Literature for gauge field theory on the lattice in geometrical formulation

Question

I have found an article by Huebschmann, Rudolph and Schmidt about "A Gauge Model for Quantum Mechanics on a Stratified Space" and I am very interested in this subject, but I don't have any background in gauge-field theory or something like that.

So my question is, if there are any good introductory books or overview articles which cover Hamiltonian (quantum) gauge field theory on the lattice in a geometrical mathematical language like the article mentioned above? Especially I am interested in references which deal with more regular cases rather than the singular cases discussed in the article above. Finally it is important that it covers this topic in a way that one can gain a bit deeper physical understanding (without giving up a clear, rigorous and geometrical mathematical language).

Added Thanks for your answers. Gauge field theory seems to be a very wide field, so I should perhaps mention why I want to learn some basics about gauge field theory beside of very strong intrinsic interest.

I am studying phase-space reduction in the context of deformation quantization of systems with finite degrees of freedom. Now I want to know if it is possible to reinterpret this situation (at least as a toy model) in some way in terms of gauge field theories. So my aim is to learn at least as much of gauge field theory that I can understand if and why such an reinterpretation is possible or how far one can go.

The idea to look for lattice gauge theories was the following quote from the paper by Huebschmann et. al.

Gauge theory in the Hamiltonian approach, phrased on a finite spatial lattice, leads to tractable finite-dimensional models for which one can analyze the role of singularities explicitly. Under such circumstances, after a choice of tree gauge has been made, the unreduced classical phase space amounts to the total space $\mathrm{T}^* (K \times \dots \times K)$ of the cotangent bundle on a product of finitely many copies of the manifold underlying the structure group $K$. Gauge transformations are then given by the lift of the action of $K$ on $K \times \dots \times K$ by diagonal conjugation. This leads to a finite-dimensional Hamiltonian system with symmetries.

Why a asked for a "geometric language" is clear because the strength of deformation quantization is in fact in the area of the quantization of systems with a more complex phase-space geometry.

Having this additional information it is perhaps easier for you to give me some hints where to start in literature.

Answer 1

Although not mathematical per se, I personally like Kogut's works for Hamiltonian lattice gauge theory: there is an old RMP article and a rather good book which I purchased specifically for its presentation of the Hamiltonian theory. Of course Kogut introduced the subject along with Wilson in a PRD article, so his presentation of the material is not particularly exotic.

I will say that the advantage of the lattice is that rigor is not really much of an issue unless one wishes to take a continuous limit. Even so the incorporation of material on the Ising theory is helpful in this regard.

Answer 2

Rudolph&Schmidt just released their second volume Differential Geometry and Mathematical Physics II: Fiber bundles, Topology and Gauge fields, and it covers exactly this stuff.

These two volumes instantly became my favorite intro to geometric methods in theoretical physics. I'm sure in a few years they'll replace the likes of Nakahara due to their incredible mathematical completeness and physical insight.

Answer 3

I'm going to guess there will be no good introductory texts at that level that will be all that useful. You will probably have to delve into the physics literature (particularly, review articles). Although, a lot of lattice QCD stuff is written for experimentalists who know little formal QCD or math, so you should be able to find something reasonably accessible for a beginning step.

The text "Differential Geometry, Gauge Theories, and Gravity" is sufficient for the geometric part of the mathematical background of gauge theories in general, and even has a paragraph about lattice QCD! And applying this formalism to lattice QCD isn't "hard" (thus, a paragraph ;)). However, the "quantum" part of QCD is not covered by the classical gauge theories described in that text, and to really understand that you have to know QCD.

As for the physical intuition of what's going on, I doubt you will get much without a reasonable understanding of quantum field theories in general. You certainly can't get an understanding of these without losing rigor, since there is not yet a rigorous version of QFT (there are some very good books out there that do rigorously what we can do rigorously, but that is not much). Lattice QCD is on much better footing, but you do lose a lot of the intuition skipping straight to that.

So I would suggest going to the arxiv, and trying to find a good review article that makes you happy, and then check out its references when you get confused. But if you want to really develop a physical intuition for what's going on, I think that would be a lot of work.

Offhand, the book mentioned by Steve looks good, but if it is good for you or not depends on your background. And by math standards it does not look particularly mathematical.

Answer 4

Tony Phillips, a topologist at Stony Brook who taught me differential topology when I was a first-year graduate student, has worked on lattice gauge theory since the mid-to-late 1980s. You can try to take a look at his papers on MathSciNet on this topic. His language is certainly geometrical.