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Hey Guys!

I've got an exciting question! It's been burning on my mind for years, but I think I can formulate it now. It's not so much a specific question, but rather a physical story which perhaps this thread can uncover.

It starts in Chapter 6 of Thornton & Marion, Classical Dynamics of Particles and Systems, the chapter on calculus of variations. This is like the "mathematics you're gonna need" for the following chapter on Lagrangian and Hamiltonian dynamics. In this chapter, they use Fermat's principle as an example: Light travels by the path which takes the least amount of time.

1) I was thinking about the propagation of a photon on a curved spacetime background, (general relativity) and, how at every location in the spacetime which is on the particle's (the photon's) classical path, (its worldline) the particle "chooses" which, possibly new, direction is, for the following infinitesimal bit of its motion *only,* the direction which will contribute the least amount of elapsed time if the particle has in its "mind" that it is trying to get to some "final" location of the spacetime. If there are no humans present to specify this "final" point, I presume the photon will merely follow the curves of the spacetime, that is, a null geodesic, at the rate of 299,792,458 m/s for all eternity, thereby taking the most time-efficient path to its "final location," which, in this case, is at infinity. (How can there be a most time-efficient path to infinity… )

2) Non-Uniqueness: I understand the classical path is not necessarily unique. In quantum mechanics, I believe these extra classical paths simply get summed over in some way in the overall path integral, and everything is all right because we have Heisenberg's uncertainty relation anyway, but what does it mean in classical mechanics? In classical mechanics, a definite path must be realized. How is this path selected amongst the solution set?

3) Path Integration: In quantum mechanics, one begins with an initial point, a final point and an elapsed time T. One then evaluates a path integral to obtain the amplitude for a particle to begin at the initial point and end at the final point a time T later. You modulus-square this (complex) amplitude to obtain the probability for the aforementioned "motion." (however one interprets what "motion" is in quantum mechanics, given the Heisenberg uncertainty relation)

I'm trying to make 1), 2) and 3) make sense simultaneously in my poor head. I'm trying to bring all these ideas together. I guess what this is driving at is a path integral formulation of general relativity which, in the classical limit and in the particular solution for a single photon, produces a single, realized path which minimizes the physical quantity, time. Again, there is this confusion in 2) about the non-uniqueness of the classical path.

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# The Classical Path, QM Path Integrals and Paths in Curved Spacetime

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