Quantum Fields and Probability

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

## Antti-Jukka Kupiainen

Julkaistu, 21.4.2018 klo 9:27

On next TUESDAY 24.4. the lecture is MOVED to 12-14 (the time of exercise session) because
of the following:

Everybody is encouraged to go to Aalto University colloquium at 15:15:

April 24th, 15-16, hall D : Prof. Clément Hongler (École polytechnique fédérale de Lausanne),

Statistical Field Theory and the Ising Model

The developments of statistical mechanics and of quantum field theory are among the major achievements of 20th century's science. In the second half of the century, these two subjects started to converge, resulting in some of the most remarkable successes of mathematical physics. At the heart of this convergence lies the conjecture that critical lattice models are connected, in the continuous limit, to conformally symmetric field theories. This conjecture has led to much insight into the nature of phase transitions and to beautiful formulae describing lattice models, which have remained unproven for decades.

In this talk, I will focus on the planar Ising model, perhaps the most studied lattice model, whose investigation has initiated much of the research in statistical mechanics. I will explain how, in the last ten years, we have developed tools to understand mathematically the emerging conformal symmetry of the model, and the connections with quantum field theory. This has led one to the proof of celebrated conjectures for the Ising correlations and for the description of the emerging random geometry. I will then explain how these tools have then yielded a rigorous formulation of the field theory describing this model, allowing one to make mathematical sense of the seminal ideas at the root of the subject of conformal field theory.

## Antti-Jukka Kupiainen

Julkaistu, 3.4.2018 klo 13:33

THIS WEEK FRIDAY April 6 we have lecture at the usual time 12-14!

## Antti-Jukka Kupiainen

Julkaistu, 3.4.2018 klo 9:56

Tuesday April 10 lecture is at 12-14 and exercise session at 14-16

## Antti-Jukka Kupiainen

Julkaistu, 20.3.2018 klo 9:16

NO LECTURE on next week Tuesday, March 27

## Antti-Jukka Kupiainen

Julkaistu, 7.3.2018 klo 10:27

SUMMER JOBS in mathematical physics available: please inquire paolo.muratore-ginanneschi@helsinki.fi

### Aikataulu

No exercise session on Tuesday 24.4.
The last exercise session will be held on FRIDAY 4.5 at 12.

On next TUESDAY 24.4. the lecture is MOVED to 12-14 (the time of exercise session) because
of the following:

Everybody is encouraged to go to Aalto University colloquium at 15:15:

April 24th, 15-16, hall D : Prof. Clément Hongler (École polytechnique fédérale de Lausanne),

Statistical Field Theory and the Ising Model

The developments of statistical mechanics and of quantum field theory are among the major achievements of 20th century's science. In the second half of the century, these two subjects started to converge, resulting in some of the most remarkable successes of mathematical physics. At the heart of this convergence lies the conjecture that critical lattice models are connected, in the continuous limit, to conformally symmetric field theories. This conjecture has led to much insight into the nature of phase transitions and to beautiful formulae describing lattice models, which have remained unproven for decades.

In this talk, I will focus on the planar Ising model, perhaps the most studied lattice model, whose investigation has initiated much of the research in statistical mechanics. I will explain how, in the last ten years, we have developed tools to understand mathematically the emerging conformal symmetry of the model, and the connections with quantum field theory. This has led one to the proof of celebrated conjectures for the Ising correlations and for the description of the emerging random geometry. I will then explain how these tools have then yielded a rigorous formulation of the field theory describing this model, allowing one to make mathematical sense of the seminal ideas at the root of the subject of conformal field theory.

PäivämääräAikaOpetuspaikka
Ti 16.1.2018
14:15 - 16:00
Pe 19.1.2018
12:15 - 14:00
Ti 23.1.2018
14:15 - 16:00
Pe 26.1.2018
12:15 - 14:00
Ti 30.1.2018
14:15 - 16:00
Pe 2.2.2018
12:15 - 14:00
Ti 6.2.2018
14:15 - 16:00
Pe 9.2.2018
12:15 - 14:00
Ti 13.2.2018
14:15 - 16:00
Pe 16.2.2018
12:15 - 14:00
Ti 20.2.2018
14:15 - 16:00
Pe 23.2.2018
12:15 - 14:00
Ti 27.2.2018
14:15 - 16:00
Pe 2.3.2018
12:15 - 14:00
Ti 6.3.2018
14:15 - 16:00
Pe 9.3.2018
12:15 - 14:00
Ti 13.3.2018
14:15 - 16:00
Pe 16.3.2018
12:15 - 14:00
Ti 20.3.2018
14:15 - 16:00
Pe 23.3.2018
12:15 - 14:00
Ti 27.3.2018
14:15 - 16:00
Pe 6.4.2018
12:15 - 14:00
Ti 10.4.2018
14:15 - 16:00
Pe 13.4.2018
12:15 - 14:00
Ti 17.4.2018
14:15 - 16:00
Pe 20.4.2018
12:15 - 14:00
Ti 24.4.2018
14:15 - 16:00
Pe 27.4.2018
12:15 - 14:00
Pe 4.5.2018
12:15 - 14:00

### Muu opetus

16.01. - 27.02.2018 Ti 12.15-14.00
13.03. - 27.03.2018 Ti 12.15-14.00
10.04. - 24.04.2018 Ti 12.15-14.00
Antti Kupiainen
Opetuskieli: englanti

### Materiaalit

Lecture notes and homework assignments

## Muu

### Kuvaus

Optional course.

Master's Programme in Mathematics and Statistics is responsible for the course.

The course belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

Master studies

(Varies with the content of the course)

Recommended time/stage of studies for completion: 1. or 2. year

Term/teaching period when the course will be offered: varying

Lectures and exercise classes

Exam and excercises, Course will be graded with grades 1-5

Exam, other methods will be described later