The course is an introduction to quantum field theory from a probabilistic point of view.

BACKGROUND

Quantum Field Theory (QFT) has become a universal framework to study physical systems with infinite number of degrees of freedom. Originally developed for high energy physics it lead to the explanation of universality in phase transitions using the Renormalization Group (RG), a powerful method to study scale invariant problems. QFT and RG ideas were thereafter applied to noisy systems, dynamical systems, non-equilibrium systems and many other problems.

Early on QFT was also acknowledged as an interesting problem for mathematicians. This lead first to axiomatic characterizations of what sort of mathematical object QFT is and in then to Constructive QFT and the use of probabilistic methods to produce concrete examples of QFT. These techniques were then used in rigorous statistical mechanics and disordered systems.

Following the pioneering work of Belavin, Polyakov and Zamoldchikov a beautiful class of QFT’s was uncovered by physicists, the two dimensional Conformal Field Theories (CFT). They have inspired a lot of new ideas in mathematics: the Schramm-Loewner Evolution (SLE) and Liouville Quantum Gravity and the theory of random surfaces.

In Stochastic Partial Differential Equations (SPDE) physicists had also used QFT to study universality, a notable example being the Kardar-Parisi-Zhang (KPZ) equation where exact non-conventional scaling behavior was uncovered. KPZ and related equations posed hard problems for analysis due to the combination of a very singular noise and nonlinearity. Quite recently QFT ideas on renormalization were introduced by Hairer to their mathematical analysis.

The course will provide an introduction to QFT from the probabilistic point of view. It is aimed at mathematicians who want to get an idea what QFT is about and to physicists who took a course in QFT and wondered what it meant mathematically.

PREREQUISITES

Probability theory including Brownian motion.

CONTENTS

1-particle Quantum mechanics

Brownian motion and Feynman-Kac formula

Thermal particle and SDE’s

Second quantization

Free quantum field

Axioms for QFT

From quantum fields to random fields

Gaussian free field

Reflection positivity: from random fields to quantum fields

SPDE’s and QFT

Interacting quantum fields = non-gaussian random fields

Perturbation theory, Feynman graphs

Perturbation theory for SPDE’s

Divergences, renormalization

Renormalization group for QFT

Renormalization group for SPDE

Hierarchical models

Critical phenomena and massless QFT

Two dimensions: Conformal invariance

Energy momentum tensor and Virasoro algebra

Free field representation

Introduction to Liouville theory