Can you consider the particles in Bohmian Mechanics mixtures?
Not the particles - but the waves. De Broglie-Bohm theory (the modern term preferred to 'Bohmian mechanics') is a statistical mechanics of waves and particles. See lecture 3 of the http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" [Broken].
DeBB theory describes the likely state of matter as it actually is, whatever processes it may be part of. If you start talking e.g. about density matrices or mixtures this refers to our partial knowledge of a system which is in itself well-defined. You need to distinguish between the case where the components waves do not physically coexist (a 'proper mixture') and the case where they do physically coexist but do not overlap (an 'improper mixture').
Can actual particle position (the one that comes out of the guiding equation) be observed/measured or even influence anything at all? If yes, how? If not, what's is good for?
Yes, it can influence the motion of other particles. Moreover, it can do it even more "strongly" than classical particles can, in the sense that it can do it nonlocally, with a strong influence at an arbitrarily large distance distance. In fact, many people dislike that theory not because a particle cannot influence other particles, but because it can influence them "too strongly" in the sense above.
Ok, the trajectory of a particle influences and is influenced by all other particles in the universe. But these trajectories cannot be observed/measured directly, they can only be inferred from the results of a measurement (which again strictly speaking depends on the position of all particles in the universe and not just the particle we want to measure).
If we knew the entire initial state we could predict the evolution at any arbitrary moment of time and the uncertainty principle would not have applied. But since we do not know the initial state, the numbers are out of our reach. And we cannot increase our knowledge through experiments because every time we do a measurement we gain some knowledge but at the same time we introduce more unknowns through the degrees of freedom of measuring apparatus. In fact we cannot learn enough of it to increase out predictive power beyond the capabilities provided by wavefunction alone (w/o guiding equation).
Does this all make sense? Is this a fair summary?
To be honest, I sort of see how it all works but I still don't see a point. All these definite trajectories sound like an attempt to describe magnetic field of a permanent magnet by postulating the existence of invisible metal shavings that line up along the field lines
Yes, very good.
Pointing to be unable to see the point is a good point.
So let us concentrate on it.
If that was a good analogy, then it would be a good argument against Bohmian trajectories. But the analogy fails because in the quantum case there is no analog of the magnetic field. In particular, the wave function is NOT analogous to the magnetic field. Namely, the magnetic field can easily be interpreted as an objectively existing entity, while the wave function cannot (due to its probabilistic interpretation, or problems with collapse, or problems with the Born rule in MWI, depending on which interpretation you prefer). So if the wave function does not objectively exist, then what does? When we observe, WHAT do we observe? THAT is the question which the Bohmian interpretation tries to answer. THAT is the point of it.
The following may also change your mind:
I'm not prepared to debate the exact meaning of "objectively existing". However in order to predict anything in QM one needs to know the wavefunction but not the definite trajectories. This gives an indication which one is more objective.
Well, that's what I'm trying to figure out. One thing we do NOT observe are the famed deterministic trajectories. What we do observe instead are macroscopic pointer states which correspond to the trajectories but in a strange roundabout ways. Not to mention [STRIKE]white elephant in the room[/STRIKE] decoherence, einselection and the like which always [STRIKE]muddy the water[/STRIKE] accompany the process of observation.
I guess my complain is that we have a guiding equation that is supposed to produce nice clean definite trajectories at all times but it is impossible to use for the stated purpose because we never have sufficient input data. And even when we do have definite trajectories, the only way we can use them is to plug them back into the same equation where they immediately get tangled with unknown/random environment to become probability distributions.
The equation certainly makes sense, it helps visualise the flow of probability, I guess it can be handy in monte-carlo simulations etc.
But there is nothing in it that requires coordinates to have definite values at all times. They might as well be distributions.
If the only objective of quantum mechanics (or of science in general) is to PREDICT, then you are right - at the moment there is no much use of trajectories. But many scientists wouldn't agree that that's all what science is supposed to do. Many scientists want EXPLANATIONS, and that's where Bohmian trajectories may be useful and more powerfull than standard form of QM.
I agree, it may be helpful, but there is nothing that requires it. But at least it is a logical possibility worthwile to explore, isn't it?
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