> If you say particles with positions do not exist, then what exactly is the relevance of this space?
That's a good question. The real answer is that no one actually knows. I think this is actually the biggest mystery in QM. But let me start with this, because I didn't make myself clear:
> It is also odd to say that position cannot be measured.
When I said that Bohmian positions are an IPU I did not intend that to mean that particle positions can't be measured. Obviously they can. The IPU-ness of Bohmian positions has to do with their ontological status, not their epistemic status. On Bohm's theory, a particle position considered along some axis is a real (in the mathematical sense) value, which is to say, it contains an infinite amount of information. But this information cannot be accessed in the same way that information stored in (say) a book can. I can open a book, even a book with an infinite number of pages, to any page and start reading it, and having read any page, I can go back and read that same page again. The information stored in Bohmian positions doesn't work that way. The laws of physics somehow conspire to hide all that information so that it can only be accessed serially and non-repeatably. The first time you measure a particle's position you get the most significant bits of its position. Those are then lost forever. You can never measure them again. The next time you measure a particle's position you get the next most significant bits of what that particle's position originally was, and so on. But you can never go back and do a second experiment to verify that the result you got for any of your measurements was actually correct and not a result of experimental error.
So the much-vaunted determinacy of Bohmian mechanics is not a reflection of the determinacy of the underlying metaphysical reality. It is really nothing more than a rhetorical trick. All the randomness is still there, it's just "pre-computed" and stored in particle positions in a way that it can only be accessed so that the world behaves exactly as if it were "really random" (whatever that means).
This same kind of trick is made manifest in a thought experiment [https://www.mathpages.com/rr/s9-07/9-07.htm] proposed by Kevin Brown. He points out that, if pi is normal (which is almost certainly is) then all of the results of all experiments ever conducted could be produced by a "cosmic Turing machine" computing the digits of pi. (See the two paragraphs beginning with "Even worse, there need be no simple rule of any kind relating the events of a deterministic universe.") Bohmian positions have exactly the same ontological status as the cosmic Turing machine. Only the window-dressing is different.
> It also helps to ask you what is real in your theory. Are wave functions real?
That's a good question. The real answer is that no one actually knows. I think this is actually the biggest mystery in QM. But let me start with this, because I didn't make myself clear:
> It is also odd to say that position cannot be measured.
When I said that Bohmian positions are an IPU I did not intend that to mean that particle positions can't be measured. Obviously they can. The IPU-ness of Bohmian positions has to do with their ontological status, not their epistemic status. On Bohm's theory, a particle position considered along some axis is a real (in the mathematical sense) value, which is to say, it contains an infinite amount of information. But this information cannot be accessed in the same way that information stored in (say) a book can. I can open a book, even a book with an infinite number of pages, to any page and start reading it, and having read any page, I can go back and read that same page again. The information stored in Bohmian positions doesn't work that way. The laws of physics somehow conspire to hide all that information so that it can only be accessed serially and non-repeatably. The first time you measure a particle's position you get the most significant bits of its position. Those are then lost forever. You can never measure them again. The next time you measure a particle's position you get the next most significant bits of what that particle's position originally was, and so on. But you can never go back and do a second experiment to verify that the result you got for any of your measurements was actually correct and not a result of experimental error.
So the much-vaunted determinacy of Bohmian mechanics is not a reflection of the determinacy of the underlying metaphysical reality. It is really nothing more than a rhetorical trick. All the randomness is still there, it's just "pre-computed" and stored in particle positions in a way that it can only be accessed so that the world behaves exactly as if it were "really random" (whatever that means).
This same kind of trick is made manifest in a thought experiment [https://www.mathpages.com/rr/s9-07/9-07.htm] proposed by Kevin Brown. He points out that, if pi is normal (which is almost certainly is) then all of the results of all experiments ever conducted could be produced by a "cosmic Turing machine" computing the digits of pi. (See the two paragraphs beginning with "Even worse, there need be no simple rule of any kind relating the events of a deterministic universe.") Bohmian positions have exactly the same ontological status as the cosmic Turing machine. Only the window-dressing is different.
> It also helps to ask you what is real in your theory. Are wave functions real?
See http://blog.rongarret.info/2015/02/31-flavors-of-ontology.ht... for my answer to this.
> Many worlds ... involves a heck of a lot of IPUs.
Yep. That's why I'm not a big fan of the MWI either. See:
http://blog.rongarret.info/2019/07/the-trouble-with-many-wor...