Sunday, September 24, 2006

Wavefunctions and hydrodynamics: crackpots vs. rational thinking

It is no secret that I consider all people whose main scientific focus is a revision of the basic postulates of quantum mechanics - and a return to the classical reasoning - to be crackpots. They just seem too stubborn and dogmatic or too intellectually limited to understand one of the most important results of the 20th century science.

Every new prediction based on the assumption that there is a classical theory that underlies the laws of quantum mechanics has been proven wrong. The local hidden variables have first predicted wrong outcomes in the EPR experiments and later they predicted the validity of Bell's inequalities and we know for sure that these inequalities are violated in Nature, just like quantum mechanics implies and quantifies. The non-local hidden variables predict a genuine violation of the Lorentz symmetry. I think that all these theories predict such a brutal violation of the Lorentz symmetry that they are safely ruled out, too. But even if someone managed to reduce the violation of the laws of special relativity in that strange framework, these theories will be ruled out in the future. Their whole philosophy and basic motivation is wrong.

The whole political movement to return physics to the pre-quantum era is a manifestation of a highly regressive attitude to science - an even more obvious crackpotism than the attempts to return physics to the era prior to string theory. But among the proposals to undo the 20th century in physics, some of the papers are even more stupid than the average.

This is also the case of the recent preprint

There are many meaningless words in that paper but let me focus on a section whose content is meant to be very clear and it is very clear, except that it is also totally dumb. The author claims that Timothy Wallstrom was wrong in his criticism of a hydrodynamic approach to the wavefunction.

What did Wallstrom point out? He looked at the theories in which

  • abs(psi)^2

is interpreted as a density of some liquid, while the usual "classical velocity" calculated from the wavefunction - the ratio of the probabilistic current and the probability density - is interpreted as an actual velocity of the same liquid.

Wallstrom said that this map might look plausible locally but it is wrong globally. The argument is trivial. Take a generic wavefunction in three dimensions. It will satisfy

  • psi=0

at a one-dimensional curve - a "cosmic string" - because the one complex or two real conditions above remove two dimensions from space. What happens with "psi" if you make a round trip around this one-dimensional curve? In quantum mechanics, you will return to the same wavefunction: the wavefunctions must be single-valued and the locus where "psi=0" is not too special, after all. Most smart high school students who are really interested in physics know that the wavefunctions are single-valued. This is the fact that underlies the quantization of the orbital angular momentum as well as other observables. It's the ultimate reason why quantum mechanics has "quantum" in its name.

On the other hand, in the hydrodynamics toy model, you can pick an arbitrary phase - which is a wrong result. The whole quantum character of quantum mechanics will disappear until you constrain the contour integrals for the velocity by a condition that is equivalent to the quantization condition in Bohr's old quantum theory, as Wallstrom pointed out.

I claim that this generic lethal flaw of the hydrodynamical model should be comprehensible to every undergraduate student who has registered for the introductory course of quantum mechanics within a few minutes after the sixth class. This argument would certainly be easy for the physicists 80 years ago, but at any rate, it has been published for 12 years. How is it possible that someone who claims to work on these things is unable to get such a simple point at least for 12 years?

The author of "Could quantum mechanics..." not only misunderstands the simple argument but he promotes this misunderstanding to a new branch of science. The author indeed admits that the phase monodromy can be arbitrary in the hydrodynamical theory - but he views it as an advantage over the conventional quantum mechanics. In other words, he indeed believes that there is no quantization of things like the angular momentum in the real world - no kidding - despite billions of experiments that say otherwise, and the hydrodynamical theory is apparently claimed to be better because it can transcend the quantization rules of quantum mechanics: the discrete spectrum of many observables is apparently another example of the sexist white male rules and stereotypes that reduce the diversity of ideas in science and that discriminate against the numbers that were not among the "priviliged" eigenvalues so far. ;-)

I just can't stand this bigotic approach. I can't stand pompous fools. The paper is an extreme example of stupidity, and no matter how many books will be written about this stupidity - books promoting the people working on similar "problems" as original scientists who are almost as good as the string theorists if not better - and no matter how many thousands of impressionable laymen and idiotic bloggers will become convinced that it is a deep idea, all these ideas will continue to be the very same patently false stupidities.

And that's the memo.

P.S.: The very idea that the wavefunction should be reparameterized into different variables by a non-linear transformation is a deeply flawed misconception. The linearity of the Hilbert space of the quantum mechanical wavefunctions is one of the key principles that allows quantum mechanics to work. If one thinks about some other variables, the devastating effect of the non-linear reparameterization will become clearest near the points where "psi=0" because this is where the non-linear transformation becomes extremely singular. This is why the places where "psi" is approximately zero could have been used by Wallstrom to show that the hydrodynamical model is wrong. The hydrodynamics toy models always create a singular earthquake near the loci of "psi=0" even though there is obviously nothing too special about these points in quantum mechanics or in reality.

But even if you picked any other model that is either redefining the wavefunction in a non-linear way or that is distinguishing priviliged operators on the Hilbert space in which your non-quantum description will be more classical, you will be able to find a proof that the theory is flawed. It's because the linearity of the wavefunction, the philosophical democracy between different observables (such as position and momentum - you can't say that one of them is classical and the other is not), and other postulates of quantum mechanics are not only beautiful and robust pillars of modern physics, but they are also experimentally proven facts.

It is fundamentally wrong to single out some observables - such as positions - to be more classical than others. In reality just like in quantum mechanics, one can talk about the spectrum of all observables, and which of them behave more classically than others is dynamically determined by the Hamiltonian - by decoherence - not by pre-established dogmas. This fact has been known at least for 20 years and everyone who understood foundations of quantum mechanics has known this fact for 20 years.

Did he know?

How could have Lee Smolin submitted such a silliness? When you try to think about his wording, it is conceivable that he does not realize that it is silly. He just thinks that the multiply-valued functions are square-integrable, and therefore they should be a part of the Hilbert space. That's of course wrong because while they might be square-integrable, they are not really functions, and therefore they are not elements of the Hilbert space. A person familiar with the mathematical terminology would know that they are not in "L^2". Most physicists would know that because they aware, unlike Smolin, of physical considerations that make them certain that the multiply-valued "functions" are not allowed.

It is also impossible to choose one value for each point which would translate multiply-valued functions on a circle to discontinuous functions. It's because the discontinuous functions wouldn't satisfy the Schrödinger equation near the discontinuity. In other words, the velocity of the liquid calculated from a discontinuous wavefunction will have an extra delta-function localized near the discontinuity, and it will thus differ from what Smolin claims to be the same thing. Equivalently, the discontinuity makes the energy diverge while the energy in the liquid picture is finite. The discontinuous wavefunctions are certainly not a part of the physically realizable Hilbert space.

Above, I assumed that Lee doesn't realize why his comments are silly. Alternatively, you might imagine that Lee Smolin realizes that what he wrote is crap, but he wants a particular preprint with a preprint number that he will cite whenever someone tells him that the classical models of quantum mechanics are impossible because of Wallstrom's argument, among other things. Lee will tell them "Wallstrom's argument has been invalidated in quant-ph/yymmnnn but unfortunately I don't have enough time now to tell you what's the argument - just read the paper". The people will eventually find out that the preprint is rubbish but Lee will earn his 15 minutes of doubts which may be enough to survive one of his public talks in which he is pumping his silliness into the audience.

Incidentally, a Harvard grad student (A.P.) has pointed out a paper by Marcel Reginatto about a very similar topic plus the Fisher information. It seems more serious. On the bottom of page 13, Reginatto also struggles with the Wallstrom's problem. As far as I can say, he also fails although not as miserably as L.S. because Reginatto at least admits that the wavefunction should be single-valued in a correct theory. ;-) Reginatto says that things look nice and simpler with a single-valued function which is not exactly what I call a physical explanation. There might be some interesting mathematical and philosophical idea in the "Fisher information" but I am probably not able to go through all the "epistemilogical" junk that has, as admitted on the bottom of page 8, no physical consequences. ;-)

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