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As a simple example, consider the following Hamiltonian:

[tex] H=\frac{p_1^2}{2m}+\frac{p_2^2}{2m}[/tex]

What does this Hamiltonian describe? Is it one free particle moving in two dimensions, or two free particles moving in one dimension? Clearly, this Hamiltonian describes both. But then, how to distinguish between these two physically different cases? Is there a FORMAL (not purely verbal!) way to distinguish the configuration space from the "physical" space?

This is essentially a classical question, but there are two reasons why I ask this question here:

First, people here are much more clever than people on the forum for Classical Physics.

The second, more important reason is that, although essentially classical, the motivation behind this question is actually quantum. Namely, the idea is that nonlocality of quantum mechanics could be avoided by noting that, ultimately, QM is nonlocal because it is formulated in the configuration space rather than in the "physical" space. For if the configuration space is reinterpreted as a "true physical" space (whatever that means), then QM becomes local in that "true physical" 3n-dimensional space, where n is the number of particles. But then the problem is to explain why the world looks to us as if it was only 3-dimensional (for simplicity, I ignore relativity). To understand that one needs first to understand what exactly makes the standard 3-dimensional physical space more "physical" than the 3n-dimensional configuration space, which is my motivation to ask the question above.