Path integral formulation

The path integral formulation is a description in quantum mechanics that generalizes the stationary 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.

This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of canonical quantization. Unlike previous methods, the path integral allows one to easily change coordinates between very different canonical descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the Lagrangian of a theory, which naturally enters the path integrals (for interactions of a certain type, these are coordinate space or Feynman path integrals), than the Hamiltonian. Possible downsides of the approach include that unitarity (this is related to conservation of probability; the probabilities of all physically possible outcomes must add up to one) of the S-matrix is obscure in the formulation. The path-integral approach has proven to be equivalent to the other formalisms of quantum mechanics and quantum field theory. Thus, by deriving either approach from the other, problems associated with one or the other approach (as exemplified by Lorentz covariance or unitarity) go away.[1]

The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s, which unified quantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition. The Schrödinger equation is a diffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possible random walks.[2]

The path integral has impacted a wide array of sciences, including polymer physics, quantum field theory, string theory and cosmology. In physics, it is a foundation for lattice gauge theory and quantum chromodynamics.[3] It has been called the "most powerful formula in physics",[4] with Stephen Wolfram also declaring it to be the "fundamental mathematical construct of modern quantum mechanics and quantum field theory".[5]

The basic idea of the path integral formulation can be traced back to Norbert Wiener, who introduced the Wiener integral for solving problems in diffusion and Brownian motion.[6] This idea was extended to the use of the Lagrangian in quantum mechanics by Paul Dirac, whose 1933 paper gave birth to path integral formulation.[7][8][9][3] The complete method was developed in 1948 by Richard Feynman.[10] Some preliminaries were worked out earlier in his doctoral work under the supervision of John Archibald Wheeler. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the Wheeler–Feynman absorber theory using a Lagrangian (rather than a Hamiltonian) as a starting point.

These are five of the infinitely many paths available for a particle to move from point A at time t to point B at time t’(>t).
  1. ^ Weinberg 2002, Chapter 9.
  2. ^ Vinokur, V. M. (2015-02-27). "Dynamic Vortex Mott Transition" (PDF). Archived from the original (PDF) on 2017-08-12. Retrieved 2018-12-15.
  3. ^ a b Hari Dass, N. D. (2020-03-28). "Dirac and the Path Integral". arXiv:2003.12683 [physics.hist-ph].
  4. ^ Wood, Charlie (2023-02-06). "How Our Reality May Be a Sum of All Possible Realities". Quanta Magazine. Retrieved 2024-06-21.
  5. ^ Wolfram, Stephen (2020-04-14). "Finally We May Have a Path to the Fundamental Theory of Physics… and It's Beautiful". writings.stephenwolfram.com. Retrieved 2024-06-21.
  6. ^ Chaichian & Demichev 2001
  7. ^ Dirac 1933
  8. ^ Van Vleck 1928
  9. ^ Bernstein, Jeremy (2010-04-20). "Another Dirac". arXiv:1004.3578 [physics.hist-ph].
  10. ^ Feynman 1948.