# A good primer for geometric quantization.

Hello everyone:

I'm searching for a good primer on geometric quantization.

I found the following:

Mathematical foundations of geometric quantization (A. Echeverria-Enriquez, et al.)

Geometric quantization (W. Ritter)

If anyone can suggest any other(s)*, I would be greatly appreciative.

Warm regards.

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The answer to your question depends on your background and on your interests. How much symplectic geometry do you know? Are your primarily interested in math or in physics? In other words: what do you know? what do you want to learn and why? – Eugene Lerman Aug 6 '12 at 14:11
A commented list of references is here: ncatlab.org/nlab/show/geometric+quantization#References – Urs Schreiber Apr 14 '13 at 13:12

There is "Geometric Quantization: A Crash Course" by Eugene Lerman.

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I learned geometric quantization from Jean-Marie Souriau — one of the initiators of the subject with Bertram Kostant — in two texts essentially:

1. "Structure des systèmes dynamiques" (chapter 5). It is in french, that's true, but there is an english version too.

2. "Construction explicite de l'indice de Maslov" (also in french without an english version).

You can Google these texts, you'll find them on the net. In the book, the first reference, you'll find the geometric prequantization construction and the quantization of elementary systems (coadjoint orbits of Poincaré group). In the second paper you'll find the geometric quantization of the harmonic oscillator with the Maslov correction. That example is not in the book. I think that that is all what geometric quantization achieves completely, in accordance with physics experiments and results. If you read these two texts you'll know as much as you can hope in the field. What has been done after in geometric quantization is essentially some refinements in representation theory, but the relation with real physical systems is not always clear (at least for me). I may be wrong, if someone else know another successful (physical) application of geometric quantization, just tell me, I'm interested.

If you are serious in your involvement and of course depending on your level in geometry, I would say that this is one year study.

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JMS doesn't do complex polarizations; those are needed to quantize e.g. coadjoint orbits of compact Lie groups (which do arise in physics). Unfortunately it's not so easy to point to a good reference to learn that. – Francois Ziegler Nov 20 '13 at 2:39
@FrancoisZiegler Agreed, even if I am not comfortable with that complexification, it is not a procedure I understand clearly the real meaning, except the fact that that gives some representation. That said, I don't understand really quantum mechanics, so... – Patrick I-Z Nov 20 '13 at 8:25
@FrancoisZiegler I have a question: what do we get from the complexification procedure for the harmonic oscillator? Does it makes sense in the first place? Do we get the right quantization? If yes, that's fine, if not why should it exist two different approaches, one for space-time extended system (non compact Lie groups and real polarizations) and another one for internal degrees of freedom (compact Lie groups and complex polarizations)? Does what I say makes sense? – Patrick I-Z Nov 20 '13 at 10:56
Yes, complex polarizations can be used for the oscillator and one still gets the correct representation (in what's called the Bargmann-Fock realization). In fact, if to the Heisenberg generators one adds just only the energy, one gets a 4-dimensional solvable Lie group (with Lie algebra span(H,P,Q,I)) having "the oscillator" as coadjoint orbit. That orbit can be quantized without pairing but ONLY if you use a complex polarization. It was the paradigmatic case for solvable groups, first done by Streater (ams.org/mathscinet-getitem?mr=207908). – Francois Ziegler Nov 20 '13 at 13:29
Oh, there are still plenty of questions that escape any framework. For example BKS pairing doesn't work for the Kepler manifold, which can be only be "dealt with" (so to speak) using a hybrid of real and complex polarizations: ams.org/mathscinet-getitem?mr=539648 – Francois Ziegler Nov 20 '13 at 14:20

Hurt's book is inspiring.

http://www.mathnet.ru/eng/intf35

Kirillov survey is short and concise.

J.Snyaticki, Geometric and Quantum Mechanics, Ap Sciences Series Vol.

http://www.amazon.com/gp/aw/d/0198502702

I do not have these ones. Woodhouse book is by Oxford university press should be good.

This has one chapter on g.q. it is quite good

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Maybe I'll elaborate a little on books written by Śniatycki and Woodhouse, because these are the ones that I've read (or rather gave up trying).

At first I picked Woodhouse, but I had to stop, probably due to weakness of my geometrical background at the time. For me, it was rather dry and I don't remember seeing there even one example completely worked out.

In the preface of Śniatycki's book it is written that its aim is performing actual computations (or something like that; I don't have it at hand). So I went the extra mile and struggled with it for a while, getting almost to the end. It was quite hard, but the reward waiting for me, a bunch of thoroughly examined examples (although I was forced to provide some details, but mainly because I wasn't aware of some standard techniques), was worth it.

It is also definitely not a bad idea to search homepage of John Baez; I remember that there are some resources concerning this topic.

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There is a set of lecture notes based on a course given by Alan Weinstein called Lectures on the Geometry of Quantization. It is part of the Berkeley Mathematics Lecture Notes series published by the AMS. You can also find the pdf freely available at Weinstein's homepage.

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Have you read it? How good is it as a first introduction? – Sadiq Ahmed Aug 6 '12 at 10:36
I haven't read it, sorry! – MTS Aug 6 '12 at 20:40
It is a great book, but it puts its focus more on the inverse direction - the semisclassical limit. I think only the last chapter is truely about Geometric Quantization, and there the presentation differs from the usual construction. – Tobias Diez Nov 20 '13 at 9:39

Teru Thomas gave some nice introductory lectures on Geometric Quantization in Edinburgh a year ago. The notes, along with notes on subsequent topics, are on his webpage here.

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The best introduction I know is last chapters of B.C.Hall's "Quantum Theory for Matematicians" book. V.P.Nair's QFT book also has a very nice chapter about it but not as much in detail.

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