155. Witten explains how to quantize gauge theory

Long time (almost 6 months?) ago Peter Woit wrote about Yau Birthday Conference and briefly mentioned the talk Edward Witten gave there. I was wondering is there preprint going to appear some day with outline of this talk, and it has finally appeared yesterday in ArXiv.

As you can imagine after reading the title and the abstract, Witten is trying to explain to a mathematician how to rigorously construct the quantum theory of a gauge field. It is especially interesting for a physicist to reiterate the procedure of making different things clean and understandable – to figure out whether the problem became clearer for us or more obscure instead.

Since a formulation in terms of functional integrals is a no go for matematicians except some really advanced ones ( like Max Kontsevich ), he is focusing on the Hamiltonian quantization, so he is inevitably to come to the Wilson lattice Hamiltonian and weak coupling expansion.

Unfortunately, it looks like he did not intend to discuss renormalization group or strong couling expansion (the things that Wilson representation is especially good for) – so YM quanta were massless till the last section of the paper Neither he wanted to discuss chaos in Yang-Mills theories  – and this topic again so naturally appears from the Hamiltonian formulation of the theory…

In the end of the paper, he mentions very briefly the mass gap problem, but does not introduce any ideas regading how one (a physicist or a mathematician) could approach it.