What is a Quantum Field Theory? — A First Introduction for Mathematicians

Introduction
Part I Basics
1 Preliminaries
1.1 Dimension
1.2 Notation
1.3 Distributions
1.4 The Delta Function
1.5 The Fourier Transform
2 Basics of Non-relativistic Quantum Mechanics
2.1 Basic Setting
2.2 Measuring Two Different Observables on the Same System
2.3 Uncertainty
2.4 Finite versus Continuous Models
2.5 Position State Space for a Particle
2.6 Unitary Operators
2.7 Momentum State Space for a Particle
2.8 Dirac’s Formalism
2.9 Why Are Unitary Transformations Ubiquitous?
2.10 Unitary Representations of Groups
2.11 Projective versus True Unitary Representations
2.12 Mathematicians Look at Projective Representations
2.13 Projective Representations of R
2.14 One-parameter Unitary Groups and Stone’s Theorem
2.15 Time-evolution
2.16 Schrödinger and Heisenberg Pictures
2.17 A First Contact with Creation and Annihilation Operators
2.18 The Harmonic Oscillator
3 Non-relativistic Quantum Fields
3.1 Tensor Products
3.2 Symmetric Tensors
3.3 Creation and Annihilation Operators
3.4 Boson Fock Space
3.5 Unitary Evolution in the Boson Fock Space
3.6 Boson Fock Space and Collections of Harmonic Oscillators
3.7 Explicit Formulas: Position Space
3.8 Explicit Formulas: Momentum Space
3.9 Universe in a Box
3.10 Quantum Fields: Quantizing Spaces of Functions
4 The Lorentz Group and the Poincaré Group
4.1 Notation and Basics
4.2 Rotations
4.3 Pure Boosts
4.4 The Mass Shell and Its Invariant Measure
4.5 More about Unitary Representations
4.6 Group Actions and Representations
4.7 Quantum Mechanics, Special Relativity and the Poincaré Group
4.8 A Fundamental Representation of the Poincaré Group
4.9 Particles and Representations
4.10 The States | p and | p
4.11 The Physicists’ Way
5 The Massive Scalar Free Field
5.1 Intrinsic Definition
5.2 Explicit Formulas
5.3 Time-evolution
5.4 Lorentz Invariant Formulas
6 Quantization
6.1 The Klein-Gordon Equation
6.2 Naive Quantization of the Klein-Gordon Field
6.3 Road Map
6.4 Lagrangian Mechanics
6.5 From Lagrangian Mechanics to Hamiltonian Mechanics
6.6 Canonical Quantization and Quadratic Potentials
6.7 Quantization through the Hamiltonian
6.8 Ultraviolet Divergences
6.9 Quantization through Equal-time Commutation Relations
6.10 Caveat
6.11 Hamiltonian
7 The Casimir Effect
7.1 Vacuum Energy
7.2 Regularization
Part II Spin
8 Representations of the Orthogonal and the Lorentz Group
8.1 The Groups SU(2) and SL(2, C )
8.2 A Fundamental Family of Representations of SU(2)
8.3 Tensor Products of Representations
8.4 SL(2, C ) as a Universal Cover of the Lorentz Group
8.5 An Intrinsically Projective Representation
8.6 Deprojectivization
8.7 A Brief Introduction to Spin
8.8 Spin as an Observable
8.9 Parity and the Double Cover SL+ (2, C ) of O+ (1, 3)
8.10 The Parity Operator and the Dirac Matrices
9 Representations of the Poincaré Group
9.1 The Physicists’ Way
9.2 The Group P ∗
9.3 Road Map
9.3.1 How to Construct Representations?
9.3.2 Surviving the Formulas
9.3.3 Classifying the Representations
9.3.4 Massive Particles
9.3.5 Massless Particles
9.3.6 Massless Particles and Parity
9.4 Elementary Construction of Induced Representations
9.5 Variegated Formulas
9.6 Fundamental Representations
9.6.1 Massive Particles
9.6.2 Massless Particles
9.7 Particles, Spin, Representations
9.8 Abstract Presentation of Induced Representations
9.9 Particles and Parity
9.10 Dirac Equation
9.11 History of the Dirac Equation
9.12 Parity and Massless Particles
9.13 Photons
10 Basic Free Fields
10.1 Charged Particles and Anti-particles
10.2 Lorentz Covariant Families of Fields
10.3 Road Map I
10.4 Form of the Annihilation Part of the Fields
10.5 Explicit Formulas
10.6 Creation Part of the Fields
10.7 Microcausality
10.8 Road Map II
10.9 The Simplest Case (N = 1)
10.10 A Very Simple Case (N = 4)
10.11 The Massive Vector Field (N = 4)
10.12 The Classical Massive Vector Field
10.13 Massive Weyl Spinors, First Attempt (N = 2)
10.14 Fermion Fock Space
10.15 Massive Weyl Spinors, Second Attempt
10.16 Equation of Motion for the Massive Weyl Spinor
10.17 Massless Weyl Spinors
10.18 Parity
10.19 Dirac Field
10.20 Dirac Field and Classical Mechanics
10.21 Majorana Field
10.22 Lack of a Suitable Field for Photons
Part III Interactions
11 Perturbation Theory
11.1 Time-independent Perturbation Theory
11.2 Time-dependent Perturbation Theory and the Interaction Picture
11.3 Transition Rates
11.4 A Side Story: Oscillating Interactions
11.5 Interaction of a Particle with a Field: A Toy Model
12 Scattering, the Scattering Matrix and Cross-Sections
12.1 Heuristics in a Simple Case of Classical Mechanics
12.2 Non-relativistic Quantum Scattering by a Potential
12.3 The Scattering Matrix in Non-relativistic Quantum Scattering
12.4 The Scattering Matrix and Cross-Sections, I
12.5 Scattering Matrix in Quantum Field Theory
12.6 Scattering Matrix and Cross-Sections, II
13 The Scattering Matrix in Perturbation Theory
13.1 The Scattering Matrix and the Dyson Series
13.2 Prologue: The Born Approximation in Scattering by a Potential
13.3 Interaction Terms in Hamiltonians
13.4 Prickliness of the Interaction Picture
13.5 Admissible Hamiltonian Densities
13.6 Simple Models for Interacting Particles
13.7 A Computation at the First Order
13.8 Wick’s Theorem
13.9 Interlude: Summing the Dyson Series
13.10 The Feynman Propagator
13.11 Redefining the Incoming and Outgoing States
13.12 A Computation at Order Two with Trees
13.13 Feynman Diagrams and Symmetry Factors
13.14 The φ 4 Model
13.15 A Closer Look at Symmetry Factors
13.16 A Computation at Order Two with One Loop
13.17 One Loop: A Simple Case of Renormalization
13.18 Wick Rotation and Feynman Parameters
13.19 Explicit Formulas
13.20 Counter-terms, I
13.21 Two Loops: Toward the Central Issues
13.22 Analysis of Diagrams
13.23 Cancellation of Infinities
13.24 Counter-terms, II
14 Interacting Quantum Fields
14.1 Interacting Quantum Fields and Particles
14.2 Road Map I
14.3 The Gell-Mann− Low Formula and Theorem
14.4 Adiabatic Switching of the Interaction
14.5 Diagrammatic Interpretation of the Gell-Mann−Low Theorem
14.6 Road Map II
14.7 Green Functions and S -matrix
14.8 The Dressed Propagator in the Källén–Lehmann Representation
14.9 Diagrammatic Computation of the Dressed Propagator
14.10 Mass Renormalization
14.11 Difficult Reconciliation
14.12 Field Renormalization
14.13 Putting It All Together
14.14 Conclusions
Part IV Renormalization
15 Prologue: Power Counting
15.1 What Is Power Counting?
15.2 Weinberg’s Power Counting Theorem
15.3 The Fundamental Space kerL
15.4 Power Counting in Feynman Diagrams
15.5 Proof of Theorem 15.3.1
15.6 A Side Story: Loops
15.7 Parameterization of Diagram Integrals
15.8 Parameterization of Diagram Integrals by Loops
16 The Bogoliubov–Parasiuk–Hepp–Zimmermann Scheme
16.1 Overall Approach
16.2 Simple Examples
16.3 Canonical Flow and the Taylor Operation
16.4 Subdiagrams
16.5 Forests
16.6 Renormalizing the Integrand: The Forest Formula
16.7 Diagrams That Need Not Be 1-PI
16.8 Interpretation
16.9 Specificity of the Parameterization
17 Counter-terms
17.1 What Is the Counter-term Method?
17.2 A Very Simple Case: Coupling Constant Renormalization
17.3 Mass and Field Renormalization: Diagrammatics
17.4 The BPHZ Renormalization Prescription
17.5 Cancelling Divergences with Counter-terms
17.6 Determining the Counter-terms from BPHZ
17.7 From BPHZ to the Counter-term Method
17.8 What Happened to Subdiagrams?
17.9 Field Renormalization, II
18 Controlling Singularities
18.1 Basic Principle
18.2 Zimmermann’s Theorem
18.3 Proof of Proposition 18.2.12
18.4 A Side Story: Feynman Diagrams and Wick Rotations
19 Proof of Convergence of the BPHZ Scheme
19.1 Proof of Theorem 16.1.1
19.2 Simple Facts
19.3 Grouping the Terms
19.4 Bringing Forward Cancellation
19.5 Regular Rational Functions
19.6 Controlling the Degree
Part V Complements
Appendix A Complements on Representations
A.1 Projective Unitary Representations of R
A.2 Continuous Projective Unitary Representations
A.3 Projective Finite-dimensional Representations
A.4 Induced Representations for Finite Groups
A.5 Representations of Finite Semidirect Products
A.6 Representations of Compact Groups
Appendix B End of Proof of Stone’s Theorem
Appendix C Canonical Commutation Relations
C.1 First Manipulations
C.2 Coherent States for the Harmonic Oscillator
C.3 The Stone–von Neumann Theorem
C.4 Non-equivalent Unitary Representations
C.5 Orthogonal Ground States!
Appendix D A Crash Course on Lie Algebras
D.1 Basic Properties and so(3)
D.2 Group Representations and Lie Algebra Representations
D.3 Angular Momentum
D.4 su(2) = so(3)!
D.5 From Lie Algebra Homomorphisms to Lie Group Homomorphisms
D.6 Irreducible Representations of SU(2)
D.7 Decomposition of Unitary Representations of SU(2) into Irreducibles
D.8 Spherical Harmonics
D.9 so(1, 3) = slC (2)!
D.10 Irreducible Representations of SL(2, C )
D.11 QFT Is Not for the Meek
D.12 Some Tensor Representations of SO↑(1,3
Appendix E Special Relativity
E.1 Energy–Momentum
E.2 Electromagnetism
Appendix F Does a Position Operator Exist?
Appendix G More on the Representations of the Poincaré Group
G.1 A Fun Formula
G.2 Higher Spin: Bargmann–Wigner and Rarita–Schwinger
Appendix H Hamiltonian Formalism for Classical Fields
H.1 Hamiltonian for the Massive Vector Field
H.2 From Hamiltonians to Lagrangians
H.3 Functional Derivatives
H.4 Two Examples
H.5 Poisson Brackets
Appendix I Quantization of the Electromagnetic Field through the Gupta–Bleuler Approach
Appendix J Lippmann–Schwinger Equations and Scattering States
Appendix K Functions on Surfaces and Distributions
Appendix L What Is a Tempered Distribution Really?
L.1 Test Functions
L.2 Tempered Distributions
L.3 Adding and Removing Variables
L.4 Fourier Transforms of Distributions
Appendix M Wightman Axioms and Haag’s Theorem
M.1 The Wightman Axioms
M.2 Statement of Haag’s Theorem
M.3 Easy Steps
M.4 Wightman Functions
Appendix N Feynman Propagator and Klein-Gordon Equation
N.1 Contour Integrals
N.2 Fundamental Solutions of Differential Equations
Appendix O Yukawa Potential
Appendix P Principal Values and Delta Functions
Solutions to Selected Exercises
Reading Suggestions
References
Index