International Society for the Advanced Study of Spacetime

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 (non-relativistic) quantum mechanics, physical space is surely three-dimensional; 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 non-relativistic quantum mechanics.

The wave function of a quantum system mathematically exists in a 3N-dimensional space, where N is the number of particles which purportedly exist in three-dimensional space. The wave function cannot exist in three-dimensional space, due to quantum-mechanical 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 3N-dimensional space, does it follow that 3N-dimensional space is real? If so, is there just 3N-dimensional space? Or are there two spaces, 3N-dimensional space and three-dimensional 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 3N-dimensional space. I find this view almost unbelievable, and I have argued for a more common-sense 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?


Albert, David (1996), “Elementary Quantum Metaphysics”, in J. Cushing, A. Fine, and S. Goldstein (eds.), Bohmian Mechanics and Quantum Theory: An Appraisal, Kluwer, pp. 277-284.

Lewis, Peter (2004), “Life in Configuration Space”, British Journal for the Philosophy of Science, forthcoming.

Monton, Bradley (2002), “Wave Function Ontology”, Synthese 130: 265-77.