Guest post: "Are classical, deterministic, field theories compatible with the predictions of quantum mechanics?"

Elliptic Composability is a blog dedicated to quantum mechanics and a recent post attracted a lot of attention and comments. In particular one of the readers, Andrei, has argued for classical physics and we have engaged in a lot of back and forth discussions about it. Although I disagree with Andrei, I have found his arguments well constructed and I thought my debate with him would be of interest to a much larger audience. So I have invited Andrei to present his best arguments in a guest post to which I will present my counter-arguments in the next post. 

Without further ado, here is Andrei's guest post:

Thank you, Florin, for inviting me to present on your blog my arguments that classical physics can still play a role in understanding nature.

The following objections are usually raised against the search for classical explanations for quantum phenomena:

Objection 1: Classical, local theories have been ruled out by Bell’s theorem.

Objection 2: Classical theories cannot explain single-particle interference (double slit experiment), quantum tunneling, the stability of atoms or energy quantification in atoms or molecules.

Objection 3: Even if one could elude the previous points, there is no reason to pursue classical theories because quantum mechanics perfectly predicts all observed phenomena.

Objection 1

It is a widespread belief that Bell’s theorem rules out local realism. As classical field theories are both local and realistic they couldn’t possible reproduce the predictions of quantum mechanics. However, a more careful reading of the theorem would be:

Assuming the experimenters are free to choose what measurements to perform, the logical conclusion is that no local and realistic theory can reproduce the predictions of quantum mechanics.

One can easily notice that, if the motion of quantum particles (including the particles the experimenter himself is made of) is described by some classical, deterministic, field theory, this freedom of choice does not make any sense.  Classical determinism implies that certain past configuration uniquely determines the future. Anything else would require a violation of the physical law.

One can still object that, even if the experimenters (say Bob and Alice) follow some deterministic process there is still no reason to assume that their behavior would be correlated. Alice could use the decay of some radioactive material to decide the measurement she performs, while Bob could let his measurement be decided by very complex random number generation software. Any assumed correlation between such unrelated processes would amount to a conspiracy no one should seriously consider.

Let’s analyze however, this experiment from the point of view of a classical field theory. One can make the following two observations:

a.       the experiment reduces, at microscopic scale, to three groups of particles: A (Alice, her radioactive material and the detector she controls), B (Bob, his computer and the detector he controls), S (the source of the entangled particles).

b.      The trajectory of a particle in A would be a function of position/momenta of particles in A,B and C; the trajectory of a particle in B would be a function of position/momenta of particles in A,B and C; The trajectory of a particle in C would be a function of position/momenta of particles in A,B and C

   From (a) and (b) it follows that Alice and Bob (and the source of the entangled particles), cannot be independent of one another. The so-called conspiracy is a direct consequence of the mathematical structure of a classical field theory.

In conclusion I have proven that Bell’s theorem cannot rule out classical deterministic field theories.

Objection 2

I will show here how some “mysterious” quantum effects are not in contradiction with what one would expect if some classical field theory describes the motion of particles at the fundamental level. I will start with the iconic double-slit experiment as it is presented by Feynman here:

http://www.feynmanlectures.caltech.edu/III_01.html

Feynman says:

“In this chapter we shall tackle immediately the basic element of the mysterious behavior in its most strange form. We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by “explaining” how it works.”

I invite everyone to read the whole description of the experiment there. Feynman chooses to use indestructible bullets as the classic analogs of electrons. The experiment is performed with the first slit open, then with the second slit open, and, in the end, with both slits open. The expected result is:

“The probabilities just add together. The effect with both holes open is the sum of the effects with each hole open alone. We shall call this result an observation of “no interference,” for a reason that you will see later. So much for bullets. They come in lumps, and their probability of arrival shows no interference.”

This is all nice, but classical physics is not the same thing as Newtonian physics of the rigid body. Let’s consider a better classical approximation of the electron, a charged bullet. The slits are made of some material that will necessarily contain a large number of charged “bullets”. As the test bullet travels, its trajectory will be determined by the field generated by the slitted barrier. The field will be a function of position/momenta of the “bullets” in the barrier. But the field produced by a barrier with two slits will be different than the field produced by a barrier with only one slit, so the effect with both holes open is NOT the sum of the effects with each hole open alone.

In this paper:

https://hekla.ipgp.fr/IMG/pdf/Couder-Fort_PRL_2006.pdf

Yves Couder has provided experimental confirmation for classical single-particle interference. The greatest mystery of quantum mechanics has been solved by the good old classical field theory.

Let’s move to another “classical impossibility”: stable atoms with quantified energy levels. It is claimed that classical electrodynamics cannot account for stable atoms, and I would accept this claim, even if the situation is not as clear as it seems. For a possible counterexample I recommend this:

https://en.wikipedia.org/wiki/Free-fall_atomic_model

The author, Gryziński, has published its model in top peer-reviewed  magazines so I guess there is some truth about it.

But, for the sake of the argument, let’s assume that classical electrodynamics cannot explain the atom. Does it prove that no classical theory could do it? I don’t think so. We know that stable groups of charges with well-defined energies are possible. They are called ionic crystals. The charged particles do not move (if thermal motion is ignored), so there is no energy loss by radiation. The geometry of the crystal dictates the energy, which has a well-defined value. The reason for the stability of the crystals stays in the repulsive force generated by the orbiting electrons. One can argue that electrons are not composite particles, but a repulsive force can be provided by some modification of the electric field.

The third example I will discuss here is tunneling. Presumably, classical physics cannot explain this phenomenon because the particle does not have enough energy to overcome the barrier. 

There are two observations I can make for this case:

a.       Neither the energy of the particle that tunnels, nor the value of the potential are accurately known at the moment of tunneling. They are average values, and these can be very different from the instantaneous ones.

b.      The actual force acting on the particle depends on the classical theory used to describe the experiment. A new theory could predict a much stronger force.

Objection 3

In the end I would like to point out a few reasons for investigating classical theories.

a.       If nature is not probabilistic after all, there is much to be discovered. Detailed mechanism behind quantum phenomena should be revealed, bringing out a deeper understanding of our universe, and maybe new physical effects.

b.      Quantum theories are not well equipped to describe the universe as a whole. There is no observer outside the universe, no measurement can be performed on it, not even in principle.

c.       Due to its inability to provide an objective description of reality, quantum mechanics may not be able to solve the cosmological constant problem. A theory that states clearly “what’s there” could provide a much better estimate of the vacuum energy. After all we are not interested in what energy someone could find by performing a measurement on the vacuum, but what the vacuum consists of, when no one is there to pump energy into it.

d.      Quantum mechanics requires an infinitely large instrument to measure a variable with infinite precision. When gravity is taken into account, it follows that local, perfectly defined properties cannot exist, because, beyond a certain mass, the instrument would collapse into a black hole.

For a detailed description of the latter two points, please read this paper by Nima Arkani-Hamed:

http://www.sns.ias.edu/ckfinder/userfiles/files/daed_a_00161%283%29.pdf

Thanks again, Florin, for this opportunity to present my arguments, and I am looking forward to seeing your opinion about them.

Andrei