Operators and Representation Theory: Canonical Models for Algebras of Operators Arising in Quantum Mechanics (Dover Books on Physics) 2nd Edition by Palle E.T. Jorgensen (PDF)

Description

Algebras of operators arise frequently in the study of representations of Lie groups, both finite-dimensional and infinite-dimensional. This book begins with extensive background material that covers definitions and terminology, operators in Hilbert space, and the imprimitivity theorem.Advancing to considerations of the algebras of operators in Hilbert space, the heart of the text examines domains of representations, operators in the enveloping algebra, and spectral theory. The final section explores covariant representations and connections, with a particular focus on infinite-dimensional Lie algebras. A helpful Appendix on the integrability of Lie algebras concludes the text.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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…Therefore, it’s great news that this little guy has returned as a Dover reprint. 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.(Disclaimer: this review is a slightly edited copy of my Amazon.com review for the out-of-print North-Holland edition of the same book)

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