First of all, I think that interpreting the action principle in quantum mechanics as a *physical* process that explores all the paths goes too far. (As a mathematical process, fine, but physical, rather not.)

The reason is that the main difficulty of the path integral formalism is the construction of the path integral measure. Doing this with a suitable level of rigor usually incorporates information from the action (Kinetic term \(\frac{m}2 \dot x^2\) for the Wiener measure, eigenvalues of the Dirac operator in QFT, etc.). When trying to incorporate other degrees of freedom like spin; one is quickly let to the see the path integral as a mnemonic device for the Trotter product formula. I would say that you have to pick some classical differential equation / operator a priori that you can use as an input to define a path integral, not the other way round.

Now the classical action principle. First, you can interpret it as a "physical" process if you really want to: The classical particle tests out all possible paths and then chooses the one that minimzes the action.

You may not agree with this formulation, though, and your reason would probably be that this looks like a non-local process: the particle has to "figure out the whole path" before going somewhere, and cannot decide "at every moment in time" where to go. However, the Euler-Lagrange equations usually *are* local, so just because the process looks non-local, it doesn't need to be inherently non-local. I'm not entirely sure what's going on myself, but the action principle is similar to Bellmann's Dynamic Programming: to go from a point \(x_0\) to a point \(x_1\), you pick an intermediate point \(x'\), solve the subproblems of going to and from there and then minimize over the intermediate point

\(S[x_0,x_1] = \text{min}_{x'} (S[x_0,x']+S[x',x_1])\)

It may be possible to obtain the classical action principle from the quantum "interference of all paths" interpretation via some sort of "tropical probability theory", but I'm not sure if that's correct and I don't know whether anyone has already tried that.

Second, I don't agree with your friend's claim that the classical action principle is "obviously" correct. I would say that it is a mysterious happenstance: Just like a law of nature, it is neither obvious nor does it follow from some higher principle. The action principle *can* be *derived* for systems with holonomic constraints, i.e. where the particles are confined to move, say, on a circle. Situations like these are usually modeled by appealing to d'Alembert's Principle but if you start with the trivial action integral for an unconstrained system, you can incorporate the constraints by simply restricting the optimization problem to trajectories that fulfill the constraints. The equivalence of these formulations requires proof.

To summarize, my points are:

- The QM explanation is not actually satisfying. (Construction of path integral measure.)
- It might be possible to derive the classical action principle from "tropical probability theory" (Conjecture!).
- The classical action principle can be given the status of a theorem (as opposed to a higher principle) for systems with constraints.