Quantum Theory, Groups and Representations: An Introduction

by Peter Woit

This text systematically presents the basics of quantum mechanics, emphasizing the role of Lie groups, Lie algebras, and their unitary representations. The mathematical structure of the subject is brought to the fore, intentionally avoiding significant overlap with material from standard physics courses in quantum mechanics and quantum field theory. The level of presentation is attractive to mathematics students looking to learn about both quantum mechanics and representation theory, while also appealing to physics students who would like to know more about the mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of the subject. The latter portions of the book focus on central mathematical objects that occur in the Standard Model of particle physics, underlining the deep and intimate connections between mathematics and the physical world. While an elementary physics course of some kind would be helpful to the reader, no specific background in physics is assumed, making this book accessible to students with a grounding in multivariable calculus and linear algebra. Many exercises are provided to develop the reader's understanding of and facility in quantum-theoretical concepts and calculations.

Genres: Physics, Science, Mathematics, Nonfiction, Quantum Mechanics, Textbooks

690 pages, Kindle Edition

First published August 28, 2014

Reviews

[Manny · Rating 4 out of 5]

This book, like nothing else I've read recently, made me feel I was eight years old again. I read the whole thing, and I sort of got it, but I know I didn't really get it. I'm hoping I'll really get it when I'm older. Well, I got some of it. The person who wrote it, Peter Woit, has for most of his life been a huge admirer of the German mathematician Hermann Weyl, who in 1929 published a book on group theory and quantum mechanics. Weyl's book is famous in physics circles, partly because it ended up being very influential, and partly because hardly anyone understands it. It's arguably the most difficult book I know. Woit has written his book to try and explain what Weyl's book is about and make it accessible to ordinary people, by which he apparently means courageous souls with at least advanced graduate knowledge of both math and physics. Woit is definitely much easier to read than Weyl, and maybe he's succeeded; it's a bit difficult to tell, partly because I'm not really in his target group (I don't know enough math or physics), and partly because the book isn't finished yet (it's a preprint which he's been using as lecture notes for the course he teaches at Columbia).

So what is the book about, you want to know? Well, I wish I could give you a proper summary, but that would involve really getting it. The crux, as the title says, is the relationship between quantum mechanics and representations of groups. Groups, if you're not a mathematician, are basically abstractions of symmetries; a representation of a group is basically when you turn it into an actual symmetry. With most groups, there's more than one way to do this. Weyl's great insight, which ended up putting 20th century physics on a new course, was that the mathematical structure of group representations is intimately linked to the mathematical structures of both quantum mechanics and classical mechanics. (For people who know what these words mean: it's not an accident that Lie brackets and Poisson brackets are both commutators). The book is full of very detailed explorations of this general idea. Some of it made good sense; for example, I finally understand what it is that makes the idea of "supersymmetry" so natural and appealing, even though there's no experimental evidence to support it. The mathematics points so clearly in a certain direction that you feel it just has to be there. Other parts made rather less sense; I still don't grasp what it means to say that a particle is essentially an irreducible representation, which disappoints me. I was hoping I'd get that.

Well, my next goal is to read Shankar's Principles of Quantum Mechanics. Woit constantly refers to him, and I can see that I should have read him first. This sort of strategy worked fine when I was eight, and I'm expecting that it'll work now; hopefully I'll be able to reread Woit a year from now and I'll wonder what it was that I couldn't understand. But right now, I'm still unhappy. It's infuriating to see that the big kids get it and I don't.
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[Update, a couple of days later]

When I'd finished, I wasn't sure how much I'd learned. But starting Shankar, I now find that the long first chapter, which gives the mathematical background, is just obvious - I can skim though it, everything makes perfect sense.

Thumbs up, Woit! I'm awarding you a fourth star.

[Wendelle · Rating 5 out of 5]

course textbook.

[Shozab Qasim · Rating 5 out of 5]

I've never been a huge fan of group theory but Woit manages to explain it wonderfully and link it to physics. While I read most sections of it - I'd definitely love to read the whole things once again sometime.