The Feynman's path integral representation gives the quantum amplitude to go from point $x$ to point $y$ as an integral over all paths.

  1. How is that idea consistent with the uncertainty principle that is considered to be fundamental? That is, having a definite, initial point $x$ is impossible for a physical particle.

  2. Probability distributions are allowed but definite values are not. Thus, can we replace definite points with well-localized distributions, together with the finite expectation values of observables, to make more physical/mathematical sense?

  3. Can we argue that the Feynman's picture is not "real" but only a way of interpreting the integral-like sum? A related idea might be the Ptolemy's picture of planetary epicycle motion that also gave correct results but for wrong reasons. A similar issue with the Huygens' principle.

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    $\begingroup$ There is no problem with a particle having a definite position as long as you don’t know anything about its momentum. That is the case here. “All paths” includes paths with every possible initial and final momentum. $\endgroup$
    – G. Smith
    Commented Dec 2, 2019 at 0:02
  • $\begingroup$ @G.Smith An infinite momentum should be constrained somehow! Mundane potentials are negligible next to infinite momentums... $\endgroup$
    – Hulkster
    Commented Dec 2, 2019 at 0:04
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    $\begingroup$ Unless I am mistaken, the path integral includes includes extremely “unphysical” paths that go faster than light, backwards in time, etc. $\endgroup$
    – G. Smith
    Commented Dec 2, 2019 at 0:16
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    $\begingroup$ Sorry, I’m not sure what you mean. I want to mention that in the path integral all the crazy unphysical paths tend to “cancel” each other, and the result is mostly from the classical path with small quantum corrections from nearby paths. But in principle all paths contribute. It shows how tidy classical behavior emerges from superposing untidy quantum behaviors. I love this insight of Feynman. $\endgroup$
    – G. Smith
    Commented Dec 2, 2019 at 0:47
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    $\begingroup$ I totally agree with G. Smith. All paths, however unphysical are included in the integral, but indeed, those who are unphysical contribute very little and tend to cancel each other. But the integral contains them all, by definition ! $\endgroup$
    – Alfred
    Commented Dec 2, 2019 at 1:04

1 Answer 1


I can't claim I understand the question, but, as you point out, the Feynman graphic visualization of Dirac's 1933 construction of QM amplitudes is an invitation for technical algorithms, allowing for creative techniques. Notably, at the Pocono conference, Bohr misunderstood it at first and rebuked RPF on that very UP ground.

  1. But... the UP certainly allows for perfectly fine transition probability amplitudes $K(x,t;x',t') = \left \langle x \mid \hat{U}(t,t') \mid x'\right \rangle,$ for perfectly precise x s, just as it does in plain vanilla Hilbert-space QM; it doesn't much care how you compute these. The very same flip-flopping between the x representation and the p representation you see in Hilbert space is utilized in defining the path integral, and proving its equivalence to standard techniques.

  2. Well, the output of the path integral is probability amplitudes! The path integral, through the magic of Gaussians, quantifies the localization of distributions, and their spread via $\hbar$ corrections. Feynman expanded on Dirac's crucial observation, and quantified departures from the "classical limit" of stationary actions. The sum over all paths is actually a technique of practically generating these distributions, which, among other things, makes lattice simulations of QFTs like QCD possible and practical!

  3. I don't understand your point and I never understood what "real" means, but, yes, the path integral is an umbrella of techniques of deriving amplitudes. Magnificent techniques, when you appreciate the functional equations and insight on ghosts it has lavished on QFT, and the practical simulation answers it makes possible in lattice gauge theory...

  • $\begingroup$ Thanks for your thoughts. $\endgroup$
    – Hulkster
    Commented Jul 30, 2020 at 12:36
  • $\begingroup$ I did not strive to over-emphasize the theme of my point, namely that the RPF imagery obscures the fact that this is conventional QM, really, in a dramatically poetic language, but all the concepts are humdrum, as Dirac introduced them, and obey the standard technical rules and restrictions of QM, once the "picture" bit is forgotten about. This is what Feynman failed to emphasize at Pocono in 1948. $\endgroup$ Commented Jul 30, 2020 at 13:39
  • $\begingroup$ Do you believe that the crux of quantum gravity is to find the right "fundamental transformation", like the Fourier transformation in QM? $\endgroup$
    – Hulkster
    Commented Aug 5, 2020 at 15:40
  • $\begingroup$ Not sure. In fact, I'm not sure the question or my answer connect to this, and how... $\endgroup$ Commented Aug 5, 2020 at 16:17

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