Lectures on Quantum Field Theory

Regularization just by quantization -- a new approach to the old problem of infinities in quantum field theory (Draft of lecture notes)

This is the current form of lecture notes on my approach to field quantization. I explain on a simple scalar-field model the physical motivation and show some preliminary applications (field produced by a pointlike charge, the vacuum-to-vacuum loop diagram, the Casimir presure). The approach is based on an appropriate construction of a representation of harmonic-oscillator Lie algebra and a sequential definition of Dirac deltas.

Lectures on Algebraic Quantum Field Theory and Operator Algebras

In this series of lectures directed towards a mainly mathematically oriented audience I try to motivate the use of operator algebra methods in quantum field theory. Therefore a title as ``why mathematicians are/should be interested in algebraic quantum field theory'' would be equally fitting. Besides a presentation of the framework and the main results of local quantum physics these notes may serve as a guide to some frontier research problems in mathematical physics with applications in particle and condensed matter physics for whose solution operator algebraic methods seem indispensable. The…

共形场论讲义

Lecture notes on conformal field theory

本文是基于P.M.2022年秋在圣母大学为数学研究生开设的课程整理的二维共形场论讲义，后续将与Nicolai Reshetikhin合作对其进行大幅修改与扩充。

Lectures on non-perturbative effects in large N gauge theories, matrix models and strings

In these lectures I present a review of non-perturbative instanton effects in quantum theories, with a focus on large N gauge theories and matrix models. I first consider the structure of these effects in the case of ordinary differential equations, which provide a model for more complicated theories, and I introduce in a pedagogical way some technology from resurgent analysis, like trans-series and the resurgent version of the Stokes phenomenon. After reviewing instanton effects in quantum mechanics and quantum field theory, I address general aspects of large N instantons and then present a d…

Dynamics and Quantum Optimal Transport: Three lectures on quantum entropy and quantum Markov semigroups

This document presents the contents of three lectures delivered by the author at the Erdős Center School ``Optimal Transport on Quantum Structures'', Septemer 19-23, 2022 in Budapest, Hungary. It presents a fairly self contained account of an active topic of current research, and this account should be accessible to most graduate students, as befits lectures for a school. The main results are known, but there a number of new proofs and some new results.

Lecture notes on the co-representation theory of the compact quantum group $SU_{q}(2).$

This is an exposition of S.L Woronowicz co-representation theory of the compact quantum group $SU_{q}(2)$ written for a seminar series.

算子代数与量子场论讲义

Lecture Notes on Operator Algebras and Quantum Field Theory

本讲义为 EMS-IAMP 春季学校准备，在算子代数框架内介绍量子场论的当代概念与方法，涵盖模理论、局部协变性及黑洞视界附近量子场的热力学应用。

Lecture notes on C*-algebras, Hilbert C*-modules, and quantum mechanics

This is a graduate-level introduction to C*-algebras, Hilbert C*-modules, vector bundles, and induced representations of groups and C*-algebras, with applications to quantization theory, phase space localization, and configuration space localization. The reader is supposed to know elementary functional analysis and quantum mechanics.

Lecture Notes on Quantum Teichmuller Theory

These notes are based on a lecture course by L. Chekhov held at the University of Manchester in May 2006 and February-March 2007. They are divulgative in character, and instead of containing rigorous mathematical proofs, they illustrate statements giving an intuitive insight. We intentionally remove most bibliographic references from the body of the text devoting a special section to the history of the subject at the end.

Lectures on Elliptic Functions and Modular Forms in Conformal Field Theory

A concise review of the notions of elliptic functions, modular forms, and theta-functions is provided, devoting most of the paper to applications to Conformal Field Theory (CFT), introduced within the axiomatic framework of quantum field theory. Many features, believed to be peculiar to chiral 2D (= two dimensional) CFT, are shown to have a counterpart in any (even dimensional) globally conformal invariant quantum field theory. The treatment is based on a recently introduced higher dimensional extension of the concept of vertex algebra.

拓扑递归与几何讲义

Lecture notes on topological recursion and geometry

本文为4小时迷你课程讲义，系统阐述拓扑递归在二维量子场论、上同调场论及曲线模空间Weil-Petersson体积计算中的体现，并从量子Airy结构视角关联Eynard-Orantin拓扑递归。

Introductory lectures on topological quantum field theory

These notes offer a lightening introduction to topological quantum field theory in its functorial axiomatisation, assuming no or little prior exposure. We lay some emphasis on the connection between the path integral motivation and the definition in terms symmetric monoidal categories, and we highlight the algebraic formulation emerging from a formal generators-and-relations description. This allows one to understand (oriented, closed) 1- and 2-dimensional TQFTs in terms of a finite amount of algebraic data, while already the 3-dimensional case needs an infinite amount of data. We evade these …

A Particle Field Theorist's Lectures on Supersymmetric, Non-Abelian Fluid Mechanics and d-Branes

This monograph is derived from a series of six lectures I gave at the Centre de Recherches Mathematiques in Montreal, in March and June 2000, while titulary of the Aisenstadt Chair.

Lectures on Quantum Mechanics for mathematicians

The main goal of these lectures -- introduction to Quantum Mechanics for mathematically-minded readers. The second goal is to discuss the mathematical interpretation of the main quantum postulates: transitions between quantum stationary orbits, wave-particle duality and probabilistic interpretation. We suggest a dynamical interpretation of these phenomena based on the new conjectures on attractors of nonlinear Hamiltonian partial differential equations. This conjecture is confirmed for a list of {\it model Hamiltonian nonlinear} PDEs by the results obtained since 1990 (we survey sketchy these …

数学物理中的十二个故事：扩展版 Heinemann 奖讲座

Twelve Tales in Mathematical Physics: An Expanded Heinemann Prize Lecture

本文以扩展版获奖讲座回顾作者在数学物理领域的主要贡献，涵盖非相对论量子力学、构造性量子场论与统计力学。

Lectures on supergeometry

Elements of supergeometry are an ingredient in many contemporary classical and quantum field models involving odd fields. For instance, this is the case of SUSY field theory, BRST theory, supergravity. Addressing to theoreticians, these Lectures aim to summarize the relevant material on supergeometry of modules over graded commutative rings, graded manifolds and supermanifolds.

量子扩散与随机矩阵理论讲义

Lecture notes on Quantum Diffusion and Random Matrix Theory

本讲义介绍一种基于随机矩阵理论研究随机薛定谔方程扩散极限的新方法，内容源自作者与 Adam Black 和 Reuben Drogin 的合作工作，并在索邦大学暑期学校讲授。

Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity

The purpose of these notes, based on a course given by the second author at Les Houches summer school, is to explain the probabilistic construction of Polyakov's Liouville quantum gravity using the theory of Gaussian multiplicative chaos. In particular, these notes contain a detailed description of the so-called Liouville measures of the theory and their conjectured relation to the scaling limit of large planar maps properly embedded in the sphere. These notes are rather short and require no prior knowledge on the topic.

Lecture Notes on Quantum Brownian Motion

Einstein's kinetic theory of the Brownian motion, based upon light water molecules continuously bombarding a heavy pollen, provided an explanation of diffusion from the Newtonian mechanics. Since the discovery of quantum mechanics it has been a challenge to verify the emergence of diffusion from the Schrödinger equation. The first step in this program is to verify the linear Boltzmann equation as a certain scaling limit of a Schrödinger equation with random potential. In the second step, one considers a longer time scale that corresponds to infinitely many Boltzmann collisions. The intuition i…

洛斯霍赫斯关于黎曼曲面模空间的讲义

Les Houches lecture notes on moduli spaces of Riemann surfaces

讲义介绍了黎曼曲面模空间的概念，涵盖递归边界结构、威滕猜想推广及JT引力与双曲几何的关系。

Lecture on Gauge Gravitation Theory. Gravity as a Higgs Field

Gravitation theory is formulated as gauge theory on natural bundles with spontaneous symmetry breaking where gauge symmetries are general covariant transformations, gauge fields are general linear connections, and Higgs fields are pseudo-Riemannian metrics.

Fibre Bundles, Jet Manifolds and Lagrangian Theory. Lectures for Theoreticians

In contrast with QFT, classical field theory can be formulated in a strict mathematical way by treating classical fields as sections of smooth fibre bundles. Addressing to the theoreticians, these Lectures aim to compile the relevant material on fibre bundles, jet manifolds, connections, graded manifolds and Lagrangian theory. They follow the perennial course of lectures on geometric methods in field theory at the Department of Theoretical Physics of Moscow State University.

Lectures on the topological recursion for Higgs bundles and quantum curves

The paper aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the topological recursion, that has its origin in random matrix theory and has been effectively applied to many enumerative geometry problems; and the other is the quantization of Hitchin spectral curves associated with Higgs bundles. Our emphasis is on explaining the motivation and examples. Concrete examples of the direct relation between Hitchin spectral curves and enumeration problems are given. A ge…

Lectures on Loop Quantum Gravity

Quantum General Relativity (QGR), sometimes called Loop Quantum Gravity, has matured over the past fifteen years to a mathematically rigorous candidate quantum field theory of the gravitational field. The features that distinguish it from other quantum gravity theories are 1) background independence and 2) minimality of structures. Background independence means that this is a non-perturbative approach in which one does not perturb around a given, distinguished, classical background metric, rather arbitrary fluctuations are allowed, thus precisely encoding the quantum version of Einstein's radi…

量子最优输运：量子耦合与多体问题

Quantum Optimal Transport: Quantum Couplings and Many-Body Problems

本文是2023年Erdös中心量子最优输运暑期学校的4.5小时课程讲义，分三讲介绍指数2的Wasserstein距离的量子对应定义、应用及三角不等式、Kantorovich对偶等核心性质。

Lectures on Yangian Symmetry

In these introductory lectures we discuss the topic of Yangian symmetry from various perspectives. Forming the classical counterpart of the Yangian and an extension of ordinary Noether symmetries, first the concept of nonlocal charges in classical, two-dimensional field theory is reviewed. We then define the Yangian algebra following Drinfeld's original motivation to construct solutions to the quantum Yang-Baxter equation. Different realizations of the Yangian and its mathematical role as a Hopf algebra and quantum group are discussed. We demonstrate how the Yangian algebra is implemented in q…

Lecture notes on operator algebras and their application in physics

Lecture notes to a one-term course on operator algebras and their application in physics. Very brief and basic introduction to the subject of Banach- and C-star algebras complemented with their appearance in physics. The course is intended at mathematics students in their master program with a basic knowledge of functional analysis and some interest in the foundations of quantum mechanics.

Lecture notes on Generalised Hydrodynamics

These are lecture notes for a series of lectures given at the Les Houches Summer School on Integrability in Atomic and Condensed Matter Physics, 30 July to 24 August 2018. The same series of lectures has also been given at the Tokyo Institute of Technology, October 2019. I overview in a pedagogical fashion the main aspects of the theory of generalised hydrodynamics, a hydrodynamic theory for quantum and classical many-body integrable systems. Only very basic knowledge of hydrodynamics and integrable systems is assumed.

Renormalization Hopf algebras and combinatorial groups

These are the notes of five lectures given at the Summer School {\em Geometric and Topological Methods for Quantum Field Theory}, held in Villa de Leyva (Colombia), July 2--20, 2007. The lectures are meant for graduate or almost graduate students in physics or mathematics. They include references, many examples and some exercices. The content is the following. The first lecture is a short introduction to algebraic and proalgebraic groups, based on some examples of groups of matrices and groups of formal series, and their Hopf algebras of coordinate functions. The second lecture presents a very…

Lectures on K-theoretic computations in enumerative geometry

These are notes from my lectures on quantum K-theory of Nakajima quiver varieties and K-theoretic Donaldson-Thomas theory of threefolds given at Columbia and Park City Mathematics Institute. They contain an introduction to the subject and a number of new results. In particular, we prove the main conjecture of arXiv:hep-th/0412021 and the conjecture of arXiv:1404.2323 in the simplest case of reduced smooth curves. We also prove the the absence of quantum corrections to the capped vertex with descendents for sufficiently large framing (and polarization), which is a property we call large framing…

Lectures on quantum energy inequalities

Quantum field theory violates all the classical energy conditions of general relativity. Nonetheless, it turns out that quantum field theories satisfy remnants of the classical energy conditions, known as Quantum Energy Inequalities (QEIs), that have been developed by various authors since the original pioneering work of Ford in 1978. These notes provide an introduction to QEIs and also to some of the techniques of quantum field theory in curved spacetime (particularly, the use of microlocal analysis together with the algebraic formulation of QFT) that enable rigorous and general QEIs to be de…

Lectures on Matrix Field Theory I

The subject of matrix field theory involves matrix models, noncommutative geometry, fuzzy physics and noncommutative field theory and their interplay. In these lectures, a lot of emphasis is placed on the matrix formulation of noncommutative and fuzzy spaces, and on the non-perturbative treatment of the corresponding field theories. In particular, the phase structure of noncommutative $ϕ^4$ theory is treated in great detail, and an introduction to noncommutative gauge theory is given.

格路径的量子解释

Quantum interpretation of lattice paths

本文通过新颖的对称lecture hall图建立Motzkin路径与lecture hall路径的双射，并据此将量子谐振子、氢原子等系统的位置算符矩解释为相应格路径的生成函数。

量子场论中的测量讲义

Lectures on measurement in quantum field theory

本文介绍量子场论测量理论，涵盖代数QFT、测量方案、态更新及Sorkin问题，并给出基于Green双曲算子的示例，强调框架的解释意义。

圈量子引力论导论：Rovelli的LQG讲义

Introduction to Loop Quantum Gravity: Rovelli's lectures on LQG

本文是Carlo Rovelli于2018年在马赛举行的圈量子引力论（LQG）讲座的LaTeX转录笔记，按原讲座顺序整理并补充了图表与方程，旨在为视频版本提供辅助学习材料。

Lectures on Factorization Homology, Infinity-Categories, and Topological Field Theories

These are notes from an informal mini-course on factorization homology, infinity-categories, and topological field theories. The target audience was imagined to be graduate students who are not homotopy theorists.