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June 21, 2011

Towards the Mathematics of Quantum Field Theory

Posted by Urs Schreiber

Currently I am in Paris, visiting Frédéric Paugam. He is in the process of finishing a book with the title

Towards the mathematics of quantum field theory.

This book makes an admirable effort of collecting together for each aspect of QFT the most modern and most elegant mathematical formalism available.

For instance

And so on.

Posted at June 21, 2011 10:07 AM UTC

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Re: Towards the Mathematics of Quantum Field Theory

Thanks!

Posted by: Thomas on June 21, 2011 6:14 PM | Permalink | Reply to this

Re: Towards the Mathematics of Quantum Field Theory

This sounds very intriguing! I would like to study (higher) categorical formulations of physics some day, and Paugam’s book looks like an excellent way to do that.

Do the relevant concepts also include the variational bicomplex? Or does the variational bicomplex drop out of higher categorical machinery?

Posted by: Tobias Fritz on June 21, 2011 7:22 PM | Permalink | Reply to this

Re: Towards the Mathematics of Quantum Field Theory

Do the relevant concepts also include the variational bicomplex?

Yes. A variational bicomplex (nLab) is the complex of “local” differential forms on a given jet bundle. As such, it naturally lives in “D-geometry” – the higher geometry over infinitesimal path ∞\infty-groupoids Π inf(X)\mathbf{\Pi}_{inf}(X): the jet bundle of any E→XE \to X is its direct image along the constant infinitesimal path inclusion X→Π inf(X)X \to \mathbf{\Pi}_{inf}(X).

The relevant notions are dicussed in chapter 2 of Beilinson-Drinfeld’s Chiral Algebra :

The decomposition of forms on 𝒟\mathcal{D}-schemes into vertical and horizontal is in 2.8.11, the interpretation of the vertical differential on jets as the variational differential is around (2.3.20.1).

In Frédéric’s book the vertical differential is around def. 10.5.1 and the horizontal part, containing the currents, appears around def. 10.3.2.

Posted by: Urs Schreiber on June 21, 2011 8:54 PM | Permalink | Reply to this

Re: Towards the Mathematics of Quantum Field Theory

cool, thanks!

Posted by: Tobias Fritz on June 21, 2011 10:05 PM | Permalink | Reply to this

Re: Towards the Mathematics of Quantum Field Theory

We are now running a “Seminar on quantum field theory” that touches on some of this stuff.

I keep topic lists and relevant remarks and pointers at this webpage, which will be updated successively in the following weeks.

Posted by: Urs Schreiber on June 22, 2011 6:33 PM | Permalink | Reply to this

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