Chapter 31 Further Reading

Primary Textbook References

Sakurai & Napolitano, Modern Quantum Mechanics (3rd ed., 2021)

  • Section 2.6: Propagators and Feynman path integrals. Sakurai gives a concise and physically motivated derivation of the path integral, starting from the propagator and working through the free-particle case. His treatment of the composition property is particularly clear.
  • Section 2.7: The free-particle propagator derived both from the path integral and from the spectral decomposition. A good consistency check for students.

Shankar, Principles of Quantum Mechanics (2nd ed., 1994)

  • Chapter 8: "The Path Integral Formulation of Quantum Theory." Shankar devotes an entire chapter to the path integral, including careful treatments of the free particle, the QHO, and the connection to statistical mechanics. His discussion of the mathematical subtleties is unusually honest for a physics textbook. This is probably the best path integral chapter in any standard graduate textbook.
  • Chapter 21: "Path Integrals: Part II." Covers the Euclidean path integral, instantons, and the connection to quantum field theory. More advanced but well motivated.

Griffiths & Schroeter, Introduction to Quantum Mechanics (3rd ed., 2018)

  • Afterword / Appendix: Griffiths includes a brief but clear discussion of the path integral in the supplementary material. It is not a complete treatment but provides a good first exposure for students who have mastered the Schrödinger formulation.

Specialized References

Feynman's Original Papers and Books

  • Feynman, R. P., "Space-Time Approach to Non-Relativistic Quantum Mechanics," Reviews of Modern Physics 20, 367 (1948): The founding paper. Feynman's writing is characteristically clear and physical. He derives the path integral, computes the free-particle propagator, discusses the classical limit, and sketches the connection to quantum electrodynamics. Essential reading for anyone serious about understanding path integrals. Remarkably accessible — an advanced undergraduate can follow most of the paper.

  • Feynman, R. P. & Hibbs, A. R., Quantum Mechanics and Path Integrals (McGraw-Hill, 1965; emended edition, Dover, 2010): The definitive textbook on path integrals by the inventor himself. Covers non-relativistic quantum mechanics, the QHO, statistical mechanics, perturbation theory, and applications to quantum electrodynamics. The Dover edition includes corrections by Daniel Styer. Dense but rewarding — every page contains insights available nowhere else.

  • Feynman, R. P., "The Principle of Least Action in Quantum Mechanics," Ph.D. dissertation, Princeton University (1942): Feynman's thesis, where the path integral first appears. Available in the collection Feynman's Thesis: A New Approach to Quantum Theory, ed. L. M. Brown (World Scientific, 2005). The thesis includes material on the action-at-a-distance electrodynamics that originally motivated Feynman's work — fascinating historical context.

Mathematical and Conceptual Treatments

  • Zinn-Justin, J., Path Integrals in Quantum Mechanics (Oxford, 2005): A careful and thorough treatment aimed at advanced graduate students. Covers the path integral derivation, Gaussian integrals, perturbation theory, instantons, the large-order behavior of perturbation series, and the connection to stochastic processes. More mathematical than Feynman & Hibbs but very well organized.

  • Kleinert, H., Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (5th ed., World Scientific, 2009): An encyclopedic reference (1624 pages) covering every conceivable application of path integrals. Includes rigorous treatments, variational methods, topological effects, and applications to polymer physics and even financial derivatives. Not a first reference, but an invaluable resource once you have the basics.

  • Schulman, L. S., Techniques and Applications of Path Integration (Wiley, 1981; Dover, 2005): A readable and physically oriented treatment that covers many topics not found elsewhere: path integrals on multiply connected spaces, the Morse index and focal points, semiclassical approximations, and spin path integrals. The discussion of the mathematical status of the path integral measure is particularly clear.

  • Simon, B., Functional Integration and Quantum Physics (2nd ed., AMS Chelsea, 2005): The rigorous mathematical treatment. Simon proves the Feynman-Kac formula, establishes the connection to the Wiener measure, and discusses the mathematical foundations of the Euclidean path integral. For mathematically inclined physicists and mathematical physicists.

The Classical Limit and Semiclassical Methods

  • Gutzwiller, M. C., Chaos in Classical and Quantum Mechanics (Springer, 1990): The definitive treatment of semiclassical quantum mechanics via the path integral. Gutzwiller's trace formula connects the quantum energy spectrum to classical periodic orbits — one of the deepest results in semiclassical physics. Essential reading for anyone interested in quantum chaos or the classical limit.

  • Berry, M. V. & Mount, K. E., "Semiclassical approximations in wave mechanics," Reports on Progress in Physics 35, 315 (1972): A beautifully written review of semiclassical methods, including the WKB approximation, turning-point corrections, and the connection to the path integral's stationary phase approximation.

Statistical Mechanics and Computational Methods

  • Ceperley, D. M., "Path integrals in the theory of condensed helium," Reviews of Modern Physics 67, 279 (1995): The landmark review on path integral Monte Carlo simulations of helium. Covers the ring polymer mapping, exchange sampling for bosons, the winding number estimator for superfluidity, and quantitative comparison with experiment. A masterclass in computational physics.

  • Chandler, D. & Wolynes, P. G., "Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids," Journal of Chemical Physics 74, 4078 (1981): The paper that introduced the ring polymer picture of quantum statistical mechanics to the chemistry community. Short, elegant, and enormously influential.

  • Craig, I. R. & Manolopoulos, D. E., "Quantum statistics and classical mechanics: Real time correlation functions from ring polymer molecular dynamics," Journal of Chemical Physics 121, 3368 (2004): The founding paper of Ring Polymer Molecular Dynamics (RPMD), now a standard tool in quantum chemistry.

Instantons and Nonperturbative Physics

  • Coleman, S., "The uses of instantons," in Aspects of Symmetry (Cambridge, 1985), Chapter 7: Sidney Coleman's Erice lectures on instantons are the clearest and most entertaining introduction to the subject. Coleman explains the double-well tunneling problem, the periodic potential, and the vacuum structure of gauge theories using the Euclidean path integral. Required reading.

  • Rajaraman, R., Solitons and Instantons (North-Holland, 1982): A systematic treatment of classical solutions in quantum field theory, including instantons, solitons, and monopoles. The early chapters on quantum mechanics path integrals provide excellent preparation for the field theory applications.

Feynman Diagrams

  • Mattuck, R. D., A Guide to Feynman Diagrams in the Many-Body Problem (2nd ed., Dover, 1992): A gentle and often humorous introduction to Feynman diagrams, starting from quantum mechanics and building up to condensed matter applications. The "doodle" approach makes the subject less intimidating.

  • Lancaster, T. & Blundell, S. J., Quantum Field Theory for the Gifted Amateur (Oxford, 2014): Chapters 16-20 provide an accessible introduction to Feynman diagrams and the path integral for quantum field theory, aimed at physicists who are not specialists in particle theory. Well-written, with many worked examples.

Lattice Gauge Theory

  • Wilson, K. G., "Confinement of quarks," Physical Review D 10, 2445 (1974): Wilson's paper introducing lattice gauge theory — one of the most important papers in theoretical physics. Wilson discretizes the Yang-Mills path integral on a spacetime lattice and shows how to formulate the confinement problem.

  • Gattringer, C. & Lang, C. B., Quantum Chromodynamics on the Lattice (Springer, 2010): A modern and complete textbook on lattice QCD, from the foundations (Euclidean path integral, lattice discretization) to state-of-the-art algorithms and results.

Historical and Biographical

  • Gleick, J., Genius: The Life and Science of Richard Feynman (Pantheon, 1992): Chapters 8-9 describe the development of the path integral, from the beer-party conversation with Jehle through the wartime thesis work and the postwar development of QED. Vivid and well-researched.

  • Schweber, S. S., QED and the Men Who Made It (Princeton, 1994): A detailed history of the development of quantum electrodynamics, with extensive treatment of Feynman's, Schwinger's, and Tomonaga's contributions. Chapter 8 covers Feynman's path integral work. The most thorough historical account.

  • Dirac, P. A. M., "The Lagrangian in quantum mechanics," Physikalische Zeitschrift der Sowjetunion 3, 64 (1933): The paper that planted the seed for Feynman's path integral. Dirac's observation that the short-time propagator "corresponds to" the exponential of the Lagrangian was the key insight. Short and profound.

For the Adventurous

  • Witten, E., "Quantum field theory and the Jones polynomial," Communications in Mathematical Physics 121, 351 (1989): Witten showed that the Jones polynomial — an invariant of knots — can be computed as the expectation value of a Wilson loop in a Chern-Simons path integral. This paper, which helped earn Witten the 1990 Fields Medal, demonstrates the extraordinary reach of path integral methods into pure mathematics.

  • Hawking, S. W., "The path-integral approach to quantum gravity," in General Relativity: An Einstein Centenary Survey (Cambridge, 1979): Hawking's introduction to the gravitational path integral and the no-boundary proposal. Speculative but visionary.

  • Polyakov, A. M., Gauge Fields and Strings (Harwood, 1987): Polyakov's lectures on the path integral in gauge theory and string theory. Compact, deep, and full of original insights. For advanced readers who want to see how the path integral serves as the foundation of modern theoretical physics.