Described here is Feynman's path integral approach to quantum mechanics and quantum field theory from a functional integral point of view. Therein lies the main focus of Euclidean field theory. The notion of Gaussian measure and the construction of the Wiener measure are covered. As well, the notion of classical mechanics and the Schrödinger picture of quantum mechanics are recalled. There, the equivalence to the path integral formalism is shown by deriving the quantum mechanical propagator from it. Additionally, an introduction to elements of constructive quantum field theory is provided for readers.
Author(s): Nima Moshayedi
Series: Springer Briefs in Physics
Edition: 1
Publisher: Springer Nature Singapore
Year: 2023
Language: English
Pages: 118
Tags: Quantum Field Theory, Feynman Path Integrals
Preface
Contents
1 Introduction
2 A Brief Recap of Classical Mechanics
2.1 Newtonian Mechanics
2.1.1 Conservation of Energy
2.2 Hamiltonian Mechanics
2.2.1 The General Formulation
2.2.2 The Poisson Bracket
2.3 Lagrangian Mechanics
2.3.1 Lagrangian System
2.3.2 Hamilton's Least Action Principle
2.4 The Legendre Transform
3 The Schrödinger Picture of Quantum Mechanics
3.1 Postulates of Quantum Mechanics
3.1.1 First Postulate
3.1.2 Second Postulate
3.1.3 Third Postulate
3.1.4 Summary of Classical and Quantum Mechanics
3.2 Elements of Functional Analysis
3.2.1 Bounded Operators
3.2.2 Unbounded Operators
3.2.3 Adjoint of an Unbounded Operator
3.2.4 Sobolev Spaces
3.3 Quantization of a Classical System
3.3.1 Definition
3.3.2 Eigenvalues of a Single Harmonic Oscillator
3.3.3 Weyl Quantization on mathbbR2n
3.4 Schrödinger Equations, Fourier Transforms and Propagators
3.4.1 Solving the Schrödinger Equation
3.4.2 The Schrödinger Equation for the Free Particle Moving on mathbbR
3.4.3 Solving the Schrödinger Equation with Fourier Transform
4 The Path Integral Approach to Quantum Mechanics
4.1 Feynman's Formulation of the Path Integral
4.1.1 The Free Propagator for the Free Particle on mathbbR
4.2 Construction of the Wiener Measure
4.2.1 Towards Nowhere Differentiability of Brownian Paths
4.2.2 The Feynman–Kac Formula
4.3 Gaussian Measures
4.3.1 Gaussian Measures on mathbbR
4.3.2 Gaussian Measures on Finite-Dimensional Vector Spaces
4.3.3 Gaussian Measures on Real Separable Hilbert Spaces
4.3.4 Standard Gaussian Measure on mathcalH
4.4 Wick Ordering
4.4.1 Motivating Example and Construction
4.4.2 Wick Ordering as a Value of Feynman Diagrams
4.4.3 Abstract Point of View on Wick Ordering
4.5 Bosonic Fock Spaces
5 Construction of Quantum Field Theories
5.1 Free Scalar Field Theory
5.1.1 Locally Convex Spaces
5.1.2 Dual of a Locally Convex Space
5.1.3 Gaussian Measures on the Dual of Fréchet Spaces
5.1.4 The Operator (Δ+m2)-1
5.2 Construction of Self-Interacting Theories
5.2.1 More Random Variables
5.2.2 Generalized Feynman Diagrams
5.2.3 Theories with Exponential Interaction
5.2.4 The Osterwalder–Schrader Axioms
5.3 QFT as an Operator-Valued Distribution
5.3.1 Relativistic Quantum Mechanics
5.3.2 Garding–Wightman Formulation of QFT
5.4 Outlook
5.4.1 Generalization and Gauge Theories
5.4.2 TQFTs and the Functorial Approach
Appendix Bibliography
