Path integral formulation

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These are just three of the paths that contribute to the quantum amplitude for a particle moving from point A at some time t0 to point B at some other time t1.

The path integral formulation of quantum mechanics is a description of quantum theory that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.


  • Since its inception in Richard Feynman’s 1942 doctoral thesis, the path integral has been a physicist’s dream and a mathematician’s nightmare. To a physicist, the path integral provides a powerful and intuitive way to understand quantum mechanics, building on the simple idea that quantum physics is fundamentally a theory of superposition and interference of probability amplitudes. The “sum over histories” offers a framework for tackling problems ranging from Feynman diagrams to lattice chromodynamics, from quantum cosmology to superfluid vortices to stock-option pricing. To a mathematician, the path integral is at best an ill-defined formal expression. It is some sort of vaguely integral-like object involving a “sum” over a badly specified collection of functions, having an undefined measure, and whose value is apparently determined by a group of unclear and perhaps incompatible limits that may or may not yield finite answers.
    • Steven Carlip: (2008). "Review of Functional Integration: Action and Symmetries Functional Integration: Action and Symmetries by Pierre Cartier and Cécile DeWitt-Morette". Physics Today 61 (10): 61–62. DOI:10.1063/1.3001873.
  • The Feynman method has the virtue that it provides us with a vivid picture of nature’s quantum trickery at work. The idea is that the path of a particle through space is not generally well defined in quantum mechanics. … So when an electron arrives at a point in space—say a target screen—many different histories must be integrated together to create this one event.
    Feynman’s so-called path-integral, or sum-over-histories approach to quantum mechanics, set this remarkable concept out as a mathematical procedure. It remained more or less a curiosity for many years, but as physicists pushed quantum mechanics to its limits— applying it to gravitation and even cosmology—so the Feynman approach turned out to offer the best calculational tool for describing a quantum universe. History may well judge that, among his many outstanding contributions to physics, the path-integral formulation of quantum mechanics is the most significant.
    • Paul Davies (1994), Introduction to Six Easy Pieces by Richard P. Feynman
  • You could not imagine the sum-over-histories picture being true for a part of nature and untrue for another part. You could not imagine it being true for electrons and untrue for gravity. It was a unifying principle that would either explain everything or explain nothing. And this made me profoundly skeptical. I knew how many great scientists had chased this will-o’-the-wisp of a unified theory. The ground of science was littered with the corpses of dead unified theories. Even Einstein had spent twenty years searching for a unified theory and had found nothing that satisfied him. I admired Dick tremendously, but I did not believe he could beat Einstein at his own game.
  • The sum-over-paths formulation is particularly convenient for integrating out one set of coordinates to concentrate on the remaining set. Thus the photon propagator in quantum electrodynamics is obtained ... by "integrating out" the photon variables, leaving electrons and positrons, both real and virtual, to interact by means of the covariant function + 1.
  • The path integral is a formulation of quantum mechanics equivalent to the standard formulations, offering a new way of looking at the subject which is, arguably, more intuitive than the usual approaches. Applications of path integrals are as vast as those of quantum mechanics itself, including the quantum mechanics of a single particle, statistical mechanics, condensed matter physics and quantum field theory. ...
    It is in quantum field theory, both relativistic and nonrelativistic, that path integrals (functional integrals is a more accurate term) play a much more important role, for several reasons. They provide a relatively easy road to quantization and to expressions for Green’s functions, which are closely related to amplitudes for physical processes such as scattering and decays of particles. The path integral treatment of gauge field theories (non-abelian ones, in particular) is very elegant: gauge fixing and ghosts appear quite effortlessly. Also, there are a whole host of nonperturbative phenomena such as solitons and instantons that are most easily viewed via path integrals. Furthermore, the close relation between statistical mechanics and quantum mechanics, or statistical field theory and quantum field theory, is plainly visible via path integrals.
  • The idea behind the Feynman path integral goes back to a paper by P. A. M. Dirac published in 1933 in Physikalische Zeitschrift der Sowjetunion. It formed the core of Richard Feynman’s space–time approach to quantum mechanics and quantum electrodynamics. Although the path integral was not mathematically well defined, it was widely used in quantum field theory, statistical mechanics, and string theory. Recently, path integrals have been the heuristic guide to spectacular developments in pure mathematics.
    • Meinhard E. Mayer: (2001). "Review of The Feynman Integral and Feynman's Operational Calculus the Feynman Integral and Feynman's Operational Calculus by Gerald W. Johnson and Michel L. Lapidus". Physics Today 54 (8): 48–50. DOI:10.1063/1.1404851.
  • In the sixties, a new field of applications of functional integrals appeared — the quantization of gauge fields. The electromagnetic field, the Einstein gravitational field, the Yang-Mills field and the chiral field can serve as examples of gauge fields. The action functionals of those fields are invariant under gauge transformations which depend on one or several arbitrary functions. From a mathematical point of view, gauge fields are fields of geometrical origin which are connected with fibrations over four-dimensional space-time. The specificity of geometrical fields has to be taken into account when quantizing them; otherwise incorrect results may be obtained.
  • Path integrals (the terminology for functional integrals, derived from their quantum mechanical origin) had been introduced by Dirac in the 1930s. However, their mathematical fuzziness (in contrast to the precise Euclidean Wiener integrals) had discouraged their serious application in quantum theory. Nonetheless, despite the absence of a mathematical definition it became apparent that path integrals in field theory were ideally suited to
       (i) implement the symmetries of the theory directly,
       (ii) incorporate constraints simply,
       (iii) explore field topology,
       (iv) isolate relevant dynamical variables,
       (v) describe non-zero temperature.
    They were key ingredients in model-making for unified theories, and by the late 1970s a working knowledge of functional integrals had become extremely useful for most field theorists.

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