Lectures on Quantum Field Theory and Functional Integration

Preface
Contents
1 Notation and Mathematical Preliminaries
	1.1 Generalized Functions (Distributions)
	1.2 Functional Differentiation
	1.3 Gaussian Integration
	1.4 Groups and Their Representations
	1.5 Exercises
2 Quantum Theory of the Scalar Free Field
	2.1 Classical Field Theory. Lagrange Equations and The Noether Theorem
	2.2 Classical Scalar Free Field
	2.3 Quantization of the Scalar Field
	2.4 The Poincare Group and Its Representations
	2.5 Functional Representation of Quantum Fields
	2.6 Exercises
3 Interacting Fields and Scattering Amplitudes
	3.1 Interaction Picture
	3.2 Correlation Functions
	3.3 Gell-Mann-Low Formula
	3.4 The Integral Kernel of an Operator
	3.5 Momentum Representation
	3.6 Coupling Constant Renormalization
	3.7 Euclidean Correlation Functions
	3.8 Dimensional Regularization and Power Counting
	3.9 Generating Functional: A Perturbative Formula
	3.10 The Euclidean Quantum Field Theory: Osterwalder-Schrader Formulation
	3.11 Heisenberg Picture: The Asymptotic Fields
	3.12 Reduction Formulas
	3.13 Exercises
4 Thermal States and Quantum Scalar Field on a Curved Manifold
	4.1 Fields at Finite Temperature
	4.2 Scalar Free Field on a Globally Hyperbolic Manifold
	4.3 Exercises
5 The Functional Integral
	5.1 Trotter Product Formula and The Feynman Integral
	5.2 Evolution for Time-Dependent Hamiltonians
	5.3 The Wiener Integral and Feynman-Wiener Integral
	5.4 The Stochastic Integral: The Feynman Integral for a Particle in an Electromagnetic Field
	5.5 Stochastic Differential Equations
	5.6 Exercises
6 Feynman Integral in Terms of the Wiener Integral
	6.1 Feynman-Wiener Integral for Polynomial Potentials
	6.2 Feynman-Wiener Integral for Potentials Which are Fourier-Laplace …
	6.3 Functional Integration in Terms of Oscillatory Paths in QFT
	6.4 Feynman-Wiener Integration in QFT in Two Dimensions
	6.5 Exercises
7 Application of the Feynman Integral for Approximate Calculations
	7.1 Semi-classical Expansion: The Stationary Phase Method
	7.2 Stationary Phase for an Anharmonic Oscillator
	7.3 The Loop Expansion in QFT
	7.4 The Saddle Point Method: The Loop Expansion in Euclidean Field Theory
	7.5 Effective Action
	7.6 Determinants of Differential Operators
	7.7 The Functional Integral for Euclidean Fields at Finite Temperature
	7.8 Exercises
8 Feynman Path Integral in Terms  of Expanding Paths
	8.1 Expansion Around a Particular Solution
	8.2 The Upside-Down Oscillator
	8.3 Solution in the Heisenberg Picture
	8.4 Quantum Mechanics at an Imaginary Time
	8.5 Paths at Imaginary Time as Euclidean Fields
	8.6 Free Field on a Static Manifold
	8.7 Time-Dependent Gaussian State in Quantum Field Theory
	8.8 Free Field in an Expanding Universe
	8.9 Free Field in De Sitter Space
	8.10 Interference of Classical and Quantum Waves
	8.11 Exercises
9 An Interaction with a Quantum Electromagnetic Field
	9.1 Functional Integral Quantization of the Electromagnetic Field
	9.2 The Abelian Higgs Model
	9.3 Euclidean Version: The Polymer Representation
	9.4 One-Loop Determinant: A Non-Perturbative Method
	9.5 Non-relativistic QED: A Charged Particle Interacting …
	9.6 Heisenberg Equations of Motion in QED Environment
	9.7 Noise in the Squeezed State
	9.8 Feynman Formula in QED with an Axion
	9.9 Decoherence in an Environment of Photons
	9.10 Entropy of Gaussian Wigner States
	9.11 Exercises
10 Particle Interaction with Gravitons
	10.1 Classical Gravity
	10.2 Quantum Geodesic Deviation
	10.3 Heisenberg Equations
	10.4 Stochastic Motion in the Thermal Environment
	10.5 Exercises
11 Quantization of Non-Abelian Gauge Fields
	11.1 Non-Abelian Gauge Theories
	11.2 The Non-Abelian Higgs Model: Symmetry Breaking and Mass Generation
	11.3 The Effective Scalar Field Action in Non-Abelian Gauge Field
	11.4 Fadeev-Popov Procedure
	11.5 The Background Field Method
	11.6 The Effective Action in Non-Abelian Gauge Theories
	11.7 Exercises
12 Lattice Approximation
	12.1 Lattice Approximation in Euclidean Scalar Field Theory
	12.2 Polymer Representation
	12.3 Lattice Approximation in Gauge Theories
	12.4 Exercises
Appendix  References
Index