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?
This post imported from StackExchange Physics at 2014-07-24 15:47 (UCT), posted by SE-user Maxim Raykin