ueit said:
OK, we need to specify the system energy as well. So what? We have a function relating the quantum potential to some well defined properties of the system evolving in 3d space + time. No need for other worlds/dimensions, no need to ascribe reality to the Hilbert space itself. If we imagine this quantum potential as a kind of space geometry, only this geometry needs to be real in BM, nothing else.
But this "geometry" needs the exact wavefunction, and can only be deduced from that wavefunction, so the wavefunction is entirely part of the "ontology" of the theory. Now, you can make extra hypotheses, which come down to specifying a specific wavefunction (like, it must be in the ground state or something), but this limits severely the applicability of BM. In short, you cannot do BM when the initial wavefunction is not given.
You can compare this with the EM field in classical physics. The fact that there are non-trivial vacuum solutions to the EM field, means that these fields have an essential existence of their own. You cannot do electrodynamics without using the EM field (or the vector potential or anything of the kind).
You could get away with it if there were purely a Coulomb interaction, because from the configuration of the charged particles, the Coulomb interaction can be derived. The E-field would then simply be a convenience, but would not be an essential part of the dynamics (and hence, of the ontology). You could, if you wanted to, eliminate the E-field in all calculations.
Even the EM field could be partly eliminated, by using the retarded potential expressions. All EM waves emitted from other charges can be eliminated that way. But what you cannot eliminate, are the initial radiative conditions. There can be initial EM waves, unrelated to any charged source. This dynamical element cannot be eliminated, and hence, the EM field has an essential dynamics to itself, which means that any theory of the EM interaction must consider that there is an ontology to the EM field, and that it is not just an intermediate variable used for convenience.
In the same way, in BM, you NEED the initial wavefunction, because it cannot be derived from the particle positions. It is not an intermediate variable which could be eliminated at leisure. It is an essential component of the dynamical formulation of BM, and hence has an ontological existence.
In conclusion, BM's ontology requires, beyond classical parameters, a parameter defining the "space quantum geometry".
Well, that's an euphemism to say that you need the wavefunction...
The details of how we perform the calculation are irrelevant to that ontology.
Not really. You cannot eliminate it, it is not an intermediate quantity just introduced by convenience but which could be eliminated entirely.
MWI, on the other side, requires that the mathematical formalism maps to some existing reality, BM doesn't need that.
yes, it does so, for exactly the same reason: it is an essential part of the dynamics.
This is a very strange objection, indeed. We are certainly aware of the quantum potential in the same way we are aware about any other potential, by observing how particles move in its presence. That we cannot directly access this potential is not in any way different from the fact that we cannot observe an electric field without at least a charged object being present. If you maintain that BM needs that postulate then you need to ask classical mechanics to add postulates for the non-observability of every classical potential as well.
Well, in classical physics, the ontology consists of particles and fields (both of them). Together they specify the configuration space (or the phase space, if you want). It is hard to say which aspect of the point in conguration space is generating a certain subjective experience: are it purely the particle states, or are it the field states, or both ? Hard to say whether it is the EM field configuration in the brain of a creature living in a classical world which is giving it his memory states, or whether it are the particle configurations ! I would say that it is the entire state which does so. But in a classical setting, this is pretty irrelevant.
1. BM doesn't look non-local from the point of view of the universal wave-function in the sense that no interaction needs to be transmitted ftl. The quantum potential evolves deterministically, regardless of particles' motion and the particles only interact locally.
No, not really. In BM, the potential is function of the wavefunction AND the positions of the remote particles. This is the non-local element: the guiding equation:
<br />
\frac{dq_k}{dt} = \frac{\hbar}{m_k}\frac{Im\left[ \Psi^* \partial_k \Psi \right]}{\Psi^* \Psi}_{q_1,q_2...,q_N}<br />
The presence, in the generalised velocity for the k-th particle, of the generalized coordinates of the other particles
at the same moment, makes this an explicitly non-local (and non-lorentz-invariant) expression.
So, whyle BM requires an absolute reference frame it doesn't seem to conflict directly with relativity (which could be formulated, I think, on an absolute RF).
Well, that's a contradiction in terms: any theory requiring an absolute reference frame is in conflict with the fundamental postulate of relativity. It is always possible to make it observationally in agreement with relativity, but it means, in that case, that one has introduced unnecessary elements which break explicit Lorentz invariance. This is the same with an ether theory, or with, say, the coulomb gauge fixing condition in the canonical quantization of QED.
2. Is there a fully relativistic QFT yet (without the non-local collapse or with the collapse relativistically treated)?
You cannot treat the collapse relativistically. What is done in QFT, is the calculation of matrix elements of the unitary evolution operator which transforms initial particle states in final particle states, for "large times". It is very similar to the U(t1,t0) operator which transforms |psi(t_0)> into |psi(t1)>, but taken in the limit where t1 goes to +infinity and t0 goes to -infinity. It is only for that case that there are approximative techniques.
This unitary operator is nothing else but the solution to the schroedinger equation, as usual. These complex numbers, squared, give the transition probabilities of the corresponding initial state in the final state.
Usually, this is done by calculating an approximation to an expression which is called a "path integral". Given that we calculate in this way, the probabilities for the transition from "long ago" into the "far future", this can be interpreted in any way you like. You can continue to consider the superposition of final states (whose coefficients are nothing else but the matrix elements calculated) a la MWI, or you can decide to project one out (in which case you do something non-local), a la Copenhagen.
3. Is MWI proven to have a mathematically rigorous relativistic extension? I've read some articles claiming that problems relating to the world splitting could appear.
The mathematical part of MWI is nothing else but standard unitary quantum theory. QFT is known not to be rigorously correct, but this is just a model as any other.
