What is Sum over histories: Definition and 11 Discussions

The path integral formulation is a description in quantum mechanics 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.
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 been proved 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.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.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. This idea was extended to the use of the Lagrangian in quantum mechanics by Paul Dirac in his 1933 article. The complete method was developed in 1948 by Richard Feynman. 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.

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  1. S

    B Is There a Cosmological Model That Considers All Possible Laws of Physics?

    Hawking and Hartle proposed a well-known model which postulated a sum over all possible histories considering all compact euclidean metrics to explain the origin of the universe (this is called the No Boundary model). I was wondering whether there is any model or theory (related to cosmology)...
  2. H

    I Sum over histories implies no definite history?

    What does Feynman's sum over histories mean to the interpretation of our world? Does it mean that we (or a particle) do not have a definite history, but only the most probable one?
  3. S

    A Wavefunction of the Universe considering all possible boundaries?

    The Hawking-Hartle no boundary condition is well known. The authors considered a many worlds/histories model considering a sum over all compact euclidean metrics. But are there any models or theories that consider a sum over all possible metrics or boundaries? And finally, if all possible...
  4. S

    A Hartle-Hawking sum over all possible metrics?

    Physicists Stephen W Hawking and James B Hartle 1 proposed that the universe, in its origins, had no boundary conditions both in space and time. To do that, they proposed a sum over all compact euclidean compact metrics. I have heard that they only considered these metrics in order to simplify...
  5. Matta Tanning

    Relation between phase space and path integral formulation?

    I am trying to conceptually connect the two formulations of quantum mechanics. The phase space formulation deals with quasi-probability distributions on the phase space and the path integral formulation usually deals with a sum-over-paths in the configuration space. I see how they both lead...
  6. B

    Sum over histories and double slit?

    I was recently studying Feynman's sum-over-histories approach to quantum probability. I also was reading an interesting paper on the double slit experiment. How do these two work together. Do some of the probability waves not have a out of phase partner to interfere with itself? On a related...
  7. J

    Confused by Feynman sum over histories

    This seems to contradict the very tools we use to perform the scientific method. Immediately after performing any experiment, the measurement becomes a historical event, and this says that we cannot say that any historical event has actually occurred. Thus, I cannot actually say that I...
  8. P

    Understanding the Strange World of Sum-over-Histories: A Brief Overview

    I'm working on an article about sum-over-histories. Could folks more knowledgeable than I review this and point out any errors? Thank you. ---- The 1920's were an exciting time, with unexplained experimental results all over the place. Eventually someone would get the math to pretty...
  9. A

    How does Feynman Sum Over Histories make sense?

    How does Feynman "Sum Over Histories" make sense? Hi. I am new to Quantum Physics and this forum as well. I was reading The Grand Design, by Stephen Hawking, and came upon the "Alternative Histories" theory in Quantum Mechanics. My questions are: 1.) How is it possible that a particle can...
  10. bcrowell

    Sum over histories, including histories that violate conservation laws?

    It's been 20 years since I took field theory in grad school, and I didn't really understand it all that well even then, so I'm basically looking for a very low-level explanation of the following issue... In the sum-over-histories approach, there is the question of which histories to include...
  11. R

    Sum over histories and relationships

    Ok, everyone knows that according to Feynman's theory a subatomic particle traverses all paths from one point to another simultaneously. My idea is to apply this behavior to relationships. For example, given a choice between five girls each choice made invokes a different actual history out of...