This book offers a concise introduction to quantum field theory and functional integration for students of physics and mathematics. Its aim is to explain mathematical methods developed in the 1970s and 1980s and apply these methods to standard models of quantum field theory. In contrast to other textbooks on quantum field theory, this book treats functional integration as a rigorous mathematical tool. More emphasis is placed on the mathematical framework as opposed to applications to particle physics. It is stressed that the functional integral approach, unlike the operator framework, is suitable for numerical simulations. The book arose from the author's teaching in Wroclaw and preserves the form of his lectures. So some topics are treated as an introduction to the problem rather than a complete solution with all details. Some of the mathematical methods described in the book resulted from the author's own research.
Author(s): Zbigniew Haba
Publisher: Springer
Year: 2023
Language: English
Pages: 235
City: Cham
Tags: Quantum Field Theory, Functional Integral, Feynman Integral, Quantization
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
