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I want to better understand what the path integral is and what it isn't. Even though I do this to learn QFT, this question is purely concerned with classical fields, no quantization is intended at all. Neither am I talking about classical particles with well-defined paths, though.

Feynman introduces the path integral with the Gedankenexperiment of expanding a double slit experiment to have an infinite amount of screens with an infinite amount of holes in each. He does this, talking about the options a particle has to travel to the screen. In QM, all possible paths then interfere and the path integral is used to calculate this interference.

What I read out there is nothing but Huygens principle.

Thus, if we have a classical field with a point source at point $A$ at time $t_0$ and we want to calculate the amplitude/phase of the field at point $B$ at time $t$, would the path integral be a proper tool for it?

If so,

  • how does it relate to Fermat's principle in this case?
  • is there any important difference in either the measure or the involved action between a classical field or a quantum particle? (differences that would apply to any classical field)

Or put differently: Is the path integral a (mathematical) tool for quantum things or for wave things? Is it linked to the probability aspect or only to the wave nature of quantum phenomena?

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  • $\begingroup$ Related: physics.stackexchange.com/q/2041/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Jul 26, 2020 at 13:37
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    $\begingroup$ Hi, thanks for your answer. :) I cannot see, however, how these two questions are related and I didn't find any useful hint in the answers or links in there, unfortunately. The question you linked to, is about how classical phenomena emerge from a quantum description. I explicitly do not ask about that, as I tried to clarify in the second sentence. I am concerned about using the mathematical tool of path integrals in the context of a purely classical field theory. $\endgroup$
    – akreuzkamp
    Commented Jul 27, 2020 at 13:36
  • $\begingroup$ I'd like to see a true answer to this, as it's something I have also pondered. $\endgroup$
    – SSD
    Commented Apr 12, 2022 at 15:09
  • $\begingroup$ Most of these questions are explained in my updated answer in the linked Phys.SE post. $\endgroup$
    – Qmechanic ♦
    Commented Apr 12, 2022 at 15:25
  • $\begingroup$ When talking about a light wave it’s best to think of individual coherent photons emitted from a common source. See “Single Edge Certainty” at Billalsept.com $\endgroup$
    – Bill Alsept
    Commented Apr 12, 2022 at 15:31

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