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Book Details1999; 723 ppMSC: Primary 81
Ideas from quantum field theory and string theory have had considerable impact on mathematics over the past 20 years. Advances in many different areas have been inspired by insights from physics.
In 1996–97 the Institute for Advanced Study (Princeton, NJ) organized a special yearlong program designed to teach mathematicians the basic physical ideas which underlie the mathematical applications. The purpose is eloquently stated in a letter written by Robert MacPherson: “The goal is to create and convey an understanding, in terms congenial to mathematicians, of some fundamental notions of physics ... [and to] develop the sort of intuition common among physicists for those who are used to thought processes stemming from geometry and algebra.”
These volumes are a written record of the program. They contain notes from several long and many short courses covering various aspects of quantum field theory and perturbative string theory. The courses were given by leading physicists and the notes were written either by the speakers or by mathematicians who participated in the program. The book also includes problems and solutions worked out by the editors and other leading participants. Interspersed are mathematical texts with background material and commentary on some topics covered in the lectures.
These two volumes present the first truly comprehensive introduction to this field aimed at a mathematics audience. They offer a unique opportunity for mathematicians and mathematical physicists to learn about the beautiful and difficult subjects of quantum field theory and string theory.
ReadershipGraduate students and research mathematicians working in various areas of mathematics related to quantum field theory.
This item is also available as part of a set: 
Table of Contents

Front Cover

Preface

Brief Contents

CrossReference Codes

Contents

Introduction

Glossary

Part 1 Classical Fields and Supersymmetry

Notes on Supersymmetry (following Joseph Bernstein)

Introduction

CHAPTER 1 Multilinear Algebra

§ 1. 1. The sign rule

§1.2. Categorical approach

§1.3. Examples of the categorical approach

§1.4. Free modules

§1.5. Free commutative algebras

§1.6. The trace

§ 1. 7. Even rules

§1.8. Examples of the "even rules" principle

§1.9. Alternate description of super Lie algebras

§1.10. The Berezinian of an automorphism

§1.11. The Berezinian of a free module

Appendix to §1: Graded super vector spaces

CHAPTER 2 Super Manifolds: Definitions

§§2.12.7. Super manifolds as ringed spaces

2 .1.

2.2 Remarks.

2.3 Examples.

2.5.

2.6 Remarks.

2. 7.

§§2.82.9. The functor of points approach to super manifolds

2.8.

2.9.

§2.10. Super Lie groups

§2.11. Classical series of super Lie groups

CHAPTER 3 Differential Geometry of Super Manifolds

§3.1. Introduction

§3.2. Vector bundles

§3.3. The tangent bundle, the cotangent bundle and the de Rham complex

§3.4. The inverse and implicit function theorems

§3.5. Distributions

§3.6. Connections on vector bundles

§3. 7. Actions of super Lie algebras; vector fields and flows; Lie derivative

§3.8. Super Lie groups and HarishChandra pairs

§3.9. Densities

§3.10. Change of variables formula for densities

§3.11. The Lie derivative of sections of Ber(Ω1M)

§3.12. Integral forms

§3.13. A second definition of integral forms

§3.14. Generalized functions

§3.15. Integral forms as functions of infinitesimal submanifold elements

CHAPTER 4 Real Structures

§§4.14.3. Real structures and *operations

4.1.

4.2.

4.3.

§4.4. Super Hilbert spaces

§4.5. SUSY quantum mechanics

§4.6. Real and complex super manifolds

§4.7. Complexifications, in infinite dimensions

§4.8. cs manifolds

§4.9. Integration on cs manifolds; examples

REFERENCES

Notes on Spinors

Introduction

CHAPTER 1 Overview

CHAPTER 2 Clifford Modules

CHAPTER 3 Reality of Spinorial Representations and Signature Modulo 8

CHAPTER 4 Pairings and Dimension Modulo 8, over C

CHAPTER 5 Passage to Quadratic Subspaces

CHAPTER 6 The Minkowski Case

REFERENCES

Classical Field Theory

Introduction

CHAPTER 1 Classical Mechanics

§ 1.1. The nonrelativistic particle

§1.2. The relativistic particle

§1.3. Noether's theorem

§1.4. Synthesis

CHAPTER 2 Lagrangian Theory of Classical Fields

§2.1. Dimensional analysis

§2.2. Densities and twisted differential forms

§2.3. Fields and lagrangians

§2.4. First order lagrangians

§2.5. Hamiltonian theory

§2.6. Symmetries and Noether's theorem

§2.7. More on symmetries

§2.8. Computing Noether's current by gauging symmetries

§2.9. The energymomentum tensor

§2.10. Finite energy configurations, classical vacua, and solitons

§2.11. Dimensional reduction

Appendix: Takens' acyclicity theorem

CHAPTER 3 Free Field Theories

§3.1. Coordinates on Minkowski spacetime

§3.2. Real scalar fields

§3.3. Complex scalar fields

§3.4. Spinar fields

§3.5. Abelian gauge fields

CHAPTER 4 Gauge Theory

§4.1. Classical electromagnetism

§4.2. Principal bundles and connections

§4.3. Pure YangMills theory

§4.4. Electric and magnetic charge

CHAPTER 5 σModels and Coupled Gauge Theories

§5.1. Nonlinear σmodels

§5.2. Gauge theory with bosonic matter

CHAPTER 6 Topological Terms

§6.1. Gauge theory

§6.2. WessZuminoWitten terms

§6.3. Smooth Deligne cohomology

CHAPTER 7 Wick Rotation: From Minkowski Space to Euclidean Space

§7.1. Kinetic terms for bosons

§7.2. Potential terms

§7.3. Topological terms and θterms

§7.4. Kinetic terms for fermions

REFERENCES

Supersolutions

Introduction

CHAPTER 1 Preliminary Topics

§ 1. 1. Super Minkowski spaces and super Poincare groups

§1.2. Superfields, component fields, and lagrangians

§1.3. A simple example

CHAPTER 2 Coordinates on Superspace

§2.1. M32 , M44, M6(8,0) and their complexifications

§2.2. Dimensional reduction

§2.3. Coordinates on M32

§2.4. Coordinates on M44

§2.5. Coordinates on M6(8,0)

§2.6. Low dimensions

CHAPTER 3 Supersymmetric σModels

§3.1. Preliminary remarks on linear algebra

§3.2. The free supersymmetric σmodel

§3.3. Nonlinear supersymmetric σmodel

§3.4. Supersymmetric potential terms

§3.5. Superspace construction

§3.6. Dimensional reduction

CHAPTER 4 The Supersymmetric σModel in Dimension 3

§4.1. Fields and supersymmetry transformations on M 32

§4.2. The σmodel action on M32

§4.3. The potential term on M32

§4.4. Analysis of the classical theory

§4.5. Reduction to M2(1,1)

CHAPTER 5 The Supersymmetric σModel in Dimension 4

§5.1. Fields and supersymmetry transformations on M44

§5.2. The σmodel action on M44

§5.3. The superpotential term on M44

§5.4. Analysis of the classical theory

CHAPTER 6 Supersymmetric YangMills Theories

§6.1. The minimal theory in components

§6.2. Gauge theories with matter

§6.3. Superspace construction

CHAPTER 7 N = 1 YangMills Theory in Dimension 3

§7.1. Constrained connections on M32

§7.2. The YangMills action on M32

§7.3. Gauge theory with matter on M32

CHAPTER 8 N = 1 YangMills Theory in Dimension 4

§8.1. Constrained connections on M44

§8.2. The YangMills action on M44

§8.3. Gauge theory with matter on M 44

CHAPTER 9 N=2 YangMills in Dimension 2

§9.1. Dimensional reduction of bosonic YangMills

§9.2. Constrained connections on M2(2,2)

§9.3. The reduced YangMills action

CHAPTER 10 N=1 YangMills in Dimension 6 and N=2 YangMills in Dimension 4

§10.1. Constrained connections on M6(B,O)

§10.2. Reduction to M48

§10.3. More theories on M44 with extended supersymmetry

CHAPTER 11 The Vector Multiplet on M6(8,0)

§11.1. Complements on M6(B,O)

§11.2. Constrained connections

§11.3. An auxiliary Lie algebra

§11.4. Components of constrained connections

REFERENCES

Sign Manifesto Pierre Deligne and Daniel S. Freed

§1. Standard mathematical conventions

§2. Choices

§3. Rationale

§4. Notation

§5. Consequences of §2 on other signs

§6. Differential forms

§ 7. Miscellaneous signs

Part 2 Formal Aspects of QFT

Note on Quantization Pierre Deligne

Introduction to QFT David Kazhdan

Introduction

LECTURE 1 Wightman Axioms

§1.0. Setup and notations

§1.1. Wightman axioms

§1.2. Wightman functions

§1.3. Reconstruction of QFT from Wightman functions

§1.4. Spinstatistics Theorem

§1.5. Mass spectrum of a theory

§1.6. Asymptotics of Wightman functions

LECTURE 2 Euclidean Formulation of Wightman QFT

§2.1. Analytic continuation of Wightman functions

§2.2. Euclidean formulation of Wightman QFT

§2.3. Schwinger functions and measures on the mapspaces

§2.4. PCT Theorem

§2.5. Timeordering

LECTURE 3 Free Field Theories

§3.1. Some examples of free classical field theories

§3.2. Clifford module

§3.3. Examples of free QFT's

§3.4. Free QFT of arbitrary spin

§3.5. Wightman functions of a free field theory; truncated Wightman functions

§3.6. Gaussian measures

§3.7. Normal ordering

LECTURE 4 Scattering Theory

§4.1. Introduction

§4.2. System of n particles (potential scattering)

§4.3. HaagRuelle theory

§4.4. Scattering matrix

LECTURE 5 Feynman Graphs

§5.1. Feynman graph expansion

§5.2. Quasiclassical (lowloop) approximations

§5.3. Effective potential

Perturbative Quantum Field Theory Edward Witten

Introduction

LECTURE 1 Renormalization of Feynman Diagrams

§1.1. Perturbative expansion of a twopoint correlation function

§1.2. The Φ3theory

§1.3. Perturbative expansion of Feynman integrals

§1.4. Computation of a Feynman integral over functions on a Minkowski space

§1.5. Renormalization of divergent graphs

§1.6. Renormalization in higher orders

LECTURE 2 Perturbative Renormalizability of Field Theories

§2.1. Renormalizability of quantum field theories

§2.2. Critical dimensions of some field theories

§2.3. Perturbative renormalization of critical theories

LECTURE 3 Composite Operators and Operator Product Expansion

§3.1. Local functionals in a classical field theory

§3.2. Quantization of local functionals in a free theory

§3.3. Multiplication of composite operators

§3.4. Operator product expansion (OPE) in the free theory

§3.5. Normal ordering and renormalization

§3.6. Composite operators in an interacting critical theory

§3.7. Stability of the classical field equations under quantization

§3.8. Operator product expansion in an interacting theory

LECTURE 4 Scattering Theory

§4.1. Nonrelativistic scattering theory: the asymptotic conditions

§4.2. Relation with experiments

§4.3. The LippmannSchwinger equation

§4.4. The Born approximation

§4.5. Feynman diagrams

§4.6. Relativistic versus nonrelativistic scattering theory: propagation of particles

§4. 7. Relativistic versus nonrelativistic scattering theory: propagation of signals

LECTURE 5 Remarks on Renormalization and Asymptotic Freedom

§5.1. Ambiguity in operator products

§5.2. Symmetry breaking

§5.3. An oversimplified version of experimental confirmation of asymptotic freedom

Index of Dirac Operators Edward Witten

Introduction

LECTURE 1 The Dirac Operator in Finite Dimensions

§1.1. Introduction

§1.2. The Dirac operator on a spin manifold

§1.3. The case of a circle action

§1.4. σmodels in 1+1 dimensions

LECTURE 2 The Dirac Operator on Loop Space

§2.1. Introduction

§2.2. The Lagrangian formulation: σmodels in two dimensions

§2.3. Quantization

§2.4. The index of Q+

§2.5. The computation around the fixed points of the S1action

§2.6. Path integral approach

§2.7. Bundles whose coupled signature or Dirac operator has constant character

§2.8. Generalization to vector bundles over the loop space

Elementary Introduction to Quantum Field Theory Ludwig Faddeev

Introduction

LECTURE 1 Basics of Quantum Mechanics and Canonical Quantization in Hilbert Space

§1.1. Observables and states

§1.2. Dynamics

§ 1.3. Quantization

LECTURE 2 The Harmonic Oscillator and Free Fields

§2.1. The harmonic oscillator

§2.2. Perturbations

§2.3. Quantum field theory

§2.4. Smatrix and Feynman diagrams

LECTURE 3 Comments on Scattering

§3.1. The Smatrix

§3.2. Mass renormalization

§3.3. Charge renormalization

LECTURE 4 Singular Lagrangians

§4.1. Lagrangian and Hamiltonian formalisms

§4.2. Constraints

§4.3. Examples

LECTURE 5 Quantization of YangMills Fields

§5.1. The physical variables

§5.2. Gauge conditions in the functional integral

Renormalization Groups David Gross

Introduction

LECTURE 1 Introduction to Renormalization Groups

§1.1. What is renormalization group?

§1.2. The general scheme of the method of renormalization group

§1.3. Wilsonian scheme for the theory of a scalar field: a mathematical description

§1.4. Applications of renormalization group theory to phase transitions

§1.5. Reminder of renormalization theory

§1.6. Dimensional regularization

LECTURE 2 The Renormalization Group Equation

§2.1. Finite renormalization

§2.2. The dimensional regularization prescription of finite renormalization

§2.3. Scaledependence of finite renormalization prescriptions

§2.4. The renormalization group flow corresponding to a scale dependent renormalization prescription

§2.5. Computation of the renormalization group flow in the 1loop approximation

§2.6. Asymptotic freedom

LECTURE 3 A Closer Look at the Renormalization Group Equation

§3.1. Dynamical patterns of the renormalization group flow

§3.2. Are there any asymptotically free theories without nonabelian gauge fields?

§3.3. Renormalization group equations with many couplings

§3.4. The renormalization group equation for composite operators

§3.5. Anomalous dimension

§3.6. The canonical part of the ßfunction

LECTURE 4 Dynamical Mass Generation and Symmetry Breaking in the GrossNeveu Model

§4.1. Dynamical mass generation

§4.2. The GrossNeveu model

§4.3. The large N limit

LECTURE 5 Wilsonian renormalization group equation

Note on Dimensional Regularization Pavel Etingof

§ 1. The Ddimensional integral

§2. Ddimensional integral with parameters

§3. Ddimensional integral of functions arising from Feynman diagrams

§4. Dimensional regularization of Feynman integrals

§5. Ddimensional Stokes formula

Homework Edward Witten

Introduction

CHAPTER 1 Problems

Problem sets from fall term

Fall exam

Superhomework

Addendum to Superhomework

CHAPTER 2 Solutions to Selected Problems

Solution to Problem FPl

Appendix to FPl: On torsion (by Pierre Deligne)

Solution to Problem FP2

Solution to Problem FP3

Solution to Problem FP4

Solution to Problem FP5

Solution to Problem FP6

Solution to Problem FP7

Solution to Problem FP9

Solution to Problem FP10

Solution to Problem FP11

Solution to Problem FP12

Solution to Problem FP14

Solution to Problem FP15

Solution to Problem FP16

Canonical quantization

Path integral on S1

Solution to Problem FP17

Solution to Problem FP18

Solution to Problem FP19

Solution to Problem FEl

Solution to Problem FE2

Solution to Problem FE3

Solution to Problem FE4

Solution to Problem FE5

Solution to Problem FE6

Solution to Problem FE7

Solution to Problem ASH1

Solution to Problems ASH2 and ASH3

Solution to Problem ASH4

Solution to Problem ASH5

Solution to Problem ASH6

Solution to Problem ASH7

Solution to Problem ASH8

Index

Back Cover


Additional Material

Reviews

An immense amount of valuable material on recent developments. The development of classical supersymmetry by Deligne and collaborators is careful and systematic ... masterful treatment ...the book is a magnificent achievement.
SIAM Review 
A concise introduction to the quantum field theory and perturbative string theory, with as much emphasis on a mathematically satisfying exposition and clarity as possible ... will be helpful to all mathematicians and mathematical physicists who wish to learn about the beautiful subject of quantum field theory.
European Mathematical Society Newsletter


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Ideas from quantum field theory and string theory have had considerable impact on mathematics over the past 20 years. Advances in many different areas have been inspired by insights from physics.
In 1996–97 the Institute for Advanced Study (Princeton, NJ) organized a special yearlong program designed to teach mathematicians the basic physical ideas which underlie the mathematical applications. The purpose is eloquently stated in a letter written by Robert MacPherson: “The goal is to create and convey an understanding, in terms congenial to mathematicians, of some fundamental notions of physics ... [and to] develop the sort of intuition common among physicists for those who are used to thought processes stemming from geometry and algebra.”
These volumes are a written record of the program. They contain notes from several long and many short courses covering various aspects of quantum field theory and perturbative string theory. The courses were given by leading physicists and the notes were written either by the speakers or by mathematicians who participated in the program. The book also includes problems and solutions worked out by the editors and other leading participants. Interspersed are mathematical texts with background material and commentary on some topics covered in the lectures.
These two volumes present the first truly comprehensive introduction to this field aimed at a mathematics audience. They offer a unique opportunity for mathematicians and mathematical physicists to learn about the beautiful and difficult subjects of quantum field theory and string theory.
Graduate students and research mathematicians working in various areas of mathematics related to quantum field theory.

Front Cover

Preface

Brief Contents

CrossReference Codes

Contents

Introduction

Glossary

Part 1 Classical Fields and Supersymmetry

Notes on Supersymmetry (following Joseph Bernstein)

Introduction

CHAPTER 1 Multilinear Algebra

§ 1. 1. The sign rule

§1.2. Categorical approach

§1.3. Examples of the categorical approach

§1.4. Free modules

§1.5. Free commutative algebras

§1.6. The trace

§ 1. 7. Even rules

§1.8. Examples of the "even rules" principle

§1.9. Alternate description of super Lie algebras

§1.10. The Berezinian of an automorphism

§1.11. The Berezinian of a free module

Appendix to §1: Graded super vector spaces

CHAPTER 2 Super Manifolds: Definitions

§§2.12.7. Super manifolds as ringed spaces

2 .1.

2.2 Remarks.

2.3 Examples.

2.5.

2.6 Remarks.

2. 7.

§§2.82.9. The functor of points approach to super manifolds

2.8.

2.9.

§2.10. Super Lie groups

§2.11. Classical series of super Lie groups

CHAPTER 3 Differential Geometry of Super Manifolds

§3.1. Introduction

§3.2. Vector bundles

§3.3. The tangent bundle, the cotangent bundle and the de Rham complex

§3.4. The inverse and implicit function theorems

§3.5. Distributions

§3.6. Connections on vector bundles

§3. 7. Actions of super Lie algebras; vector fields and flows; Lie derivative

§3.8. Super Lie groups and HarishChandra pairs

§3.9. Densities

§3.10. Change of variables formula for densities

§3.11. The Lie derivative of sections of Ber(Ω1M)

§3.12. Integral forms

§3.13. A second definition of integral forms

§3.14. Generalized functions

§3.15. Integral forms as functions of infinitesimal submanifold elements

CHAPTER 4 Real Structures

§§4.14.3. Real structures and *operations

4.1.

4.2.

4.3.

§4.4. Super Hilbert spaces

§4.5. SUSY quantum mechanics

§4.6. Real and complex super manifolds

§4.7. Complexifications, in infinite dimensions

§4.8. cs manifolds

§4.9. Integration on cs manifolds; examples

REFERENCES

Notes on Spinors

Introduction

CHAPTER 1 Overview

CHAPTER 2 Clifford Modules

CHAPTER 3 Reality of Spinorial Representations and Signature Modulo 8

CHAPTER 4 Pairings and Dimension Modulo 8, over C

CHAPTER 5 Passage to Quadratic Subspaces

CHAPTER 6 The Minkowski Case

REFERENCES

Classical Field Theory

Introduction

CHAPTER 1 Classical Mechanics

§ 1.1. The nonrelativistic particle

§1.2. The relativistic particle

§1.3. Noether's theorem

§1.4. Synthesis

CHAPTER 2 Lagrangian Theory of Classical Fields

§2.1. Dimensional analysis

§2.2. Densities and twisted differential forms

§2.3. Fields and lagrangians

§2.4. First order lagrangians

§2.5. Hamiltonian theory

§2.6. Symmetries and Noether's theorem

§2.7. More on symmetries

§2.8. Computing Noether's current by gauging symmetries

§2.9. The energymomentum tensor

§2.10. Finite energy configurations, classical vacua, and solitons

§2.11. Dimensional reduction

Appendix: Takens' acyclicity theorem

CHAPTER 3 Free Field Theories

§3.1. Coordinates on Minkowski spacetime

§3.2. Real scalar fields

§3.3. Complex scalar fields

§3.4. Spinar fields

§3.5. Abelian gauge fields

CHAPTER 4 Gauge Theory

§4.1. Classical electromagnetism

§4.2. Principal bundles and connections

§4.3. Pure YangMills theory

§4.4. Electric and magnetic charge

CHAPTER 5 σModels and Coupled Gauge Theories

§5.1. Nonlinear σmodels

§5.2. Gauge theory with bosonic matter

CHAPTER 6 Topological Terms

§6.1. Gauge theory

§6.2. WessZuminoWitten terms

§6.3. Smooth Deligne cohomology

CHAPTER 7 Wick Rotation: From Minkowski Space to Euclidean Space

§7.1. Kinetic terms for bosons

§7.2. Potential terms

§7.3. Topological terms and θterms

§7.4. Kinetic terms for fermions

REFERENCES

Supersolutions

Introduction

CHAPTER 1 Preliminary Topics

§ 1. 1. Super Minkowski spaces and super Poincare groups

§1.2. Superfields, component fields, and lagrangians

§1.3. A simple example

CHAPTER 2 Coordinates on Superspace

§2.1. M32 , M44, M6(8,0) and their complexifications

§2.2. Dimensional reduction

§2.3. Coordinates on M32

§2.4. Coordinates on M44

§2.5. Coordinates on M6(8,0)

§2.6. Low dimensions

CHAPTER 3 Supersymmetric σModels

§3.1. Preliminary remarks on linear algebra

§3.2. The free supersymmetric σmodel

§3.3. Nonlinear supersymmetric σmodel

§3.4. Supersymmetric potential terms

§3.5. Superspace construction

§3.6. Dimensional reduction

CHAPTER 4 The Supersymmetric σModel in Dimension 3

§4.1. Fields and supersymmetry transformations on M 32

§4.2. The σmodel action on M32

§4.3. The potential term on M32

§4.4. Analysis of the classical theory

§4.5. Reduction to M2(1,1)

CHAPTER 5 The Supersymmetric σModel in Dimension 4

§5.1. Fields and supersymmetry transformations on M44

§5.2. The σmodel action on M44

§5.3. The superpotential term on M44

§5.4. Analysis of the classical theory

CHAPTER 6 Supersymmetric YangMills Theories

§6.1. The minimal theory in components

§6.2. Gauge theories with matter

§6.3. Superspace construction

CHAPTER 7 N = 1 YangMills Theory in Dimension 3

§7.1. Constrained connections on M32

§7.2. The YangMills action on M32

§7.3. Gauge theory with matter on M32

CHAPTER 8 N = 1 YangMills Theory in Dimension 4

§8.1. Constrained connections on M44

§8.2. The YangMills action on M44

§8.3. Gauge theory with matter on M 44

CHAPTER 9 N=2 YangMills in Dimension 2

§9.1. Dimensional reduction of bosonic YangMills

§9.2. Constrained connections on M2(2,2)

§9.3. The reduced YangMills action

CHAPTER 10 N=1 YangMills in Dimension 6 and N=2 YangMills in Dimension 4

§10.1. Constrained connections on M6(B,O)

§10.2. Reduction to M48

§10.3. More theories on M44 with extended supersymmetry

CHAPTER 11 The Vector Multiplet on M6(8,0)

§11.1. Complements on M6(B,O)

§11.2. Constrained connections

§11.3. An auxiliary Lie algebra

§11.4. Components of constrained connections

REFERENCES

Sign Manifesto Pierre Deligne and Daniel S. Freed

§1. Standard mathematical conventions

§2. Choices

§3. Rationale

§4. Notation

§5. Consequences of §2 on other signs

§6. Differential forms

§ 7. Miscellaneous signs

Part 2 Formal Aspects of QFT

Note on Quantization Pierre Deligne

Introduction to QFT David Kazhdan

Introduction

LECTURE 1 Wightman Axioms

§1.0. Setup and notations

§1.1. Wightman axioms

§1.2. Wightman functions

§1.3. Reconstruction of QFT from Wightman functions

§1.4. Spinstatistics Theorem

§1.5. Mass spectrum of a theory

§1.6. Asymptotics of Wightman functions

LECTURE 2 Euclidean Formulation of Wightman QFT

§2.1. Analytic continuation of Wightman functions

§2.2. Euclidean formulation of Wightman QFT

§2.3. Schwinger functions and measures on the mapspaces

§2.4. PCT Theorem

§2.5. Timeordering

LECTURE 3 Free Field Theories

§3.1. Some examples of free classical field theories

§3.2. Clifford module

§3.3. Examples of free QFT's

§3.4. Free QFT of arbitrary spin

§3.5. Wightman functions of a free field theory; truncated Wightman functions

§3.6. Gaussian measures

§3.7. Normal ordering

LECTURE 4 Scattering Theory

§4.1. Introduction

§4.2. System of n particles (potential scattering)

§4.3. HaagRuelle theory

§4.4. Scattering matrix

LECTURE 5 Feynman Graphs

§5.1. Feynman graph expansion

§5.2. Quasiclassical (lowloop) approximations

§5.3. Effective potential

Perturbative Quantum Field Theory Edward Witten

Introduction

LECTURE 1 Renormalization of Feynman Diagrams

§1.1. Perturbative expansion of a twopoint correlation function

§1.2. The Φ3theory

§1.3. Perturbative expansion of Feynman integrals

§1.4. Computation of a Feynman integral over functions on a Minkowski space

§1.5. Renormalization of divergent graphs

§1.6. Renormalization in higher orders

LECTURE 2 Perturbative Renormalizability of Field Theories

§2.1. Renormalizability of quantum field theories

§2.2. Critical dimensions of some field theories

§2.3. Perturbative renormalization of critical theories

LECTURE 3 Composite Operators and Operator Product Expansion

§3.1. Local functionals in a classical field theory

§3.2. Quantization of local functionals in a free theory

§3.3. Multiplication of composite operators

§3.4. Operator product expansion (OPE) in the free theory

§3.5. Normal ordering and renormalization

§3.6. Composite operators in an interacting critical theory

§3.7. Stability of the classical field equations under quantization

§3.8. Operator product expansion in an interacting theory

LECTURE 4 Scattering Theory

§4.1. Nonrelativistic scattering theory: the asymptotic conditions

§4.2. Relation with experiments

§4.3. The LippmannSchwinger equation

§4.4. The Born approximation

§4.5. Feynman diagrams

§4.6. Relativistic versus nonrelativistic scattering theory: propagation of particles

§4. 7. Relativistic versus nonrelativistic scattering theory: propagation of signals

LECTURE 5 Remarks on Renormalization and Asymptotic Freedom

§5.1. Ambiguity in operator products

§5.2. Symmetry breaking

§5.3. An oversimplified version of experimental confirmation of asymptotic freedom

Index of Dirac Operators Edward Witten

Introduction

LECTURE 1 The Dirac Operator in Finite Dimensions

§1.1. Introduction

§1.2. The Dirac operator on a spin manifold

§1.3. The case of a circle action

§1.4. σmodels in 1+1 dimensions

LECTURE 2 The Dirac Operator on Loop Space

§2.1. Introduction

§2.2. The Lagrangian formulation: σmodels in two dimensions

§2.3. Quantization

§2.4. The index of Q+

§2.5. The computation around the fixed points of the S1action

§2.6. Path integral approach

§2.7. Bundles whose coupled signature or Dirac operator has constant character

§2.8. Generalization to vector bundles over the loop space

Elementary Introduction to Quantum Field Theory Ludwig Faddeev

Introduction

LECTURE 1 Basics of Quantum Mechanics and Canonical Quantization in Hilbert Space

§1.1. Observables and states

§1.2. Dynamics

§ 1.3. Quantization

LECTURE 2 The Harmonic Oscillator and Free Fields

§2.1. The harmonic oscillator

§2.2. Perturbations

§2.3. Quantum field theory

§2.4. Smatrix and Feynman diagrams

LECTURE 3 Comments on Scattering

§3.1. The Smatrix

§3.2. Mass renormalization

§3.3. Charge renormalization

LECTURE 4 Singular Lagrangians

§4.1. Lagrangian and Hamiltonian formalisms

§4.2. Constraints

§4.3. Examples

LECTURE 5 Quantization of YangMills Fields

§5.1. The physical variables

§5.2. Gauge conditions in the functional integral

Renormalization Groups David Gross

Introduction

LECTURE 1 Introduction to Renormalization Groups

§1.1. What is renormalization group?

§1.2. The general scheme of the method of renormalization group

§1.3. Wilsonian scheme for the theory of a scalar field: a mathematical description

§1.4. Applications of renormalization group theory to phase transitions

§1.5. Reminder of renormalization theory

§1.6. Dimensional regularization

LECTURE 2 The Renormalization Group Equation

§2.1. Finite renormalization

§2.2. The dimensional regularization prescription of finite renormalization

§2.3. Scaledependence of finite renormalization prescriptions

§2.4. The renormalization group flow corresponding to a scale dependent renormalization prescription

§2.5. Computation of the renormalization group flow in the 1loop approximation

§2.6. Asymptotic freedom

LECTURE 3 A Closer Look at the Renormalization Group Equation

§3.1. Dynamical patterns of the renormalization group flow

§3.2. Are there any asymptotically free theories without nonabelian gauge fields?

§3.3. Renormalization group equations with many couplings

§3.4. The renormalization group equation for composite operators

§3.5. Anomalous dimension

§3.6. The canonical part of the ßfunction

LECTURE 4 Dynamical Mass Generation and Symmetry Breaking in the GrossNeveu Model

§4.1. Dynamical mass generation

§4.2. The GrossNeveu model

§4.3. The large N limit

LECTURE 5 Wilsonian renormalization group equation

Note on Dimensional Regularization Pavel Etingof

§ 1. The Ddimensional integral

§2. Ddimensional integral with parameters

§3. Ddimensional integral of functions arising from Feynman diagrams

§4. Dimensional regularization of Feynman integrals

§5. Ddimensional Stokes formula

Homework Edward Witten

Introduction

CHAPTER 1 Problems

Problem sets from fall term

Fall exam

Superhomework

Addendum to Superhomework

CHAPTER 2 Solutions to Selected Problems

Solution to Problem FPl

Appendix to FPl: On torsion (by Pierre Deligne)

Solution to Problem FP2

Solution to Problem FP3

Solution to Problem FP4

Solution to Problem FP5

Solution to Problem FP6

Solution to Problem FP7

Solution to Problem FP9

Solution to Problem FP10

Solution to Problem FP11

Solution to Problem FP12

Solution to Problem FP14

Solution to Problem FP15

Solution to Problem FP16

Canonical quantization

Path integral on S1

Solution to Problem FP17

Solution to Problem FP18

Solution to Problem FP19

Solution to Problem FEl

Solution to Problem FE2

Solution to Problem FE3

Solution to Problem FE4

Solution to Problem FE5

Solution to Problem FE6

Solution to Problem FE7

Solution to Problem ASH1

Solution to Problems ASH2 and ASH3

Solution to Problem ASH4

Solution to Problem ASH5

Solution to Problem ASH6

Solution to Problem ASH7

Solution to Problem ASH8

Index

Back Cover

An immense amount of valuable material on recent developments. The development of classical supersymmetry by Deligne and collaborators is careful and systematic ... masterful treatment ...the book is a magnificent achievement.
SIAM Review 
A concise introduction to the quantum field theory and perturbative string theory, with as much emphasis on a mathematically satisfying exposition and clarity as possible ... will be helpful to all mathematicians and mathematical physicists who wish to learn about the beautiful subject of quantum field theory.
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