# Rough paths and QFT

1. Oct 22, 2014

### atyy

Because of the Feynman path integral, QFT can be made into a statistical field theory. In rigourous relativistic field theories, this is formalized by the Osterwalder-Schrader conditions. At any rate, there are well established links between quantum field theory and statistical physics.

A famous equation in statistical physics is the KPZ equation, which appears to involve the product of distributions and so isn't obviously well defined, which of course hasn't stopped physicists from finding it meaningful. One of this year's Fields Medals was given to Martin Hairer, apparently for being able to make rigourous sense of the KPZ equation using Terry Lyons's "rough paths" theory.

In this abstract for a talk by Lyons http://www.oxford-man.ox.ac.uk/events/what-can-rough-paths-do-for-you, it is mentioned that rough paths theory is related via Hairer's work to "John Cardy's work on constructive conformal field theory". Is there any simple introduction to what this means?

2. Oct 23, 2014

### atyy

http://arxiv.org/abs/1012.3873
From constructive field theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics
Jacques Magnen, Jérémie Unterberger

http://iecl.univ-lorraine.fr/~Jeremie.Unterberger/book-rough-paths.pdf
Rough path theory
Jérémie Unterberger

http://arxiv.org/abs/1303.5113v4
A theory of regularity structures
Martin Hairer
"This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a "renormalisation group" which is defined canonically in terms of the regularity structure associated to the given class of PDEs.
As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the $\Phi^{4}_{3}$ Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of 3-dimensional ferromagnets near their critical temperature."

http://www.hairer.org/notes/Regularity.pdf
Introduction to regularity structures
Martin Hairer

These are short notes from a series of lectures given at the University of Rennes in June 2013, at the University of Bonn in July 2013, at the XVIIth Brazilian School of Probability in Mambucaba in August 2013, and at ETH Zurich in September 2013. They give a concise overview of the theory of regularity structures as exposed in the article [Hai14]. In order to allow to focus on the conceptual aspects of the theory, many proofs are omitted and statements are simplified. We focus on applying the theory to the problem of giving a solution theory to the stochastic quantisation equations for the Euclidean $\Phi^{4}_{3}$ quantum field theory.

Last edited: Oct 24, 2014