It seems appropriate to start my contribution to quantum foundational debates at fqxi with one of the most contentious issues in the subject - the status of the wave-vector. There are essentially two common views: either the wavefunction is ontological or it is epistemic. The ontological view holds that the wave-vector should be interpreted realistically, i.e. whatever things can actually exist in reality - the possible ontological states - are in one to one correspondence with wave-vectors. In contrast, the epistemic view holds that the wave-vector is a state of knowledge/information/belief (delete as appropriate) and its role more analogous to that of a probability distribution in classical physics rather than that of a point in phase space.
Let me briefly review the arguments for each position, with the underlying assumption that a realist view of quantum theory is desirable.
The Case for Ontological Wave-vectors:
- We know that any "completion" of quantum mechanics, i.e. a hidden-variable theory, has to have very bizzare properties, e.g. nonlocality, so it seems that there is no mathematical object other than the wave-vector that we could reasonably attach to reality.
- Almost all interpretations of quantum theory that make well-defined statements about what exists in reality, e.g. many-worlds, Bohmian mechanics and spontaneous collapse models, take the wave-vector to be an ontological object. There are simply no viable realist interpretations that don't take this view.
- Interference would be very difficult to explain if the wave-vector were a purely epistemic object. It seems that we must conceive of it as more like a real wave in 3d space in order to understand this, despite the fact that it is actually defined on configuration space.
The Case for Epistemic Wave-vectors:
- The measurement problem is arguably the most important interpretational question, but it poses no problem if the wave-vector is epistemic. We are very used to the idea that epistemic states can change radically when new information is aquired without having any major implications for what is going on in reality, e.g. updating a probability distribution via Bayes' rule.
- Many of the phenomena of quantum information theory have perfectly straightforward explanations if the wave-vector is epistemic, but seem bizzare if it is ontological. For example, teleportation is closely analogous to the transfer of a probability distribution from one point to another using classical correlations (a.k.a. a secret key), but seems to require instantaneous transfer of an infinite amount of information if the wave-vector is ontological.
- The reason why there are no realist interpretations in which the wave-vector is epistemic is just that we simply haven't thought hard enough about what the possible ontologies underlying quantum theory are.
In my opinion, the case for epistemic wave-vectors is gathering pace, but the question I want to ask here is whether this is really an all-or-nothing issue? Could it not be the case that some parameters of the wavefunction are ontological, i.e. enough of them to explain interference, and others are epistemic, i.e. enough to give a neat resolution of the measurement problem and to explain the analogies to classical probability theory?
The most simplistic version of this might take the modulus squared of the wavefunction expressed in the position basis to be epistemic and the phase to be ontological. Of course, this is not particularly compelling because it breaks the symmetry between position and momentum. Both interference and the analogies to probability theory can exist with respect to all bases, so this hardly solves the problem.
Still, it suggests that perhaps we should ask whether we can define in a more rigorous way what it means for a parameter of a theory to be ontological or epistemic. This might suggest new possible interpretations of quantum theory or, if not, at least it might lead to some no-go theorems that explain why our current realist interpretations look the way they do.