Quantum mechanics without wavefunctions...

cgk

...that's the title of a recent communication in the Journal of Chemical Physics[1] (as communication it is free access to everyone):

http://jcp.aip.org/resource/1/jcpsa6/v136/i3/p031102_s1

The authors perform some rather clever looking algebraic transformations and arrive at a formulation of nonrelativistic spinless QM (without fermionic or bosonic symmetry constraints, though) which looks a lot like a classical Hamilton-Jacobi formulation of mechanics with one additional quantum coordinate/momentum pair (additionally to the classical position/momentum coordinates) for each particle. That's basically the hidden variable, as far as I understand.

I was wondering if any of you had an opinion on this approach, especially the experts on QM interpretations. I am myself not quite sure what to make of that. On the one hand it does look very clever and leads to an entirely real formulation (i.e., no complex numbers) of QM without wave functions and with a clear action principle. On the other hand, it looks a bit like just replacing the N-dimensional complex Schroedinger equation by a 2N-dimensional set of real differential equations. The chief advantage seems to be that the latter is easier to combine with classical approximations and easier to integrate numerically. But is this also a novel way of looking at things? Is there more to it? (The authors seems to think so and make some rather grand claims normally not found in J. Chem. Phys.). Any opinions welcome.


[1]: If you are concerned about the journal: J. Chem. Phys. is highly respected and the #1 journal for chemical physics and many parts of molecular physics---in particular for all kinds of numerical quantum dynamics and quantum mechanics applied to real atomic and molecular systems. If the authors had actually found a reformulation of QM which is helpful for numerical computations, it would make a lot of sense to publish it there first, because that is the journal the people read who would likely use it.

lugita15

Hmm, without actually opening the paper, it seems reminiscent of Bohmian mechanics: my understanding is that the starting point for Bohmians is exactly the Hamilton-Jacobi equation. Also, I think the whole Bohmian framework is based on breaking up the complex Schrodinger equation into real components (either the real part and imaginary part, or the modulus and the argument of the complex number in polar form; I don''t recall which). So this may be a rediscovery of Bohm.

cgk

Not directly: The authors are aware of Bohmian mechanics. But in Bohmian mechanics there are still wave functions (in the form of pilot waves) and complex amplitudes. This reformulation does not have that. There are no wave functions, no complex numbers, no Schroedinger equation.

Btw: I just realized that my original post looks like an advertisement for that paper. That was not intended. I am neither one of the authors, nor in any way affiliated with them (in fact, I have never heard of them before).

lugita15

In Bohmian mechanics, the complex numbers are broken up into real components. The complex wave function or amplitude is not there, and the pilot wave is entirely real. The complex Schrodinger equation is broken up into two real equations, the one governing the quantum potential and the one governing the pilot wave. I still don't see any advantages over Bohm.

martinbn

It seems that the paper you should link to is their citation 7. It seems that they are citing the result from there.

arkajad

From citation 7:

Bohmian mechanics without pilot waves

Bill Poirier
Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech University, Box 41061, Lubbock, TX 79409-1061, United States
Abstract

In David Bohm’s causal/trajectory interpretation of quantum mechanics, a physical system is regarded as consisting of both a particle and a wavefunction, where the latter “pilots” the trajectory evolution of the former. In this paper, we show that it is possible to discard the pilot wave concept altogether, thus developing a complete mathematical formulation of time-dependent quantum mechanics directly in terms of real-valued trajectories alone. Moreover, by introducing a kinematic definition of the quantum potential, a generalized action extremization principle can be derived. The latter places very severe a priori restrictions on the set of allowable theoretical structures for a dynamical theory, though this set is shown to include both classical mechanics and quantum mechanics as members. Beneficial numerical ramifications of the above, “trajectories only” approach are also discussed, in the context of simple benchmark applications.
Keywords

Time-dependent quantum mechanics;
Trajectory formulation of quantum mechanics;
Interpretation of quantum mechanics;
Bohmian mechanics;
Pilot wave;
Quantum trajectory methods

cgk

I read through ref 7, but it is only concerned with 1-dimensional quantum mechanics. Also, the treatment of the time-dependent 1D case is much more elegant in the current paper (but ref 7 goes into more detail about the example applications).

What is really striking to me about the current paper is that is does not feel wrong.

bohm2

I posted this paper in another thread previously. I found this comment near the end of the paper interesting:

Demystifier

They don't have a wave function, but they do have a quantum potential Q. In Bohmian mechanics there is also a quantum potential (different from theirs) which, more or less, is equivalent to the wave function. It seems that their quantum potential does not depend on the wave function, which would be good if there were no other problems with the theory. But there are! I don't see that they explain why their trajectories give the same statistical predictions as standard QM (which ARE given by the wave function). Without explaining why it makes the same statistical predictions as standard QM, their theory makes absolutely no sense.

cgk

The authors are calculating Bohmian trajectories, and from those the wave function can be reconstructed (up to a phase factor). This process is described in detail for the time-dependent 1-D case in their ref 7 (there they also exactly reproduce standard QM results of 1D scattering simulations in order to prove their theory's suitability for numerics).

In the current paper they don't give the explicit construction of the wave function for the multi-dimensional case (probably due to space constraints); but given the other details I guess we can take as a working hypothesis that this reformulation is not completely broken. At least until there is at least some indication that this is the case.

bohm2

Do they explain the existence/source of trajectories in the first place? Without a wave function/pilot wave what "grounds" the trajectories? Unless, I'm misunderstanding (very likely), it's kind of like describing the orbit of the earth around the sun without gravitation.

Demystifier

OK, I guess I have to see their ref. 7. When I see it, I will make further comments.

Demystifier

OK, I have seen now ref. 7, so here are my comments.

The paper seems technically correct. It is mathematically possible to rewrite the equations in that form. However, similarly to the path-integral method, it does not seem to suggest a new interpretation of QM, or more precisely, a new resolution of the measurement problem in QM. That's because the theory deals with an INFINITE ENSEMBLE of trajectories, rather than only one trajectory as in Bohmian case. With such an ensemble of trajectories you cannot choose one as "the real one", so you cannot explain why in experiments only one of many possible outcomes realizes. In other words, you cannot explain the appearance of "wave function collapse".

Demystifier

There is a better analogy in classical physics. In classical theory of electrostatics, you need to calculate electric field at all points of space. So, electric field is analogous to a wave function. From electric field you can calculate the lines of force
http://en.wikipedia.org/wiki/Line_of_force
known also as field lines
http://en.wikipedia.org/wiki/Field_line
which are analogous to the set of all Bohmian trajectories. These field lines are nothing but curves tangential to the direction of electric field; they do not necessarily represent the trajectories of physical particles. It is possible to rewrite the equations determining the electric field into equations that determine the field lines DIRECTLY, without first calculating the electric field. In this way, the electric field can be eliminated from fundamental equations.