

Quantum Mechanics and the Dimensionality of Space
Bradley Monton
What is the dimensionality of space according to quantum mechanics? One might think that, according to (nonrelativistic) quantum mechanics, physical space is surely threedimensional; it’s only when one gets to theories which attempt to unify quantum mechanics and general relativity that the dimensionality of space becomes an open question. But in fact the dimensionality of space is an open question even for nonrelativistic quantum mechanics.
The wave function of a quantum system mathematically exists in a 3Ndimensional space, where N is the number of particles which purportedly exist in threedimensional space. The wave function cannot exist in threedimensional space, due to quantummechanical holism: the quantum state of a system cannot always be fully described by the quantum states of its subsystems. Given that the wave function exists in 3Ndimensional space, does it follow that 3Ndimensional space is real? If so, is there just 3Ndimensional space? Or are there two spaces, 3Ndimensional space and threedimensional space? If the latter, are there causal or metaphysical connections between the two spaces, or do the spaces evolve metaphysically independently?
David Albert (1996) has argued that according to quantum mechanics, there is just 3Ndimensional space. I find this view almost unbelievable, and I have argued for a more commonsense ontology in Monton 2102. Peter Lewis (2104) has replied to my arguments, defending a position intermediate between Albert’s and mine. So this leaves an open question: who is right; what is the dimensionality of space according to quantum mechanics?
References
Albert, David (1996), “Elementary Quantum Metaphysics”, in J. Cushing, A. Fine, and S. Goldstein (eds.), Bohmian Mechanics and Quantum Theory: An Appraisal, Kluwer, pp. 277284.
Lewis, Peter (2004), “Life in Configuration Space”, British Journal for the Philosophy of Science, forthcoming.
Monton, Bradley (2002), “Wave Function Ontology”, Synthese 130: 26577.


