[Paper Review] Path Integrals in Quantum Physics

L. Feld|arXiv (Cornell University)|Sep 6, 2012

Particle physics theoretical and experimental studies65 references3 citations

TL;DR

This paper provides a comprehensive, accessible introduction to path integral methods in quantum physics, covering non-relativistic quantum mechanics, many-body systems, and quantum field theory. It presents heuristic derivations, numerical implementations via Fortran codes, and solutions to problems, making path integrals approachable for graduate students across diverse physics disciplines.

ABSTRACT

These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, many-body physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions, quantization of gauge theories etc.) as well as specialized topics (scattering, dissipative systems, spin \& color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions.

Motivation & Objective

• To equip graduate students with practical working knowledge of path integral methods across quantum physics domains.
• To present path integrals in a heuristic, non-mathematical manner suitable for immediate application to physical problems.
• To bridge theoretical concepts with numerical computation through embedded Fortran programs for Monte Carlo and variational evaluation.
• To include detailed problem sets with solutions to reinforce learning and application.
• To extend coverage to specialized topics such as gauge theories, dissipative systems, spin, and lattice methods.

Proposed method

• Uses heuristic, intuitive derivations of path integrals without requiring advanced mathematical formalism.
• Applies path integrals to standard systems like quadratic Lagrangians, tunneling, and harmonic oscillators.
• Introduces numerical evaluation of Euclidean path integrals using Monte Carlo and variational methods via simple Fortran codes.
• Covers quantization of gauge theories, bosons, fermions, and spin/color degrees of freedom in the path integral framework.
• Employs lattice methods to discretize path integrals for numerical computation.
• Incorporates problem sets with detailed solutions to illustrate methodological application.

Experimental results

Research questions

• RQ1 How can path integral methods be systematically introduced to graduate students without requiring advanced mathematical rigor?
• RQ2 What are the key numerical techniques for evaluating path integrals in quantum systems, and how can they be implemented on a PC?
• RQ3 How do path integrals describe non-perturbative phenomena such as tunneling and dissipative systems?
• RQ4 How can fermionic and bosonic degrees of freedom be consistently incorporated into the path integral formalism?
• RQ5 What is the role of gauge symmetry and how is it preserved in path integral quantization?

Key findings

• Path integrals can be effectively taught and applied in a heuristic, non-technical way to a broad range of quantum physics problems.
• Numerical evaluation of path integrals using Monte Carlo and variational methods is feasible and instructive, as demonstrated by embedded Fortran programs.
• The formalism successfully describes tunneling, dissipative systems, and systems with spin and color degrees of freedom.
• Gauge theories can be consistently quantized using path integrals, with proper treatment of constraints and symmetry.
• The inclusion of problem sets with solutions enhances pedagogical utility and reinforces conceptual and computational understanding.
• The final English version (v4) includes a complete set of solutions, significantly increasing its value for self-study and classroom use.