Does the Hilbert space of the universe have to be infinite dimensional to make sense of quantum mechanics?

Question

Does the Hilbert space of the universe have to be infinite dimensional to make sense of quantum mechanics? Otherwise, decoherence can never become exact. Does interpreting quantum mechanics require exact decoherence and perfect observers of the sort which can only arise from exact superposition sectors in the asymptotic future limit?

Answer 1

With a finite-dimensional Hilbert space, the whole apparatus of practical QM is lost. Very little is left - no continuous spectra, no scattering theory, no S-matrix, no cross sections. No Dirac equation, no relativity theory, no relation between symmetry and conservation laws, no quantum fields. Almost the whole of the achievements of modern physics would be ruined.

Already the Hilbert space of a single oscillating mode is infinite-dimensional, and the universe contains zillions of them. Fortunately, zillions times infinite is still infinite, but...

Due to superselection sectors, the Hilbert space of QED is already nonseparaple (i.e., has an uncountable basis). The physical Hilbert space of a quantum field theory is the direct integral of the Hilbert spaces corresponding to the different superselection sectors. The direct integral is mathematically well defined, http://en.wikipedia.org/wiki/Direct_integral and gives a nonseparable space once the integral is over a continuum.

In QED it is (at least) the continuum of directions in 3-space. One needs this nonseparable space to define Lorentz transformations of charges states, as charged states moving in different directions are in different superselection sectors. Thus the dimension of the Hilbert space of the universe should be at least the cardinality of the continuum.

Now QED describes the universe with gravitation, weak and strong forces ignored. Unfortunately, very little is known about the Hilbert space of nonabelian gauge theories and quantum gravity, so it is not that clear which cardinality the Hilbert space of the universe will have once we know whether the universe is described by one.

On the other hand, the interpretation of quantum mechanics cannot depend on exact models, as our models of the real world are never exact replicas of the latter.

Answer 2

The universe probably has an infinite dimensional Hilbert space. However, finite dimensions suffice to "make sense of" quantum mechanics. Or at least finite dimensions are sufficient to gain intuition and to expose the philosophical difficulties of the various interpretations of quantum mechanics. A cat lives in a very high dimensional (or infinite) Hilbert space, but the essence of the Schrodinger's cat paradox can be grasped by just considering it to be a two-state system.

And finite dimensions are sufficient for perfect decoherence. An example is the $|+> \otimes |0>$ state fed into a controlled-not gate in the context of quantum computing. In fact, that simple circuit is actually a pretty good model for understanding the role of observers (the second qubit "observes" or copies the state of the first qubit, thereby causing decoherence). Finite dimensions are a lot easier to think about. You may of course not want to think of you as an observer as being a qubit entangled with the object you were measuring, but that is a can of worms that philosophers will probably still be working out for a very long time.