantum mechanics that can be a valuable resource for anyone who is interested in mathematical physics.
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