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I would be grateful if someone could explain me this argument from Philosophy of Physics in plain English. I'm not sure how Albert arrives at his conclusion and I lack the mathematical skills to follow the original argument in the 1996 paper. It purports to show why although the configuration space of QM states is high-dimensional (3N-dimensional for N particles), it appears to us that particles, and hence everyday objects 'live' in a 3D space. Here is a description of it:

[Albert, 1996] begins by considering an “NN-dimensional classical-mechanical configuration space, in which a single world-particle is floating around” (p. 280). He goes on to ask how such a situation could come to have the appearance of a collection of particles in a lower-dimensional space. He begins with a world-particle governed by a free Hamiltonian, whose trajectory would therefore be a straight line in the confguration space, traversed at constant speed.

And note that the trajectory of a world-particle like this one can patently contain no suggestion whatever as to whether we are dealing here with a single material particle moving freely in an N-dimensional physical space, or (say) N/3 distinct material particles moving freely in the three-dimensional physical space, or N distinct particles moving in a one-dimensional physical space. Nothing about a trajectory like that (to put it slightly differently) can make it natural or make it plausible or make it reasonable or make it simple or make it elegant or make it any other desirable thing to suppose that any one of those possibilities, as opposed to any of the others I mentioned, or any one of the others I didn’t mention, actually obtains. (Albert 1996, p. 280)

Albert concludes that the appearance of a low-dimensional space must depend on the existence of something other than a free Hamiltonian (i.e., on what are normally called interaction terms in the Hamiltonian), and he goes on to consider just what sort of Hamiltonian might make the postulation of a low-dimensional physical space plausible.

The original argument is in: Albert D.Z. (1996) Elementary Quantum Metaphysics. In: Cushing J.T., Fine A., Goldstein S. (eds) Bohmian Mechanics and Quantum Theory: An Appraisal. Boston Studies in the Philosophy of Science, vol 184. Springer, Dordrecht

The description I gave here is from: Maudlin, T. (2013) The nature of the Quantum State in Ney, Alyssa et al., 2013. The Wave Function: Essays on the Metaphysics of Quantum Mechanics, Oxford University Press. (page 141)

  • He does not really give an argument for it but simply relies on the standard physical heuristics. The interaction terms he adds are two-point interaction potentials, in the classical limit they will correspond to particles with the number of coordinates equal to the number of terms under the sum interacting via forces depending on Euclidean distances between them. If n is 2, 3 or some other dimension we can always split N=kn into k particles moving in nD space, but for interactions to look "natural" nD distances must occur in the interaction terms. – Conifold Dec 27 '17 at 20:38

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