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Quantum Fields and Strings: A Course for Mathematicians: Volume 1
 
Quantum Fields and Strings: A Course for Mathematicians
Softcover ISBN:  978-0-8218-2012-4
Product Code:  QFT/1.S
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Quantum Fields and Strings: A Course for Mathematicians
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Quantum Fields and Strings: A Course for Mathematicians: Volume 1
Softcover ISBN:  978-0-8218-2012-4
Product Code:  QFT/1.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $55.20
eBook ISBN:  978-1-4704-7390-7
Product Code:  QFT/1.E
List Price: $48.00
MAA Member Price: $43.20
AMS Member Price: $38.40
Softcover ISBN:  978-0-8218-2012-4
eBook ISBN:  978-1-4704-7390-7
Product Code:  QFT/1.S.B
List Price: $117.00 $93.00
MAA Member Price: $105.30 $83.70
AMS Member Price: $93.60 $74.40
  • Book Details
     
     
    1999; 723 pp
    MSC: 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 year-long 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.

    Readership

    Graduate 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
    • Cross-Reference 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
    • §§2.1-2.7. Super manifolds as ringed spaces
    • 2 .1.
    • 2.2 Remarks.
    • 2.3 Examples.
    • 2.5.
    • 2.6 Remarks.
    • 2. 7.
    • §§2.8-2.9. 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 Harish-Chandra 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
    • §§4.1-4.3. 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 energy-momentum 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 Yang-Mills theory
    • §4.4. Electric and magnetic charge
    • CHAPTER 5 σ-Models and Coupled Gauge Theories
    • §5.1. Nonlinear σ-models
    • §5.2. Gauge theory with bosonic matter
    • CHAPTER 6 Topological Terms
    • §6.1. Gauge theory
    • §6.2. Wess-Zumino-Witten 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 θ-terms
    • §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. M3|2 , M4|4, M6|(8,0) and their complexifications
    • §2.2. Dimensional reduction
    • §2.3. Coordinates on M3|2
    • §2.4. Coordinates on M4|4
    • §2.5. Coordinates on M6|(8,0)
    • §2.6. Low dimensions
    • CHAPTER 3 Supersymmetric σ-Models
    • §3.1. Preliminary remarks on linear algebra
    • §3.2. The free supersymmetric σ-model
    • §3.3. Nonlinear supersymmetric σ-model
    • §3.4. Supersymmetric potential terms
    • §3.5. Superspace construction
    • §3.6. Dimensional reduction
    • CHAPTER 4 The Supersymmetric σ-Model in Dimension 3
    • §4.1. Fields and supersymmetry transformations on M 3|2
    • §4.2. The σ-model action on M3|2
    • §4.3. The potential term on M3|2
    • §4.4. Analysis of the classical theory
    • §4.5. Reduction to M2|(1,1)
    • CHAPTER 5 The Supersymmetric σ-Model in Dimension 4
    • §5.1. Fields and supersymmetry transformations on M4|4
    • §5.2. The σ-model action on M4|4
    • §5.3. The superpotential term on M4|4
    • §5.4. Analysis of the classical theory
    • CHAPTER 6 Supersymmetric Yang-Mills Theories
    • §6.1. The minimal theory in components
    • §6.2. Gauge theories with matter
    • §6.3. Superspace construction
    • CHAPTER 7 N = 1 Yang-Mills Theory in Dimension 3
    • §7.1. Constrained connections on M3|2
    • §7.2. The Yang-Mills action on M3|2
    • §7.3. Gauge theory with matter on M3|2
    • CHAPTER 8 N = 1 Yang-Mills Theory in Dimension 4
    • §8.1. Constrained connections on M4|4
    • §8.2. The Yang-Mills action on M4|4
    • §8.3. Gauge theory with matter on M 4|4
    • CHAPTER 9 N=2 Yang-Mills in Dimension 2
    • §9.1. Dimensional reduction of bosonic Yang-Mills
    • §9.2. Constrained connections on M2|(2,2)
    • §9.3. The reduced Yang-Mills action
    • CHAPTER 10 N=1 Yang-Mills in Dimension 6 and N=2 Yang-Mills in Dimension 4
    • §10.1. Constrained connections on M6|(B,O)
    • §10.2. Reduction to M4|8
    • §10.3. More theories on M4|4 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. Spin-statistics 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 map-spaces
    • §2.4. PCT Theorem
    • §2.5. Time-ordering
    • 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. Haag-Ruelle theory
    • §4.4. Scattering matrix
    • LECTURE 5 Feynman Graphs
    • §5.1. Feynman graph expansion
    • §5.2. Quasi-classical (low-loop) approximations
    • §5.3. Effective potential
    • Perturbative Quantum Field Theory Edward Witten
    • Introduction
    • LECTURE 1 Renormalization of Feynman Diagrams
    • §1.1. Perturbative expansion of a two-point correlation function
    • §1.2. The Φ3-theory
    • §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 Lippmann-Schwinger equation
    • §4.4. The Born approximation
    • §4.5. Feynman diagrams
    • §4.6. Relativistic versus non-relativistic scattering theory: propagation of particles
    • §4. 7. Relativistic versus non-relativistic 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. σ-models in 1+1 dimensions
    • LECTURE 2 The Dirac Operator on Loop Space
    • §2.1. Introduction
    • §2.2. The Lagrangian formulation: σ-models in two dimensions
    • §2.3. Quantization
    • §2.4. The index of Q+
    • §2.5. The computation around the fixed points of the S1-action
    • §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. S-matrix and Feynman diagrams
    • LECTURE 3 Comments on Scattering
    • §3.1. The S-matrix
    • §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 Yang-Mills 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. Scale-dependence 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 1-loop 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 ß-function
    • LECTURE 4 Dynamical Mass Generation and Symmetry Breaking in the Gross-Neveu Model
    • §4.1. Dynamical mass generation
    • §4.2. The Gross-Neveu model
    • §4.3. The large N limit
    • LECTURE 5 Wilsonian renormalization group equation
    • Note on Dimensional Regularization Pavel Etingof
    • § 1. The D-dimensional integral
    • §2. D-dimensional integral with parameters
    • §3. D-dimensional integral of functions arising from Feynman diagrams
    • §4. Dimensional regularization of Feynman integrals
    • §5. D-dimensional 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
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1999; 723 pp
MSC: 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 year-long 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.

Readership

Graduate 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:
  • Front Cover
  • Preface
  • Brief Contents
  • Cross-Reference 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
  • §§2.1-2.7. Super manifolds as ringed spaces
  • 2 .1.
  • 2.2 Remarks.
  • 2.3 Examples.
  • 2.5.
  • 2.6 Remarks.
  • 2. 7.
  • §§2.8-2.9. 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 Harish-Chandra 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
  • §§4.1-4.3. 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 energy-momentum 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 Yang-Mills theory
  • §4.4. Electric and magnetic charge
  • CHAPTER 5 σ-Models and Coupled Gauge Theories
  • §5.1. Nonlinear σ-models
  • §5.2. Gauge theory with bosonic matter
  • CHAPTER 6 Topological Terms
  • §6.1. Gauge theory
  • §6.2. Wess-Zumino-Witten 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 θ-terms
  • §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. M3|2 , M4|4, M6|(8,0) and their complexifications
  • §2.2. Dimensional reduction
  • §2.3. Coordinates on M3|2
  • §2.4. Coordinates on M4|4
  • §2.5. Coordinates on M6|(8,0)
  • §2.6. Low dimensions
  • CHAPTER 3 Supersymmetric σ-Models
  • §3.1. Preliminary remarks on linear algebra
  • §3.2. The free supersymmetric σ-model
  • §3.3. Nonlinear supersymmetric σ-model
  • §3.4. Supersymmetric potential terms
  • §3.5. Superspace construction
  • §3.6. Dimensional reduction
  • CHAPTER 4 The Supersymmetric σ-Model in Dimension 3
  • §4.1. Fields and supersymmetry transformations on M 3|2
  • §4.2. The σ-model action on M3|2
  • §4.3. The potential term on M3|2
  • §4.4. Analysis of the classical theory
  • §4.5. Reduction to M2|(1,1)
  • CHAPTER 5 The Supersymmetric σ-Model in Dimension 4
  • §5.1. Fields and supersymmetry transformations on M4|4
  • §5.2. The σ-model action on M4|4
  • §5.3. The superpotential term on M4|4
  • §5.4. Analysis of the classical theory
  • CHAPTER 6 Supersymmetric Yang-Mills Theories
  • §6.1. The minimal theory in components
  • §6.2. Gauge theories with matter
  • §6.3. Superspace construction
  • CHAPTER 7 N = 1 Yang-Mills Theory in Dimension 3
  • §7.1. Constrained connections on M3|2
  • §7.2. The Yang-Mills action on M3|2
  • §7.3. Gauge theory with matter on M3|2
  • CHAPTER 8 N = 1 Yang-Mills Theory in Dimension 4
  • §8.1. Constrained connections on M4|4
  • §8.2. The Yang-Mills action on M4|4
  • §8.3. Gauge theory with matter on M 4|4
  • CHAPTER 9 N=2 Yang-Mills in Dimension 2
  • §9.1. Dimensional reduction of bosonic Yang-Mills
  • §9.2. Constrained connections on M2|(2,2)
  • §9.3. The reduced Yang-Mills action
  • CHAPTER 10 N=1 Yang-Mills in Dimension 6 and N=2 Yang-Mills in Dimension 4
  • §10.1. Constrained connections on M6|(B,O)
  • §10.2. Reduction to M4|8
  • §10.3. More theories on M4|4 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. Spin-statistics 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 map-spaces
  • §2.4. PCT Theorem
  • §2.5. Time-ordering
  • 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. Haag-Ruelle theory
  • §4.4. Scattering matrix
  • LECTURE 5 Feynman Graphs
  • §5.1. Feynman graph expansion
  • §5.2. Quasi-classical (low-loop) approximations
  • §5.3. Effective potential
  • Perturbative Quantum Field Theory Edward Witten
  • Introduction
  • LECTURE 1 Renormalization of Feynman Diagrams
  • §1.1. Perturbative expansion of a two-point correlation function
  • §1.2. The Φ3-theory
  • §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 Lippmann-Schwinger equation
  • §4.4. The Born approximation
  • §4.5. Feynman diagrams
  • §4.6. Relativistic versus non-relativistic scattering theory: propagation of particles
  • §4. 7. Relativistic versus non-relativistic 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. σ-models in 1+1 dimensions
  • LECTURE 2 The Dirac Operator on Loop Space
  • §2.1. Introduction
  • §2.2. The Lagrangian formulation: σ-models in two dimensions
  • §2.3. Quantization
  • §2.4. The index of Q+
  • §2.5. The computation around the fixed points of the S1-action
  • §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. S-matrix and Feynman diagrams
  • LECTURE 3 Comments on Scattering
  • §3.1. The S-matrix
  • §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 Yang-Mills 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. Scale-dependence 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 1-loop 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 ß-function
  • LECTURE 4 Dynamical Mass Generation and Symmetry Breaking in the Gross-Neveu Model
  • §4.1. Dynamical mass generation
  • §4.2. The Gross-Neveu model
  • §4.3. The large N limit
  • LECTURE 5 Wilsonian renormalization group equation
  • Note on Dimensional Regularization Pavel Etingof
  • § 1. The D-dimensional integral
  • §2. D-dimensional integral with parameters
  • §3. D-dimensional integral of functions arising from Feynman diagrams
  • §4. Dimensional regularization of Feynman integrals
  • §5. D-dimensional 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.

    European Mathematical Society Newsletter
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