Questions about Quantization and Noncommutative Geometry

Question

I am trying to orient myself among the vast amount of literature, trying to study the prerequisites necessary for gauge theory and theoretical physics. I have an undergraduate degree in mathematics and I am now studying manifold theory and geometry of bundles. I have an understanding of quantum mechanics but have never fully understood the mathematical structure of quantum field theory, though I have come to know that neither does anybody really.

I am particularly interested in the techniques of principle bundles in gauge theory, but I am bewildered by the physics books on gauge theory and quantum field theory, they make me feel very uncomfortable coming from a mathematician's perspective. I want to be on the path to be able to understand the paper's from Ed Witten, not so much on string theory but with respect to quantum field theory. I also am curious about noncommutative geometry.

I am not sure if it is better to post here or on the math stack exchange, but I will give physics a try first.

Now for my question:

• In regular quantum mechanics the procedure for quantization as far as I know is setting up the classical correspondence of the poission bracket to the quantum mechanical operator commutator. In the canonical quantization procedure for quantum field theory, a similar process is done with the fields as operators. That is as far as I have understood, but when it comes to the gauge theory of principle bundles and the U(1)xSU(2)xSU(3) groups, I have heard something about gauge quantization and the names of BRST, and I am just not sure what is the significance of this. I don't know why but I have got the idea from somewhere that people don't actually know how to quantize gauge theories. Why is quantization needed on these gauge groups? Does noncommutative geometry attempt to solve this problem of quantization?

I have looked at Quantization of Gauge Fields by Henneaux and it is unintelligible to me. I am looking for a resource similar to Gauge Theory and Variational Principles by Bleecker, but one that explains this so called gauge quantization. I realize noncommutative geometry is probably out of my reach until I become more proficient in geometry and topology, but I would like to know at least if this field attempts to answer the gauge quantization problem.

In summary,

• What is gauge quantization and how does it relate to noncommutative geometry?

I apologize if I have made mistakes, I am still very new to this field and quite disoriented. My understanding is only very rudimentary of principal bundles and quantum field theory. If it is better for me to post in the math stack exchange please let me know. Thank you!

Answer 1

Here is an answer which might be more at home on MathOverflow: I don't think there's any significant relationship between non-commutative geometry and gauge quantization. They're not really the same kind of thing. NCG is a language, while gauge quantization is a technique.

Quantum mechanics and classical gauge theory are basically orthogonal generalizations of classical mechanics theory.

In classical mechanics, the space of configurations is a manifold, whose functions form a commutative algebra. In classical gauge theory, the space of configurations is a derived manifold; the space of functions on this space is not a vector space, but instead a differential complex.

In quantum physics, another kind of geometry comes up, where the algebra of functions gets replaced with a non-commutative algebra. Said differently, non-commutative geometry studies operator algebras by analogy with geometry. This is relevant to any quantum system, and particularly relevant for quantum theories which have a classical limit. Classical theories typically have geometric descriptions, so it's nice to study them with a language that's indifferent to the classical/quantum distinction.

Obviously, you can try to make both generalizations at once. However, quantizing classical gauge theories is a somewhat delicate procedure. One major problem is that you can easily destroy the differential structure on the algebra of functions by deforming it to something non-commutative. Gauge quantization is a set of techniques for attempting this deformation in a controlled fashion. The constructions can probably be described in a language like non-commutative geometry, but it's not clear to me that using NCG provides any additional benefit.

For references, I'd suggest browsing Urs Schreiber's http://ncatlab.org and seeing what strikes your fancy.