This idea violates the superposition principle of quantum mechanics, and picks out a preferred basis. As such, it wrecks the property of QM that the state space is invariant under arbitrary unitary transformations, only certain dicrete unitaries will preserve the phase space structure. Such an idea is a new thing, and you need to check that it makes sense, because on its face, such a thing will lead to irreversible decoherence in any finite state space.
For example, suppose I place a spin 1/2 particle in a magnetic field which I control. By using a weak field for long times, I can make the pure state point in any direction I want in the Block sphere. Saying that only a discrete set of states are allowed to be pure will break the rotation symmetry into something else, some directions of spin orientation will be allowed to be pure, while others will not. This can only happen if the effective pure states are extremely dense on the Block sphere.
If you insist that the discrete transformations of the Block sphere form a discrete subgroup of SU(N) (where N is the number of basis quantum states, and SU(N) is the full unitary group on these), then I think you are in trouble. The discrete subgroups of SU(N) are highly constrained, for SU(2) you essentially get the symmetry group of the Platonic solids, while for SU(N) it's open, but you would need that the discrete groups approximate SU(N) arbitrarily well at large N, which doesn't look likely to me. Anyway, with the Platonic solid groups, you would not be able to reproduce rotational invariance at all, since the electron would only be pure in a set of directions forming a dodecahedron (at best) and this is not a very good approximation to a sphere.
If you drop the insistence that the symmetry of the state space forms a group, then you might be ok, but then it is not clear how you make a mathematical structure which realizes this idea. If you say that only the full quantum state space of the whole universe is discrete, then you need to check that if you isolate a single electron, you can make an approximate block sphere for the single electron. These are difficult problems, and without a resolution to these, I think the idea must remain in the realm of the very speculative.
The discreteness ideas from quantum gravity come from Banks's bound on the universe's information content from the cosmological horizon entropy. If you take this seriously, the effective dimension of the Hilbert space that describes our causal patch (the universe) is finite and growing. This is inconsistent with unitary evolution. One way around this is to declare that the true string theory Hilbert space is defined in the asymptotic future, when the universe decays into a zero temperature vacuum. This seems hard to say, since we need to know what vacuum it will pick, and this is certainly not determined from our current observations.
My own speculation is that you need to think of the cosmological state as a classsical probability distribution over the different possible string theory end-states, with no coherence between the different possible end-states. This is extremely speculative, because there is no direct matching between density matrices and cosmological evolution known. Then the finite information content of the universe would not be so mysterious, it would just say, given that we know so little about the end state, we can't fit too much information into the universe, since each bit we have inside the cosmological horizon is a bit we must know about the endstate already today, since it is encoded in the asymptotic future. This is just me musing about an unsolved problem, I don't know the answer, but I think that this sort of thing is more firmly grounded in established physics than any sort of naive discretization of Hilbert space.
This post imported from StackExchange Physics at 2014-04-11 15:48 (UCT), posted by SE-user Ron Maimon
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