# Maxim Raykin's “solution to the measurement problem” using infinitely many derivatives

Recently I was made aware of the following arXiv preprint by Maxim Raykin: Analytical Quantum Dynamics in Infinite Phase Space. As far as I understand it, Raykin's idea is to reinterpret quantum mechanics as a theory of ordinary classical particles moving around in R3 -- except with the twist that now the differential equation governing the particles takes as input not only their positions and the first derivatives of the positions, but all the higher derivatives as well. Once you decide to look for this sort of "evolution rule," Raykin says that you can find a not-even-very-complicated one that perfectly reproduces all the predictions of QM, at least in the (special?) case that the wavefunctions are analytic. Indeed, the differential equation that you get, describing the particle trajectories, ends up being identical to the guiding equation in Bohmian mechanics. However, the key difference from Bohmian mechanics is that here one makes no explicit reference to any "wavefunction" or "guiding field" -- only to all the higher derivatives of the "actual" particle positions.

On this account, Raykin wants to explain "quantum nonlocality" as arising from the fact that an analytic function is determined by its collection of higher derivatives at any point. The idea, I suppose, is that you can't specify all the higher derivatives of the positions of the particles in Alice's lab, without also knowing what's going on in Bob's lab. While Raykin doesn't discuss this, I imagine one could also explain the exponentiality inherent in (say) quantum computing, in terms of the fact that given n particle positions x1,...,xn, the number of tuples (k1,...,kn) for which one can form the combined derivative

$\frac{d^{k_1}}{dx_1^{k_1}} ... \frac{d^{k_n}}{dx_n^{k_n}}$

increases exponentially with n, even after a finite cutoff is imposed on max{k1,...,kn}. In any case, there has to be some explanation, since (just like in Bohmian mechanics) this account is specifically constructed to give exactly the same predictions as standard QM.

Look: if you follow the quant-ph arXiv, you see another revolutionary solution to the conceptual problems of QM trumpeted every week. Most of those solutions can safely be rejected on the ground that they contain no new idea, not even a terrible idea. But I was unable to reject the present idea on that ground: this way of thinking about QM seems bizarre to me, but it's not one that I've seen before, nor would it ever have occurred to me.

I can envision at least five possible reactions to Raykin's idea:

(0) There's some error that prevents it from working.

(1) There's no error, but it's a completely uninteresting triviality. Because an analytic function is completely determined by its higher derivatives at any point, Raykin's "differential equation" is really just an elaborate way of stating the tautology that a particle's future trajectory is completely determined by its future trajectory. Obviously, that would be true regardless of what the trajectories were (provided only that they're analytic): nothing specific about QM is being used here. Differential equations that depend on infinitely many derivatives simply lack predictive power. And as for the use of analytic functions' global nature to "explain quantum nonlocality" --- it's no better an explanation than Gerard 't Hooft's "superdeterminism." In both cases, the cosmic conspiracy one needs to posit, with every particle's trajectory mysteriously constrained by every other's since the beginning of the universe, is astronomically worse than the relatively-benign nonlocality (e.g., Bell inequality violations) that one was trying to explain in the first place.

(2) Sure, the fact that QM can be formulated in this way is a tautology, but it's a kind of cute tautology, one that could conceivably provide useful insights.

(3) This is a genuinely new reformulation of quantum mechanics, in the same sense that (say) Bohmian mechanics was -- so it's important in a similar way, regardless of whether you like it philosophically.

(4) [Raykin's own view, apparently] This is the true, unique solution to the measurement problem, which resolves all known conceptual problems with QM, is a necessary stepping-stone to quantum gravity, etc. etc.

Of the five views above, the only one that I feel extremely confident in rejecting is (4). Since Raykin's paper appears to have gotten zero (public) reaction so far, my question is the following:

Does anyone else have thoughts or observations that would support or rule out reactions (0), (1), (2), or (3)?

-
I agree with Ron it can't be a new thing. The wave function must be there and act as a pilot wave – he may at most use different words for it (e.g. claim that the laws of physics are affected by it, instead of just the guiding wave). If you only had $x(t)$ etc., it simply couldn't remember the information about a function of many variables, the wave function. Perfect smoothness can't save you: we're usually considering perfectly smooth trajectories (and in many cases, wave functions), anyway. –  Luboš Motl Sep 23 '12 at 6:25
Thanks Luboš, but your answer seems to disagree with Ron's on a central point! Ron says this is equivalent to Bohm since from the infinite derivative data you can reconstruct the wavefunction, whereas you say that x(t) etc. can't remember the information about the wavefunction. It would be interesting to know who's right (or whether you're right under certain assumptions and Ron under others). –  Scott Aaronson Sep 23 '12 at 14:56

Thanks, Scott for posting your entry and for positive comments in it. One small note is due here: you are writing about first and higher derivatives of position (which may be erroneously understood as, say, first and higher derivatives of coordinates over time) while the derivatives discussed in the paper are the derivatives of an action function with respect to coordinates (as in the expression in your post). Just wanted to clarify it to avoid a possible confusion.

Now to the main question of your post: is my theory a rediscovery of de Broglie-Bohm theory (DBBT) (or “exactly the same as DBBT” as Ron put it) or anything else? Let’s compare. I will discuss a one-particle case for simplicity. Then the Bohm’s story goes like that: The physical reality consists of two things – the wave and the particle. The wave is described by the wave function, and the particle – by its position q, so that the state of such one-particle system is described by the wave function and the particle’s coordinates. The wave function evolves according to the Schrodinger equation, while the particle moves along the trajectory q(t) with a velocity given by the “guiding equation” v=h*grad(Im(log(psi)))/m (this is a modern version of DBBT, usually called “Bohmian mechanics.” It is essentially equivalent to the original Bohm’s version with the “quantum potential”). Along with the Schrodinger equation, the guiding equation is a separate and independent postulate of the theory. If we additionally postulate the initial probability distribution of particle’s position, then the experimental predictions of conventional quantum mechanics (QM) will be reproduced.

Now my story. The physical reality consists of one thing – the particle itself. The state of the particle is described by its coordinates and momentums, the evolution of which is described by first-order ordinary differential equations of motion (ODEOM) expressing time derivatives of coordinates and momentums through the derivatives of some Hamiltonian function. These ODEOM may be obtained from the variational principle, as a condition of a stationarity of the action that is equal to an integral along a trajectory from the Lagrangian function, a Legendre transformation of the Hamiltonian one. Now if we have a family, or ensemble of particle’s trajectories satisfying ODEOM, originating in every point of space, and characterized by the dependence of initial momentums from initial position, and if for every trajectory of the family we calculate corresponding action, then we will obtain the action function that describes the dependence of so calculated action from the final point of trajectory and the time when this point was reached. This action function satisfies some partial differential equation (PDE). If solved, this PDE may be used for obtaining information about the particle’s motion. Namely, it may be proven that the particle’s momentums in every point of every trajectory of the family are equal to corresponding space partial derivatives of the action function; as particle’s velocity is expressed through momentums by ODEOM, we thus obtain the expression for velocity in every point through the derivatives of the action function, and then by integrating velocity we can obtain trajectories themselves without solving ODEOM.

Here I must say that I agree with Ron that this sounds “exactly the same as” something very familiar. I just don’t think that this “something” is Bohmian mechanics; for me, it is rather the classical one. For those who didn’t see the paper (http://arxiv.org/abs/1204.1540 hereafter called AQD) I’ll now resolve the mystery. It is shown in section 2 of AQD, that there exists a generalized Hamiltonian formalism (GHF) that connects a large class of PDEs, interpreted as PDEs for an action functions, to corresponding ordinary differential equations (ODEs), interpreted as ODEOM, in the way described above. Like in a classical Hamiltonian formalism (CHF), in this generalized formalism ODEOM define the motion uniquely (as they are just first-order ODEs!) without any mentioning of corresponding PDEs, but the full theory that includes both ODE and PDE sides is more powerful, rich and interesting. As was described above, in GHF, as in CHF, the momentums on trajectories which belong to some family are equal to space partial derivatives of the action function that describes this family, but the difference is that in GHF the set of these space partial derivatives includes all of them, rather than only first derivatives as in CHF. Consequently while in CHF the number of momentums is equal to dimensionality of configuration space, in GHF this number is infinite, and each momentum is identified by multi-index that specifies to which partial derivative of the action function this momentum will be equal to if we decide to include the action function into consideration (but we may also decide not to!). Let me define the rank of momentum as an order of corresponding partial derivative; the usual momentums of CHF will then have rank 1, while in GHF we have momentums of arbitrary rank. The Hamiltonian function is a function of (time, coordinates and) momentums, and it is demonstrated in section 2 of AQD that there are two variants of GHF. If the Hamiltonian function depends on momentums of a rank higher than 1, then by ODEOM the expression for time derivative of momentum of any given rank k includes the momentums of ranks higher than k, and so ODEOM in this case are the infinite system of equations for the infinite series of momentums. As was just said, the solution of this system always exists (at least locally) and is unique. If the system describes something physical, then nature knows how to solve it; even we mortals can do it although not without pain – see, say, “Quantum Mechanics” by Landau and Lifschitz which is full of examples of reconstruction of Taylor series from the equations on their coefficients. If, on the other hand, the Hamiltonian function depends only on momentums of the rank 1 (i.e. on the first derivative of an unknown action function on the PDE side of the story) then ODEOM express the time derivative of momentum of a given rank k through momentums of the same and lower ranks (and also, in general, through time and coordinates). Consequently, the usual momentums (of rank 1) and coordinates are ruled by ODEOM that decouple from equations for momentums of higher ranks and so may be solved separately. These decoupled ODEOM along with corresponding PDE constitute CHF, which therefore is a subtheory of GHF for Hamiltonians depending on momentums of rank 1, which in turn is a special case of the whole GHF for Hamiltonians depending on momentums of arbitrary rank.

To return to physics we note that classical mechanics is the theory of the second kind. The corresponding equation on the PDE side of the theory, i.e. the Hamiltonian-Jacobi equation, depends on the first derivatives of the action function, and correspondingly the ODE side of the theory reduces to the finite set of ODEOM, namely, to the Hamilton equations. Our understanding of classical mechanics is that the Hamilton equations constitute the physical basis of the theory, while the action function is an interesting and useful but purely mathematical entity, which allows obtaining solutions of the Hamilton equations without really solving them, by solving instead the Hamilton-Jacobi equation as was described above. Now for QM, the equation for the logarithm of the wave function happens to belong to the class of equations covered by the theory of section 2 of AQD, but in this case the Hamiltonian function depends on the second derivatives of this logarithm. An inevitable idea that then emerges is to treat this logarithm as an action function in quantum theory, and investigate the theory based on a corresponding infinite set of ODEOM. This is what I did in AQD. Note that ODEOM are obtained from the equation for the logarithm of an action function (which I called Quantum Hamilton-Jacobi equation or QHJE) unambiguously, in particular unambiguous is an equation for velocity, and so in my work this equation is not postulated, but derived. This equation happens to coincide with the guiding equation of DBBT; consequently the trajectories of particles in AQD coincide with Bohmian trajectories. The whole Bohmian mechanics now appears as a mathematical method of finding solution of an infinite system of ODEOM (which are considered as the physical basis of the theory, exactly like the Hamilton equations in classical mechanics) without really solving them, in full analogy with the Jacobi method in classical mechanics.

To summarize, although I started with different physical ideas, and ODEOM in AQD are obviously new, the particles in my theory move along the same trajectories as in Bohmian mechanics. The related physical picture, however, is completely different, and as I just wrote the guiding equation is not postulated but derived. So does this all mean that I rediscovered Bohmian mechanics? While with Bohmian mechanics my theory shares the particle’s trajectories, with conventional QM it shares the equation for the wave function (i.e. the exponent of an action function) – the Schrodinger equation. Does it mean that I also rediscovered QM? And the whole structure of the theory is the same as that of classical mechanics. Should I conclude that I rediscovered classical mechanics as well? I think the question of novelty is not that important and interesting. Let me formulate what I think is an important and really interesting question. I argue in the introduction to AQD that QM should be considered a semi-phenomenological theory (like some other great theories of the past, such as Kepler’s laws, thermodynamics, Mendeleev’s periodic table, or Ginzburg-Landau theory of superconductivity). I believe that really fundamental theory should give satisfactory answers to all conceptual questions, something that conventional QM fails to do. One such unresolved conceptual issue is a measurement problem, which Scott used in the title of his post. Another is the nature of nonlocal correlations between space-separated, but entangled particles: although QM correctly predicts the value of correlations, it doesn’t explain the mechanism that produces them, and we don’t reward a student with “A” for a correct answer without any hint on the solution. Yet another correct answer that came out of the blue is the standard expression for the probability distribution – if there is any randomness, the goal of every fundamental theory should be to derive corresponding probabilities, not postulate (which means – take from experiment) them. Thus we are in a search of a fundamental theory, from which QM will follow, and which is supposed to resolve all these conceptual problems. Presented in AQD is an attempt on such a theory, which seems to resolve all above (and others not mentioned here) issues (in particular the form of the standard quantum probability distribution is shown to follow from the theory’s ODEOM in the same way as the form of the microcanonical distribution follows from the Hamilton equations in classical statistics). This theory employs elements of other theories, but presents them in a different view and uses them in a different way, and so objections to these previous theories may be inapplicable to mine. Consequently, rather than discuss whether my theory is completely or incompletely new, I’d like to ask if this theory taken as itself does indeed resolve the conceptual issues of QM, whether this resolution appears natural and aesthetically acceptable, and whether the whole thing appears to fit plausibly into the structure of theoretical physics. And if not, then what, where and why?

-
The only difference with DeBroglie-Bohm is in philosophy, unfortunately. I am sure you are well intentioned, and that you got this from independent motivations, but the philosophical interpretation of deBroglie-Bohm is not so important, the computational ingredients that go in and the way it reproduces quantum mechanics is the most important thing, and in this, I can't see any real (meaning operational) difference between your theory and Bohm's. This is not a criticism of your efforts, I appreciate rediscovery, but call it what it is. –  Ron Maimon Sep 25 '12 at 9:00
This is very interesting. If it has something in common with dBBT, this doesn't pull down AQD. In fact, even if it would be a "rediscovery" of dBBT, it would add important value to it. I personally don't believe dBBT, but I consider it important, and not only philosophically and historically. And AQD seems to remove many concerns about dBBT. Some seem to react like "I'm not impressed, nothing new, shut up and calculate, etc". Their loss;) –  Cristi Stoica Sep 30 '12 at 19:05
@ CristiStoica: Thanks, Cristi. That’s exactly the point: even if we start the consideration of AQD from DBBT (which is absolutely not the angle I see it from!) we will find that “AQD seems to remove many concerns about dBBT” as you put it. My hope is that in fact it removes so many concerns about DBBT (and about conventional QM as well!), that there remains none. At least I don’t see any, and my question to everybody is if they do – I would really appreciate specific examples! –  Maxim Raykin Oct 1 '12 at 17:42

This is not a new thing--- it's exactly the same as DeBroglie Bohm theory. If you look at DeBroglie Bohm, the equation of motion has the wavefunction in it, so if you look at the initial jet (the position and all its derivatives), you uniquely determine the wavefunction and all its derivatives (at least along the trajectory, but probably over all the configuration space), and under the condition of analyticity, you can reconstruct the wavefunction from the infinite derivative data.

This is not surprising at all, the motivation of deBroglie Bohm is extending Einstein's identification of the Hamilton Jacobi function with the semiclassical log of the wavefunction, it's the same motivation. So it's a reformulation. It is noting that within Bohmiam mechanics, the particle positions and infinitely many derivatives are equivalent to the wavefunction information. I would guess that this is a genuine rediscovery of Bohm, rather than a rip-off, just from the amount of effort here.

-
Thanks, Ron. This was my first reaction too on reading the paper: "well, it looks like he's just rediscovered deBroglie-Bohm theory!" Later, after discussing with Raykin, I decided to be more charitable and say that at worst, this is an "alternative interpretation of the deBroglie-Bohm interpretation." For example, Raykin claims that getting rid of explicit reference to the wavefunction makes the compatibility with special relativity much more manifest than it is in ordinary Bohmian mechanics. I'm skeptical of that but honestly don't know. –  Scott Aaronson Sep 23 '12 at 15:07
More generally, a "genuine rediscovery of Bohm" from different motivations sounds pretty interesting in itself! But you claim that this is a rediscovery of Bohm from the same motivations. –  Scott Aaronson Sep 23 '12 at 15:09
@ScottAaronson: I don't see any difference between his theory and DeBroglie-Bohm--- the wavefunction still appears as the solution to the quantum Hamilton-Jacobi equation, he is just claiming that you don't need to know it off trajectory, and so what. You still know it obeys the Schrodinger equation, and it's still just as nonlocal and hideous. To make a relativistic Bohmian mechanics, you do Bohm on fields, but you do it in a preferred frame which defines the wavefunction. You don't do particle paths. Same work, same motivation, but a lot of independent work, so the guy's probably annoyed. –  Ron Maimon Sep 24 '12 at 6:35
Maxim Raykin's theory seems to have some advantages over de Broglie-Bohm's. So, if they turn out to be equivalent, this improves de Broglie-Bohm's theory. Why making this equivalence to be something bad about Maxim Raykin's theory, it seems to me it is something positive. And, to avoid any suspicion, I am not Bohmian ;) –  Cristi Stoica Sep 30 '12 at 16:53
@CristiStoica: There are no advantages--- it's the same theory. It's only a semi-new philosophy regarding the status of the wavefunction, something which Bohmians sort of believe anyway, that it's just a generalized Hamilton Jacobi function. But a theory is defined by how you simulate it, and simulating this is equivalent to simulating Bohm. –  Ron Maimon Sep 30 '12 at 20:06

To Ron’s comment. First, about the opening statement (My confidence is from the fact that if I wanted to simulate your theory on a computer, I would end up simulating Bohm):

1. This is the same discussion of a novelty question. It seems I failed to make myself clear again. Another attempt: whether or not my work is a “rediscovery of Bohmian mechanics” is not an issue. My theory has something in common with Bohmian mechanics, but does not include all of it, and adds something new, so these two theories are different. My statements are, first, that this difference makes my theory physically acceptable and, second, that my theory resolves all problems listed in the introduction to my paper. The really interesting/important questions are, first, whether the above two statements are correct, and, second, whether there are other conceptual problems in quantum mechanics which I forgot to include and which cannot be resolved in the framework of my approach. Besides,
2. It is not a fact that you necessarily end up simulating Bohm. You may also solve the equations of motion analytically. The infinite number of equations means only that the solution should be searched in a form of a formula, which will turn all equations into identities. I already referred to “Quantum mechanics” by Landau and Lifschitz, where many examples of so obtained formulas may be found. Moreover, whether we, humans, may solve these equations or not is not so important: they are first order ordinary differential equations, which have one and only one solution that defines the motion unambiguously; what really important is that nature knows how to find this solution.

Second, the rest of the comment is a speculation about my future work. Is this speculation reliable? What if I’ll find a different way to proceed? What if I won’t be naïve? I think it will be more productive to discuss the work which is already finished. In section 9 of my paper I present an explicit model which (a) is nonlocal, i.e. allows instantaneous influence over the space, and (b) is frame-independent. Are there any problems with this model?

-

Ron, I afraid I didn’t make myself clear. Let me try again. The discussion of a question whether my theory is completely or incompletely new, is or is not a rediscovery of some earlier theories would make sense only if I claimed that it is absolutely new and includes nothing from existing physics. Since I, of course, don’t make any such claims, this discussion is pointless. Obviously and inevitably, my work intersects with some previous theories and adds something new (but it is not an eclectic mixture either; the essence of my theory is in a generalization of the classical Hamiltonian formalism, the generalized formalism being able to give Hamiltonian description to both classical and quantum mechanics as two specific examples of the general theory. These two examples differ only by the form of their Hamiltonian functions and, therefore, by resulting equations of motion, the difference between classical and quantum physics being tracked down to this difference between equations of motion). As I wrote, since my theory only intersects with previous ones and does not include them as a whole, the objections against these previous theories may be inapplicable to mine. That’s why I asked to base a possible criticism on my work and only on it. Your comment is a perfect illustration of this point. Your opening sentence “The only difference with DeBroglie-Bohm is in philosophy, unfortunately” contains two statements I disagree with. First, de Broglie-Bohm theory (DBBT) is a rare example of a situation where philosophy is really important and works, I think, against it, so I consider the philosophical difference of my theory with DBBT as something very welcomed and not unfortunate at all. Let me explain. The classical limit of Bohmian mechanics is as follows: the physical reality consists of particles and, independently, some function (i.e., physical field) S (which is a classical action function of course). These entities evolve in the following way: the field S is governed by the Hamilton-Jacobi equation, and then the velocity v of the particle is equal to grad(S)/m. It is of course a philosophical prejudice which forces me to prefer to this description of classical mechanics the usual one where the evolution of particles is described by Newton’s or Hamilton’s ordinary differential equations of motion (ODEOM), while considering the statement above to be simply a formulation of a mathematical method (namely, Jacobi’s) of indirect solution of these equations. This prejudice, although mere philosophical, is nevertheless so dear to me that it alone was always sufficient to prevent me from considering DBBT as a satisfactory alternative to conventional quantum mechanics. Now with the new interpretation of the wave function as an exponent of the action function developed in my work (by the way, as I argue in the introduction to my paper, some nonstatistical interpretation of the wave function should be present in a fundamental theory but is absent from conventional quantum mechanics) the philosophical objection to Bohmian mechanics in the quantum regime becomes identical to the objection to its classical limit: DBBT declares a purely mathematical entity, the action function (or its exponent, the wave function) to be an element of physical reality, and promotes a (generalized Jacobi’s) mathematical method of solving ODEOM into the status of a physical theory. Contrary to that, my theory is based on ODEOM and has a standard classical mechanics as its classical limit; every good mathematical method of finding the solution of these equations, including the receipt of DBBT, becomes then acceptable and welcomed.

The second statement in your opening with which I disagree is that the philosophical difference of my theory with DBBT, fortunate or not, is the only one. Contrary to conventional theory, DBBT explains the mechanism of nonlocal correlations between results of experiments with space-separated but entangled particles. However, in order to do it DBBT requires a selection of a preferred reference frame, thus conflicting with Lorentz invariance. My theory also explains nonlocal correlations, but does it in a frame-independent way (the requirement of analyticity plays here a crucial role). Consequently, here we have a second example where an objection to Bohmian mechanics is inapplicable to my theory in spite of the fact that the two theories share the same trajectories (and by the way, the very existence of these examples proves that my theory is not “exactly the same as DBBT.” In fact, they specify precisely the two main differences between DBBT and my work: if you wish to obtain the latter from the former, you should add to DBBT the requirement of analyticity and the generalized Hamiltonian formalism. But then, as I just explained, the whole “guiding” philosophy of DBBT will disappear, and it will turn into a welcomed and unobjectionable mathematical part of the formalism on its PDE side).

The same request goes to Scott: since you didn’t elaborate, I can only guess (and please correct me if this guess is wrong) that your confidence in rejecting your option (4) is based on the fact that my work shares something with Bohmian mechanics. As I just demonstrated this mere fact is not sufficient. On the other hand, the critique based exclusively on the contest of my work is welcomed and solicited.

-
My confidence is from the fact that if I wanted to simulate your theory on a computer, I would end up simulating Bohm. The idea that you somehow got out of specifying a frame is not right--- you are going to specify field trajectories if you want to make your theory covariant, and these field trajectories are over field phase space. If you use particle phase space, you can't do it in a naive way using forward in time evolution. –  Ron Maimon Sep 26 '12 at 7:24

Just a comment on the following part of reaction (1): "Because an analytic function is completely determined by its higher derivatives at any point, Raykin's "differential equation" is really just an elaborate way of stating the tautology that a particle's future trajectory is completely determined by its future trajectory. Obviously, that would be true regardless of what the trajectories were (provided only that they're analytic): nothing specific about QM is being used here. Differential equations that depend on infinitely many derivatives simply lack predictive power."

I would disagree with such an opinion. If the trajectories are indeed analytic (it looks like that, but I am not quite sure), for example, if spatial coordinates are analytic functions of time, then a particle's future trajectory is completely determined by its past trajectory as well, furthermore, it is completely determined by an arbitrarily small part of the past trajectory (by the way, for this reason, I also tend to disagree with Raykin's opinion that "an analytic function is a fundamentally nonlocal object" - if we accepted this, we would have to accept that Newtonian trajectories are fundamentally nonlocal, as they can also be analytic). I would also add that while "Differential equations that depend on infinitely many derivatives simply lack predictive power", they still have some predictive power, as a truncated Taylor series provides a decent approximation for the nearest future.

-

Several persons asked me to present my work (http://arxiv.org/abs/1204.1540) in a historic, rather than logical, perspective, i.e. to specify explicitly what differs my approach from existing ones. I think this forum is a good place to answer this request. The first difference, which may be considered as the first postulate of the theory, is a restriction of admissible wave functions to analytic ones. Consequently, I consider only a tiny subset of a usually considered functional space; however, since this subset is everywhere dense in the set of smooth (and even continuously differentiable, moreover, even generalized) functions, I don't see any physical objections to my approach from this side. The restriction to analytic functions is a huge departure from the conventional theory: while a nonanalytic function really is a function in a sense that in order to define it one has to specify its value everywhere in the whole domain, an analytic function probably does not deserve a name of a function at all, for its definition reduces to specification of the center and (countable number of) coefficients of its Taylor series. I then consider the logarithm of the wave function, which I call the action function, and which now for every moment of time is an analytic function of particle’s position. The action function’s time evolution follows from the Schrodinger equation and is described by a partial differential equation (PDE) which I call quantum Hamilton-Jacobi equation (QHJE). By analyticity, this evolution reduces to time evolution of the center and coefficients of this function’s Taylor expansion, i.e. its expansion in powers of space coordinates. This time evolution of the center and coefficients is described by corresponding ordinary differential equations of motion (ODEOM), and the second difference between my approach and others is that I concentrate on these ODEOM instead of Schrodinger’s equation or QHJE. Obviously, the law of expansion center’s motion may be chosen arbitrary; the time evolution of the expansion coefficients will then adjusts itself in such a way that the whole function evolves according to QHJE. Now although by this logic all laws of expansion center’s motion are equal, there is one law that is much more equal than others, for it guarantees that ODEOM obtain a perfectly Hamiltonian form (see Eq. (2.28) in the paper) and that the motion satisfies a least action principle, where for each trajectory the corresponding action is defined, as usual, as an integral from a Lagrangian function (a Legendre transformation of a Hamiltonian one, which is extracted from QHJE) along the trajectory. This law is given by Eq. (2.22), and the question that everyone now must ask is what are the properties of a theory that declares this unique law, accompanied by corresponding equations of evolution for Taylor expansion coefficients (which I call momentums), to be the law of particle’s motion; this declaration may be considered as the second postulate of my theory, and the paper is just a detailed answer to the above inevitable question.

We have, therefore, the theory that is defined by ODEOM. Like Hamilton’s equations in classical mechanics these equations of motion constitute the physical basis of the theory; they also turn into Hamilton’s equations in the classical limit h->0. In the best traditions of theoretical physics these equations of motion are obtainable from the least action principle. Now as in classical mechanics, we can consider a family, or ensemble of trajectories defined by our ODEOM, for each trajectory calculate a corresponding action, and consider this action as a function of coordinates of trajectories’ ends. This function is the action function discussed above; along with its exponent, the wave function, it describes the just introduced family and evolves according to QHJE. The analysis of the theory is then simplified by the following fact. Equation (2.22) expresses the particle’s velocity through space derivatives of the action function; this expression is a basis of a Jacobi’s method of integration of ODEOM: rather than obtain trajectories by solving these equations directly, we can solve PDE for an action function, obtain particle’s velocity field from Eq. (2.22) by a simple differentiation, and then obtain the trajectories of the family by an easy integration of so obtained velocities. Now for a spinless particle in an external potential Eq. (2.22) gives the same expression for velocity as a “guiding equation” of the de Broglie-Bohm theory (DBBT). One of the strongest objections against DBBT was always that its guiding equation is something unheard of and appears in the theory completely ad hoc. We have here the first difference between my approach and DBBT: in my work Eq. (2.22) is just one of Hamilton’s equations, the basis of an old and perfectly well understood mathematical Jacobi’s method of obtaining the solution of ODEOM without really solving them, rather than a new physical principle as in DBBT. The second difference with DBBT, as with conventional quantum mechanics, is in the discussed above analyticity, which in particular allowed me to explain the nonlocal correlations between results of measurements with entangled particles in a frame-independent way (see section 9 of the paper), something that in DBBT could not be done (I’d also want to add here that the analyticity, besides being the basis of the whole theory, allowed a simple mathematical explanation of the wave-particle duality in the end of section 4). On the other hand, the fact that particle’s trajectories in my theory coincide with those in DBBT allowed me to borrow from it the theory of von Neumann’s measurements: as I wrote in section 8, the corresponding part of my paper is just a translation of Bohm’s treatment to the language of my work.

This exposition, hopefully, makes it clear where my theory intersects with DBBT and where it differs from it. The differences with conventional quantum mechanics are clear from the table in the paper’s conclusion, of them most notable is that in my theory the standard statistical interpretation is derived from the equations of motion rather than being postulated. My hope is that the physical picture which emerges from my work is free from conceptual difficulties. The objections to this optimistic opinion are very welcomed.

-