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Title: Operators and Representation Theory: Canonical Models for Algebras of Operators Arising in Quantum Mechanics by Palle E.T. Jorgensen ISBN: 0-444-70321-7 Publisher: Elsevier Science Ltd Pub. Date: December, 1987 Format: Paperback List Price(USD): $144.50

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Average Customer Rating: 4.5 (2 reviews)

Rating: 5
Summary: Symmetry in Quantum Mechanics and Beyond...
Comment: The idea of symmetry is tantamount to physics. This has become all the more true in Quantum Physics, where symmetries manifest themselves in highly nontrivial ways. The mathematics of the implementation of symmetries at the quantum level, however, possesses great subtlety, as seen from the work of Wigner, Mackey, Bargmann, Nelson... - Just to cite the simplest example, the nontrivial transition between ray (projective) representations and unitary representations in a Hilbert space. The quantization of symmetries had also a fundamental role in Relativistic Quantum Physics: superselection sectors, anomalies, spontaneous symmetry breaking...

Palle's book comprises, from the viewpoint of a mathematician, the underlying machinery for probing the quantum manifestations of symmetry: induced representations and central extensions of Lie algebras, representations of Lie Algebras as unbounded operators in a Hilbert space and their associated spectral theory, and Mackey's systems of imprimitivity. In this sense, the book's programme is somewhat similar to Barut & Raczka's "Group Representations for Physicists", although in a much more concise manner and more centered on Hilbert space aspects than the latter. Conciseness in a topic requiring resources from so varied mathematical areas (functional analysis, homological algebra, Lie groups and Lie algebras, representation theory) and, at the same time, keeping track of the physical motivations, requires skill - here the budding mathematical physicist won't be disappointed.

Last chapter uses the technology developed in previous chapters to probe cutting-edge topics such as representations of Virasoro algebras (conformal field theory, string theory) and the study of Connes-like differential-geometric structures in C*-algebras (noncommutative tori, etc.). Summing up, it's a blend of topics of central and lasting interest in contemporary quantum physics that is hard to find elsewhere in a single tome. All it's needed from the reader is some acquaintance with basic functional analysis (Banach and Hilbert spaces, measure theory), Lie group theory, and the notion of a C*-algebra, besides a good course in Quantum Mechanics.

The only problem here is this great book being out-of-print...

Rating: 4
Summary: Operators and Representation Theory : Canonical Models for A
Comment: This book is an excellent introduction to representation theory in quantum mechanics and outlines abstract topics in a pedagogically sound manner. I would recommend this book for beginning graduate students in physics or for anyone interested in learning about how to work with Heisenberg, Schroedinger, etc. representations and how to apply that to intersting problems. Knowledge of undergraduate quantum mechanics is required.