1. In the 'Undivided Universe', Bohm & Hiley seem to be saying that the particle is not only guided by the quantum potential (which isn't exactly the wavefunction, but very related), but even more importantly, by the classical potential. The quantum potential is usually very small compared to the classical potential. As far as I understood.
Yes, that is to say that in many cases, we can just do classical physics. The quantum potential is only important in those cases where we have a deviation between classical and quantum predictions.
2. In measurement, it would seem to me, the measurement device's visible output is (also) determined by the measured particle, not only the measured wavefunction. Measurement devices in BM are not special, they are physical objects like any other.
This is the circular thing in BM: by definition, one calls the "measurement output" as the *particle state* of the measurement apparatus. Its quantum state is as in MWI: that is: the wavefunction associated to the measurement act undergoes, just as in MWI, a superposition of outcomes. Only, the *particle state* of the measurement apparatus, which is a function of the wavefunction of the "device under test", its own wavefunction and its (unknowable) initial particle states, will go into a particular particle configuration. It is this particle configuration which is called "the outcome of measurement". If we would have called its wavefunction the "outcome of measurement" we would be in exactly the same situation as in MWI. It is the fact of focussing on the particle state of a measurement apparatus that we are able, in BM, to give a definite outcome to a measurement.
My question here: won't then the consequences also have an influence on the wavefunction, that is, even if the particle does not have an influence on its own wavefunction, won't it, via the measurement, which can potentially drive any decision? I haven't yet found a text in the above book that would be specific on this question (not looking at the mathematics).
No, the wavefunction is strictly independent of the particle world. The wavefunction in BM follows ALWAYS the schroedinger equation (just as in MWI), and never undergoes any "projection" (which would in that case give to BM exactly the same conceptual problems as in CI: why a "different" dynamics for a "measurement" than for a "physical interaction" ?).
But in BM, the wavefunction branches that are not "in resonance" with the particle state will not influence (in most cases) the particle dynamics anymore. So, concerning the particle dynamics, we can, after a measurement, just as well consider a projection: it won't influence the particle dynamics (the quantum potential) much.
This is a bit similar as considering the EM interaction of one charged particle upon another: the EM radiation that has been emitted "away" from the other particle won't influence it anymore, so in a calculation of the motion of the particles, we can just as well ignore this EM radiation. But does that mean that it doesn't exist ? I ask the same question to the Bohmians: does the fact that certain branches in the wavefunction have no influence anymore on the particle dynamics imply that they don't exist ?
3. In general, it seems that the quantum potential is a "guiding principle", but not "just" a guiding principle. As such, it plays a major role in the book. But I personally see this as similar to the stars and planets in relation to gravity. The "real" thing are the stars and planets, but that doesn't mean that gravity isn't also real and important. (Even though B&H don't use this example.)
It is stronger than this, because (at least in a Newtonian frame) the gravity interaction can be DEDUCED from the particle dynamics. There is no dynamical state to gravity itself. But not so for the wavefunction in BM: its dynamical state is NOT "recorded" in the particle states and hence, it lives its own life. But more so: in Newtonian dynamics, gravity is influenced by the particles. Not so in BM: the wavefunction is not influenced by the particles.
4. It does appear, in BM, with its emphasis on ontology, that "Hilbert space" specifically is regarded an arbitrary mathematical tool to *describe* that which eventually leads to the quantum potential, rather than thought of as a real physical space. However the quantum potential is thought of as related to a physical "force".
I know, but I think that this is a rethorical weakness of BM: IF one admits the ontological existance of the wavefunction itself, then the particle dynamics doesn't solve all of the conceptual difficulties of MWI, because that wavefunction with all its branches is "still out there". So in BM, one tries to minimise the ontology of the wavefunction and to maximize the particle ontology.