Lumo's attack on dBB continues

[Schmelzer]

With the post Bohmians' self-confidence evaporates as soon as they're expected to calculate anything continues his attack against dBB theory.  It contains, as usual, a lot of name-calling, now even personally against me - I'm named "crank", and a link to this forum posted in the comments was rewritten so that one cannot directly click it.  Whatever, this is what has to be expected.

He obviously meant that it was done in proper quantum field theory governed by the standard, "Copenhagen" postulates of quantum mechanics (at most reformulated with a different "accent" but not a different "content"). And because he must believe that Bohmian mechanics has "conquered" the standard quantum mechanics and may claim credit for all of the successes of quantum mechanics (while taking no responsibility for the alleged drawbacks), he just doesn't need to write anything, he believes.

Not that I believe that one does not need to write anything - what I believe is that one has to prove an equivalence theorem. Only if such an equivalence theorem is proven, one can use the computations made in one theory in the other theory too.

I agree, if one thinks that dBB makes somewhere different experimental predictions than standard quantum theory, then one has to make all the computations to find out where the predictions are indistinguishable and where one can distinguish them and find out which theory is true. But this is not the case. Where it is possible to construct a dBB theory, it is, in its experimental predictions, equivalent to standard quantum theory.

What are, in this case, the advantages of dBB theory? They are conceptual. The Copenhagen interpretation subdivides the world into a classical and a quantum part. Which is unproblematic from a pragmatic point of view, but not really nice. I prefer a theory without such an artificial subdivision. Which is dBB.

These Bohmian people often make claims such as "all the physics of QFT works just fine in Bohmian theory". References to incoherent preprints that make similar claims are the "evidence" you may get. All these preprints contain some "extra" (and very ugly) mathematical constructions that are absolutely different from the standard QM/QFT and it's obvious that they can't be producing the same predictions in general. But if someone claims that the Bohmian theory makes sense, shouldn't he be able to write the "proper modern [Bohmian]" textbook replacing the existing textbooks of quantum field theory? At least a few chapters, up to a calculation of some annihilation processes of QED.

The last time I have seen a really serious objection - serious enough to question the equivalence of dBB field theory and its viability - I have made some computations and published them, here is the reference:
I. Schmelzer, Overlaps in pilot wave field theories, Foundations of Physics vol. 40 nr. 3, 289-300 (2010), arXiv:0904.0764 [quant-ph].

The point of mentioning some, ugly or not, extra constructions is beyond me. Of course, once there is an equivalence theorem, there is no need to repeat standard QFT computations. And for a textbook about QFT, fine, a good idea. Except that there is no need to rewrite a lot of the QFT texts themself. In such a book I would start with the basic definitions based on the dBB interpretation, so that the initial part would indeed differ. But then I would focus on the development of the standard mathematical apparatus of QT resp. QFT. There would be some shift toward lattice regularizations, away from, say, dimensional regularization, because for lattice regularizations everything is nice, we have a well-defined dBB interpretation as well as a well-defined quantum lattice theory without infinities.

By its definition, the Bohmian mechanics must have a result for the measurement of the "photon position" that is ready before the measurement. Except that there can't be any equations – at least not local or otherwise natural equations – that could govern the motion of such "real Bohmian photons".

The point being? The dBB picture I prefer is not using photon positions as beables, but, instead, the EM field. There are problems introducing photon trajectories into a dBB picture? Fine, so don't do it.

Has anybody cared if one can define phonon trajectories in quantum condensed matter theory? Would this be an argument against a dBB variant for condensed matter theory based on trajectories for the atoms?

Well, it's simple. In quantum mechanics, the energy conservation follows from the collapse of the wave function and by the very definition of the Bohmian mechanics, Bohmian mechanics avoids the collapse at the moment of the measurement. In any Bohmian picture, the measured values must be already prepared a femtosecond before the measurement. ...
The Bohmian theory never collapses the atom's wave function to an energy eigenstate. In fact, in the Bohmian theories, the "real particle" doesn't influence the pilot wave at all!

No, dBB theory does not avoid the collapse, it describes the collapse, giving the evolution equation for the effective wave function of the subsystem in terms of the global wave function (which contains the macroscopic measurement device too) and the (macroscopically observable) trajectory of the measurement device, by the formula:
\[ \psi^{eff}(q_{sys}, t) = \psi^{full}(q_{dev}(t),q_{sys}, t).\]
As one can see from the formula, it is not the trajectory of the "real particle", which would be \(q_{sys}(t)\), which defines the collapse. But the trajectory of the macroscopic device, \(q_{dev}(t)\), which defines the result of the collapse. But don't forget that \(q_{dev}(t)\) is defined by the guidance equation, and, via the guidance equation, influenced by \(q_{sys}(t)\). And, indeed, the global wave function \(\psi^{full}(q_{dev},q_{sys}, t)\) is not influenced nor by \(q_{dev}(t)\), nor by \(q_{sys}(t)\). But the effective wave function \(\psi^{eff}(q_{sys}, t)\) already depends on \(q_{dev}(t)\), thus, is influenced by the "real particle" \(q_{sys}(t)\) too.

This is a rather brutal feature of the Bohmian theory: the theory is very loud about the influence of the pilot wave on the particle but it basically assumes that the particle doesn't affect the pilot wave at all. You should always be suspicious about theories with similar "asymmetric" influences. They sound like a theory about God who can influence everyone else but can't be influenced. In proper physics at the fundamental level, all interactions go in both ways.

This is, indeed, a strange feature of dBB theory. One which is worth to be considered and discussed. But it is, in fact, only relevant for a hypothetical, theoretical entity: the wave function of the whole universe.

The effective wave function of a subsystem is, as we have seen, influenced by the trajectory, whenever the subsystem interacts with its environment.

The absence of the collapse in Bohmian mechanics means that the atom can simply never collapse to an energy eigenstate, even though the photon that has known about the atom's energy has gone through the prism and was detected. The wave functions of the Bohmian mechanics never really collapse.

But the effective wave function of the atom collapses. It is only the wave function of atom + EM field, and later of atom + EM field + prism + detector, and a later the wave function of the whole universe, which does not collapse.

And the positions of the electron and proton aren't affected by the detection of the photon – even though they should really be correlated with the photon's energy.

That's wrong. Once there is a nontrivial interaction between the atom and the EM field, the trajectory of the configuration of the EM field is influenced by the trajectory of the atom. How? By the guiding equation. Because to define the velocity \(\dot{q}_{atom}(t)\) by the guiding equation, we need the full wave function \(\psi^{full}(q_{EM},q_{atom}, t)\) as well as the actual configurations \(q_{EM}(t),q_{atom}(t)\) of all relevant parts at that moment of time.

Of course, the influence matters only as long as there is an interaction between the system and the measurement device. If there is no such interaction, and \(\psi^{full}(q_{dev},q_{sys}, t)=\psi^{dev}(q_{dev},t) \psi^{sys}(q_{sys},t)\), there will be no such influence between the two trajectories. There is also some influence if the two systems are in a superpositional state \[\psi^{full}(q_{dev},q_{sys}, t)= \psi_1^{dev}(q_{dev},t) \psi_1^{sys}(q_{sys},t)+\psi_2^{dev}(q_{dev},t) \psi_2^{sys}(q_{sys},t).\]
If, say, the wave functions \(\psi_{1/2}^{dev}\) do not overlap, and \(q_{dev}(t)\) is inside the support of \(\psi_{1}^{dev}\), then the trajectory \(q_{sys}(t)\) will be the same as if guided by \(\psi_1^{sys}\), else as if guided by \(\psi_2^{sys}\). So, there is an influence.

All this has nothing to do with any "ad hoc fixes", it is simply the application of the standard equations of dBB theory.

---

PS: I see that secur has been banned there too:

BTW I saw your explanations on Schmelzer's crackpot website how you play with your nicknames and why you came to my server. This is in clear violation of the basic integrity rules required on this server so you were banned.

Looks quite natural: If one has only bad arguments, one has to ban those who may easily refute them.

Last but not least, there was a time when he felt more secure about his arguments, when he has given me the participate on his blog with an Argumentation about de Broglie-Bohm pilot wave theory. He has, it seems, learned the lesson that his arguments are too weak, so that he cannot allow such counterarguments on his blog.

I can. I have no problem with lumo coming here for "debunking" all this "crackpot nonsense" here. He would have to restrict himself to arguments about the content, personal attacks would not be allowed, this is all. I would guess, if he has some arguments about the content, he would make them here too. But if there are no such counterarguments about the content, it would be unreasonable for him to appear here. We will see

[Schmelzer]

I have found here a strange quote from lumo:

BTW as I said many times, Feynman could have sold the path-integral formulation as something much more, a new interpretation that is meant to "replace" Copenhagen, fix it, and so on. But he was modest and he knew better which is why he preferred to say that it was physically equivalent - and he could give a proof. The physical equivalence depends on a careful isolation of what is physical/measurable and what is just a calculational/linguistic convention. Many others ("interpreters") are much less careful about the "physicality" yet much less modest.

Strange, because this is what is done by dBB theory too: For those cases where one can construct a dBB interpretation, it is, in the quantum equilibrium, physically equivalent to standard quantum theory. And they can and do prove it. Without such a proof - which was the actual situation between de Broglie's presentation at the Solvay conference 1927 and Bohm's paper 1952 where the equivalence proof was given - dBB theory would be as dead as possible for a physical theory.

At http://physics.stackexchange.com/, where I cannot post too simply because I have created an account only now and therefore do not have any reputational points yet, I have found some other points worth to be replied to:

The more realistic field theories that we would wish to discuss, simply have not been defined mathematically in a way which would allow concrete Bohmian calculations to be exhibited. Practical QFT can ignore that constraint because of the EFT philosophy, but Bohmian field theory cannot.

Why this? The EFT philosophy is nicely compatible with dBB theory.

What I propose is to consider lattice regularizations of QFT. With a finite number of lattice nodes this fits nicely into classical quantum theory, and, as well, into classival dBB theory too. The relativistic Hamiltonian, regularized on a lattice, has the form \(H=\sum_n \pi_n^2 + V(\varphi)\), where the potential \(V(\varphi)\) depends only on the configuration variables, which is what we need to apply the standard construction of a dBB theory. So, we have a dBB lattice theory equvialent to the corresponding lattice QFT. So, as far as such a lattice regularization for QFT is effectively equivalent to QFT (however defined), we have also effective equivalence of that lattice dBB theory to that QFT (however defined). Of course, lattice regularizations are not unproblematic (fermion doubling, chiral lattice theory problems), but these are problems one would like to solve anyway to get a correctly defined QFT based on lattice regularizations.

Gauge symmetry might be a more serious problem. Bohmian mechanics can deal with the problem of special relativity by just picking a reference frame and saying that's the real one. That's a kind of gauge-fixing and it's going to work. I have no similar confidence that gauge-fixing would work for "Bohmian gauge field theory".

First, for the classical gauge theory there is no problem at all, for the gauge condition we have a very natural candidate, the Lorenz gauge \(\partial_\mu A^\mu = 0\). Classically, there is no difference.

Then, in quantum theory, there may be some differences, related with Gribov copies. But so what? This would be a point where it would be interesting if this would really lead to observable effects or not, and, if it leads, which theory is better.

But this is about another hidden variable - the gauge potential.

I know that Lubos blogged on another occasion that Bohmian mechanics absolutely couldn't deal with fermion fields, because they are based on Grassmann variables and you can't have "Grassmann beables". I don't know if that argument is valid; if it is, maybe you could still get by just with beables for the bosonic fields;

About how to handle fermion fields, see my post A dBB theory for fermion fields. But it is worth to note that an incomplete configuration space - say, based on bosons only - would be sufficient for all practical purposes too. What one needs to prove an equivalence theorem is, essentially, only that different macroscopic states of the measurement instruments can be distinguished already by this incomplete part of the configuration. I have to admit that I have never liked this idea and cannot give now a good reference to this or provide the relevant formulas, but I have seen papers where this has been proposed, as simply for particles with spin, where only the position was used, and for field theory, where only bosons were used.

[Schmelzer]

Wow, another continuation of the attack titled Bohmists' inequivalence & dishonesty.

The physically interesting part of the answer I have put into a separate thread.

I have to thank him for providing the link to http://www.bohmianmechanics.org/, nice site, I have not seen this link before. And, good work, if one wants to discredit a site, the best place to look for would be something satirical, like the following "advice for Bohmians":

Relax about it. It is unlikely that you will ever convince anyone. The result either will be “You can’t say that!” or “Sounds nice, but there must be something wrong with it otherwise the physics community would embrace it.” Seriously. You are not going to win an argument.

Still here? Wow, you are a glutton for punishment.

and try to find an objectionable joke, which can be presented out of context as evil.

Motl has changed his policy toward this forum, and linked it himself. Fine. The point for linking it was quite laughable: To support a conspiracy theory that secur and me are the same person. Together with some user7348 on stackexchange. LOL. To fake users to create the impression that there are more supporters of dBB than in reality? Sorry, this is something I leave to democrats, who think that the number of supporters of some nonsense matters. But even if I would, it would be more plausible to invent a supporter of my ether theories, where I'm really alone yet, instead of some dBB supporters, which is, in comparison, mainstream with a lot of supporters. Whatever, it is not an argument about physics at all.

Then there is a link to an, it seems, unpublished paper Pisin Chen, Hagen Kleinert, Deficiencies of Bohmian Trajectories in View of Basic Quantum Principles, which is so dubious that even Motl doubts the main technical claim. Then, we see a string theorist lamenting about the Bohmist referees rejecting anti-Bohmian papers:

By the way, it seems clear to me what happens when people – including Hagen Kleinert, a collaborator of Feynman in his last years and a co-author of an ingenious path-integral solution to the hydrogen atom – send a paper criticizing Bohmian mechanics to a journal. The editor almost certainly sends such a paper to some Bohmist referees. And what a surprise, Bohmists are dishonest Marxist aßholes who will prevent the publication even if they know that the paper is correct. The very existence of this would-be "subfield" of physics is a problem that should have been prevented.

I have emphasized the last sentence, because I think this describes the main difference between his (string-theoretic?) view and my own: I would not object against the very existence of string theory, even if I think this direction is an impasse. He thinks that the very existence of alternatives is a problem which should be prevented.

Then, there is a lot of argumentation about some thermodynamical arguments. In principle, I had planned to answer them, but I have seen that this requires more work, because I would have to cover also a lot of the conceptual foundations of thermodynamics. The problem is that about these foundations I tend to favor now the modern, Bayesian approach, described, for example, by Caticha, Entropic Inference and the Foundations of Physics, USP Press, Sao Paulo, Brazil 2012. So I guess this would open another can of worms of disagreement, now about what is entropy.

Whatever, once the microscopic theory is proven to be equivalent, the macroscopic approximation has to be equivalent too. If not, there is some error in the macroscopic approximation.

After this, Motl quotes some thread on stack exchange without giving a link. So, here is what he has not quoted.

I think it is possible to reduce it to the quantum mechanical calculations. I would acknowledge that what I have written here is an oversimplification, and that this problem needs a more detailed discussion. Of course, the von Neumann entropy is 0 for a pure state, but a measurement transforms a pure state into a mixed one, and a position measurement would transform it into a state such that the entropy I have given would be the von Neumann entropy, so it is something close enough. The details have to be discussed. But I doubt here would be the right place. – Schmelzer

Ilja, none of the statements you are making is correct. For example, in your latest comment, it is not true that a "measurement transforms a pure state to a mixed state". Where did you get this misconception? Your formulae for the entropy aren't "oversimplifications". Instead, they're so wrong that they have nothing at all to do with the correct result. Just try to calculate the numerical magnitude of the heat capacities - which you were supposed to do, anyway. You will see that they have nothing to do with the tiny measured heat capacities. – Luboš Motl

If one ignores the result of the measurement, the result is a mixed state. It is the same mixed state which you can predict before the measurement, if all you know is that the measurement will be done. – Schmelzer

But, of course, if one would quote such things as me saying "Of course, the von Neumann entropy is 0 for a pure state", one could no longer make such claims as

Instead, the entropy is S=0 for any pure state |ψ⟩|ψ⟩ while for a mixed state, we must use the von Neumann entropy ... You may see that Schmelzer doesn't understand statistical physics (and probably quantum mechanics) at all. He confuses pure states and mixed states (mixed states are absolutely needed to meaningfully discuss any nonzero values of entropy etc.) and does many other stupid things.

So, I can understand that Motl preferred not to link the discussion itself.

[Schmelzer]

And yet another attack by Lubos Motl: "Slowly for Peter Shor: Page 1 of Dirac", this time against Peter Shor. The case is rather funny: Motl thinks that some quotes from Dirac provide a sufficient argument against any classical theory. But even the consideration of the context raises doubt: Dirac has written this text at a time when de Broglie has given up his pilot wave theory (because he was unable to find the measurement theory which has been found later by Bohm). So, that he thinks that a return to a classical worldview is impossible is a triviality - else, he would have published a paper containing Bohm's theory or something similar himself. In such a situation, one has to expect that there is also some problem on the way which seems unsolvable. And that one, if one writes a textbook, writes about this problem.

The Dirac textbook quotes themself are irrelevant about dBB, no wonder, given they were written not about dBB by someone who did not even know that something like dBB can exist at all. In the remaining text I have found nothing that has not already been answered here. Or, maybe this?

I must emphasize that if you extend the atom's |ψ⟩ to the quantum field theory's |Ψ⟩ that you rebrand as a "second-quantized pilot wave", you may obviously extend the theory to "mathematically coincide" with the whole quantum field theory. But the meaning of the symbol |Ψ⟩ is completely different than in quantum field theory and if you want to preserve the "realist" i.e. classical character of your theory, you must ultimately say that some observable quantities are functions of the classical degrees of freedom |Ψ⟩.
In this way, you're just postponing the moment when you admit that it doesn't work at all. At the end, you know that the only correct interpretation of the complex numbers in |ψ⟩ or |Ψ⟩ is that they are probability amplitudes that determine the probabilities of otherwise random outcomes, via the Born rule. But Bohmian mechanics ultimately wants to say that |ψ⟩ or |Ψ⟩ are classical degrees of freedom that should be observable "directly", in a single experiment.

Hm, what dBB-like theories explicitly say is that the configuration is what we immediately observe around us. The status of the wave function is much less clear. But the majority opinion is clearly that the wave function is a or defines some really existing object.
But Bohmians are certainly not primitive positivists, who refuse to give something the status of existence if they are unable to observe it directly. Do they want to? I would guess, nobody would object if it would be given for free, but I doubt anyone would really try hard to reach this. What one wants to have is a consistent set of equations, where the future of all the really existing things is defined by really existing things only, instead of ghosts or Gods or so. So, once the evolution equation for the really existing configuration depends, via the guiding equation, on the wave function, there is a point to award the status of real existence to the wave function too.
But, let's also note that this is not at all obligatory. All the wave functions which appear in real life are, from point of view of dBB theory, only effective wave functions. Now, effective wave functions are defined by the wave function of the universe by

\[ \psi^{eff}(q_{sys},t) = \psi^{universe}(q_{everything\, else}(t), q_{sys}, t)\]

thus, it is, by definition, something which depends not only on the wave function of the universe, but also on trajectories \(q_{everything\, else}(t)\) which, without doubt, are real, and, moreover, define (via the preparation procedure) the effective wave function completely.
Whatever, it does not matter at all what the status is - ontological, epistemic, or some strange mixture of above. Once Bohmists are not positivists, they will not require a direct observability of the wave function.

Or, maybe, I have misunderstood Motl here? Maybe he sees a conflict that QM defines via the Born rule probabilities only, but dBB assumes the wave function is real? This interpretation would require that Motl has not understood that in dBB theory there is a difference between the observable probabilities for trajectories and the wave function, and that the Born rule holds only for a particular subclass of dBB states, namely states in quantum equilibrium. So, there is no such conflict at all. Probabilities for trajectories are one thing, the wave function is another thing.

At any rate, Bohmian mechanics is a theory where the pilot wave and the particle position are equally "real". So if a physical system reaches the equilibrium, it spends an "equal amount of time" at every possible region of the total phase space.

So? I think there are a lot of assumptions about how classical physics looks like which one has to use to reach such conclusions. Motl uses here the term "phase space". Hm. Sounds like he presupposes that the equations are of Hamiltonian form, so that one can define a microscopic entropy \(S=\int \rho \ln \rho dp dq\) which is preserved in time during the evolution, not?

Another line of argumentation which can be easily seen to be invalid in this form, but which is worth to be commented:

In a classical theory describing the pilot wave |ψ⟩, a set of classical degrees of freedom, there is really no reason whatever to assume that the fundamental equations controlling these degrees of freedom are linear. Everything that is not forbidden is allowed. And because this non-linearity isn't reducing the consistency of this non-quantum theory at all, one must assume that it exists – both in the normal evolution of the pilot wave as well as in its impacts on other parts of the system that we use to measure the atom (e.g. the electromagnetic field). This is an example of Dirac's claim that quantum mechanics is far less arbitrary than classical physics: the linearity of the evolution and other operators (observable) is needed for consistency (because the linearity basically coincides with the linearity rules in the probability calculus) while no similar linearity constraint may ever be justified in a classical theory.

Why this is invalid? Because the fundamental equation for \(\psi\) is nothing derived from something more fundamental, it is simply postulated. It is the Schrödinger equation. End of discussion. (This is not string theory, it is physics, where we have well-defined equations.)

But does this mean that one would not prefer to understand why the equation is the Schrödinger equation? Of course. It would be nice to be able to derive the Schrödinger equation from something else, preferably something more simple, easier to understand or so. Of course, linearity is so easy to understand - for almost everything small variations are approximately linear - that it is not even a problem. But there is, of course, not only linearity which one would like to understand there.

And now note the possibility mentioned above - that the wave function is not something really existing, but epistemic, describing our knowledge of reality, not reality itself. Such an interpretation requires a very different consideration. In particular, in this case one cannot simply say that the guiding equation defines the velocity. Our knowledge about reality cannot define how reality behaves - except in wishful thinking. Thus, the guiding equation obtains a different status - it becomes a sort of consistency condition between the real velocity and our knowledge about it defined by the wave function. Or it also obtains a status of knowledge - as the average velocity for a given state of knowledge. There are logical consistency rules, which distinguish rational, consistent sets of knowledge. And if one assumes that the wave function describes such a state of knowledge, this clearly enforces quite rigorous restrictions on the equations of evolution of such knowledge.

And, indeed, these restrictions would be more rigorous than the restrictions which exist if one assumes that the wave function is some really existing object.

In this sense, assuming that the wave function is a really existing object is a cheap solution. If one uses it, almost every system of equations would be fine: All one needs for a realistic theory is that the evolution equations depend only on really existing (according to the theory) things. Which can be simply done, by giving all the items which appear in the equations the status of really existing things. This simple solution has it advantages - it produces a simple realistic and causal theory, to counter quantum mysticism, anti-realism and rejection of causality.

But it does not promise a deeper understanding, an explanation, of the nature of the Schrödinger equation.

[John Duffield]

"But it does not promise a deeper understanding, an explanation, of the nature of the Schrödinger equation".

I think it does. But what you end up with isn't de Broglie - Bohm theory any more. As for what it is, I don't know. But see https://arxiv.org/pdf/1103.3506v2.pdf on page 16 where you quote Bell?

"Is it not clear from the smallness of the scintillation on the screen that we have to do with a particle? And is it not clear, from the diffraction and interference patterns, that the motion of the particle is directed by a wave?"

The scintillation is not the particle, just as the eye of the storm is not the storm.

[Schmelzer]

Of course, what I will end up if the program is successful will not be dBB. It differs in such a central question as what defines the complete state of reality. It is for this reason I have invented a new name for it, paleoclassical interpretation, for this program.

Actually I think Caticha's entropic dynamics is quite close to this program too. So I am quite close to giving this own program up and supporting Caticha. But I have yet some ideas beyond Caticha. Not yet written down, but work in progress.

[John Duffield]

Don't give up your own program to support Caticha. See things like this from http://arxiv.org/abs/1601.01708 :

"Our goal has been to extend entropic dynamics to curved spaces which is an important preliminary step toward an entropic dynamics of gravity."

I'm confident that this is a step in the wrong direction.

[Schmelzer]

Of course, I do not have to support every movement in every direction.

[user7348]

I'm looking at Bohm (1952). On page 171 he outlines 3 ad hoc assumptions which are needed to guarantee equivalence with quantum mechanics. (3) says that the probability density of position is given by the usual square of the magnitude of the wave function. It then says that this is due to not knowing the initial conditions of the particle. I do not understand why the initial conditions of the particle are unknown. This seems like another assumption 3b. Furthermore, if the wavefunction is a position eigenstate which always occurs after a position measurement, then P(x) is 1 at a specific location and 0 everywhere else so that the precise location of the position is well defined which undermines the reasoning of (3).

[Schmelzer]

I'm looking at Bohm (1952). On page 171 he outlines 3 ad hoc assumptions which are needed to guarantee equivalence with quantum mechanics. (3) says that the probability density of position is given by the usual square of the magnitude of the wave function. It then says that this is due to not knowing the initial conditions of the particle. I do not understand why the initial conditions of the particle are unknown.

Yes, Bohm 1952 requires the assumption that the wave function is initially in quantum equilibrium.

But this is no longer an independent assumption. Today we have the Valentini's subquantum H-theorem, which shows that the quantum equlibrium will be reached in analogy to Boltzmann's H-theorem for thermal equilibrium. This has been also supported by numerical computations by Valentini and Westman. See https://en.wikipedia.org/wiki/Antony_Valentini for references.

Furthermore, if the wavefunction is a position eigenstate which always occurs after a position measurement, then P(x) is 1 at a specific location and 0 everywhere else so that the precise location of the position is well defined which undermines the reasoning of (3).

Of course, the wavefunction is not a "position eigenstate". dBB theory is a realistic theory, so it is assumed to describe what really exists, not what is measured.