Does Dirac's argument against classical mechanics stand in contradiction to Bohm's theory?

Question

In his book on Quantum Mechanics, P.A.M. Dirac talks about the stability of the atom as a means of demonstrating the need for quantum mechanics. He writes:

The necessity for a departure from classical mechanics is clearly shown by experimental results. In the first place the forces known in classical electrodynamics are inadequate for the explanation of the remarkable stability of atoms and molecules, which is necessary in order that materials may have any definite physical and chemical properties at all. The introduction of new hypothetical forces will not save the situation, since there exist general principles of classical mechanics, holding for all kinds of forces, leading to results in direct disagreement with observation. For example, if an atomic system has its equilibrium disturbed in any way and is then left alone, it will be set in oscillation and the oscillations will get impressed on the surrounding electromagnetic field, so that their frequencies may be observed with a spectroscope. Now whatever the laws of force governing the equilibrium, one would expect to be able to include the various frequencies in a scheme comprising certain fundamental frequencies and their harmonics. This is not observed to be the case. Instead, there is observed a new and unexpected connexion between the frequencies, called Ritz's Combination Law of Spectroscopy, according to which all the frequencies can be expressed as differences between certain terms, the number of terms being much less than the number of frequencies. This law is quite unintelligible from the classical standpoint.

Since de-Broglie Bohm theory essentially introduces a quantum force to remedy classical mechanics, does Dirac's argument stand in contradiction to Bohm's theory?

Answer 1

Dirac does not intend classical to mean non-quantum mechanics; he intends classical to mean pre-quantum mechanics¹. So no, this says nothing about de Broglie–Bohm theory.

Dirac opens his paragraph with

The necessity for a departure from classical mechanics is clearly shown by experimental results.

Dirac is not talking about any theory that one might call "classical" today. He is talking about the theory of classical mechanics that existed before quantum mechanics was formulated. If you read his paragraph, all his criticisms are directed against that particular theory. For example, he says

the forces known in classical electrodynamics

and

if an atomic system has its equilibrium disturbed in any way and is then left alone, it will be set in oscillation and the oscillations will get impressed on the surrounding electromagnetic field.

These properties are specific to pre-quantum classical mechanics. His argument thus says nothing about de Broglie–Bohm theory. In fact, if I suspect that if you had asked Dirac, he would have said that de Broglie–Bohm theory was not classical mechanics.

EDIT: I want to address Lubos's comments, because some people seem to be agreeing with him and downvoting me. Further, I was asked in the comments to explain why Dirac's criticism doesn't apply to Bohmian mechanics. If you think Lubos is a genius and take everything he says uncritically, go ahead and downvote. If you actually can reason critically and think indepedently, please read my rebuttal to Lubos's comments before downvoting.

Lubos is wrong about almost everything he says about Bohm's theory in his comments. Lubos claims that the pilot wave in Bohm's theory is observable. The pilot wave in Bohm's theory is exactly the same as the Schroedinger wave function, so if the wave function in quantum mechanics is unobservable, so is the pilot wave in Bohmian mechanics. Furthermore, Bohm's theory has exactly the same predictions as quantum mechanics, so it is clear that in Bohm's theory, a system will not "be set in oscillation and the oscillations will get impressed on the surrounding electromagnetic field," simply because it isn't in standard quantum mechanics.

The only way I can see to save Lubos' statements is to realize that Bohm's theory doesn't actually work relativistically², and then to reason that because photons are intrinsicially relativistic, photons don't actually work in it. But even though Dirac was an incredibly smart man, it is completely ridiculous to claim that

1. Dirac intended his argument to work for all non-quantum theories and
2. he foresaw Bohm's theory and was clever enough to realize that his argument would be saved by the fact that Bohm's theory doesn't work relativistically.

Dirac was smart enough to know that you can't prove a theorem like this (i.e., no theory except quantum mechanics can explain observation) without stating the hypotheses. His hypotheses, which I am sure he thought he had clearly stated, were that physics behaves along the lines of pre-1900 physics ... e.g., the dictionary definition of classical as traditional in style and form. In 1930, I don't believe anybody was using "classical physics" to mean "non-quantum". (I'll retract this statement if you can find an example of "classical" clearly meaning "non-quantum" rather than "pre-quantum" before 1940.) They were using "classical" to mean "pre-1900 physics", i.e., physics before quantum mechanics and relativity came along.

¹ A standard definition of classical is: traditional in style or form.

² Bohm's theory's adherents may claim otherwise, but I don't believe anybody has demonstrated that one can have a Bohmian theory with Lorentz-invariant particle trajectories and general relativity, which means that relativistically, the theory has severe drawbacks.

Answer 2

Yes, absolutely, Dirac's argument shows that one could never construct a complete theory – which specifies the rules for evolution as well as predictions for the measurement and what happens after the measurement – that would be compatible with the basic facts about the atoms.

This no-go theorem applies to Bohmian mechanics because it is just another classical theory, indeed. Bohmian mechanics is a classical theory composed of a classical field $\psi$, the pilot wave, which is said to obey the same equations as the wave function in quantum mechanics but has a completely different interpretation, plus some extra beables – typically classical particle positions $x_i(t)$. These positions are influenced by the pilot wave but the opposite reaction doesn't exist.

Like any normal enough classical system, the configuration space of Bohmian mechanics is continuous so one may always divide a small perturbation by two and what this perturbation evolves out of this perturbation is the original outcome divided by two, too.

That's in direct contradiction with the facts such as the discrete spectra of atoms etc. (The energy eigenstates of atoms may be present mathematically but Bohmian mechanics won't allow any mechanism that could imprint the spectra to the electromagnetic radiation of emitted light.) The dynamical equation (i.e. Schrödinger's equation in particular) in quantum mechanics is continuous but the meaning of the wave function is probabilistic so a very tiny perturbation of $\psi$ means a very small probability of a new (but finitely strong) new process, not a guaranteed presence of a very weak process. This is a key different of quantum mechanics from any classical theory, including Bohmian mechanics.

Dirac's textbook starts with several other arguments that instantly rule out classical theories including Bohmian mechanics, especially the argument about the observer low heat capacity of all atoms. If Bohmian theory could be extended to a full theory of interacting atoms, they would be described by a huge phase space because they have many classical plus new degrees of freedom. If the thermal equilibrium were possible in such a theory, the entropy of an atom (the logarithm of the volume of the phase space of states that are accessible at the thermal equilibrium) would be huge due to these extra degrees of freedom, in a direct conflict with experiments (which shows that the atom's heat capacity is always comparable to $k_B$).

Quantum mechanics allows these correct predictions of the low heat capacity because the state of the bound states is basically unique or has low degeneracy when the energy is required to be close to the ground state energy. Small perturbations of the ground state don't refer to mutually exclusive states. Instead, quantum mechanics says that a state that is mutually exclusive with the ground state – so that it could contribute to the entropy – has to be orthogonal to the ground state, i.e. very different. This is equivalent to the usual "quantization of the phase space" that is effectively divided to "cells". That's how quantum mechanics manages to produce a "small number of cells" which is needed e.g. for the low heat capacities.

Advocates of Bohmian mechanics never discuss any of these elemenentary, but still advanced relatively to the "Bohmian demos" issues – thermal equilibrium of objects, heat capacity, but also propagation of photons and other bosons, emission of sharp spectral lines, the observed collapse of the atoms to energy eigenstates once the energy of the photon is measured, the existence of fermionic quantum fields, and so on. And the main reason is that all these important parts of physics as we know it are irreconcilable with Bohmian mechanics – and every other "realist" i.e. classical theory.

Answer 3

Since de-Broglie Bohm theory essentially introduces a quantum force to remedy classical mechanics,

That isn't what dBB theory does. It introduces a state-dependent force. Which means it introduces states (waves) and then the wave exerts a quantum force.

does Dirac's argument stand in contradiction to Bohm's theory?

No. And not for the reasons above. The dBB theory is for non relativistic quantum mechanics. And that means it doesn't really address electromagnetism the same way the Schrödinger equation doesn't address electromagnetism.

When you solve for, say the energy levels of hydrogen, you don't use electromagnetic fields you just use a scalar potential. Even when you solve it as a two particle problem (so a wave in a 6d configuration space) you still just write a scalar potential like $$V(x_e,y_e,z_e,x_p,y_p,z_p)=\frac{-k^2e^2}{\sqrt{(x_e-x_p)^2+(y_e-y_p)^2+(z_e-z_p)^2}}$$ which is just the electrostatic potential. No magnetism for instance, and no electrodynamics. If you have an external classical potential you can put that in, but this isn't the full classical electromagnetism in particular you aren't assigning a state to the electromagnetic field, one a state to the particles.

So dBB is really incomplete, it doesn't have a full proper QFT version.

But all Dirac was saying is that a force doesn't change the nature that a disequilibrium in classical physics imprints into the wider world. But in dBB you essentially have a radical departure from classical physics in that in addition to a configuration you have a state, a wave. And the state dependent wave acts as a pilot wave exert state dependant forces. Which allows you to have different states exert different forces.

That's the change from classical physics and since dBB does it, dBB isn't a classical theory and Dirac is right, you have to do something different to get quantum mechanics.

In you wanted to fully explore how Dirac's comments apply to dBB you should look a deviation from equilibrium, i.e. a state that isn't an energy eigenstate and see how it evolves and look for an imprint. And the naive nonrelativistic Schrödinger equation might have each energy eigenstate evolve just a phase and so by linearity the superposition evolve just as the sum of each.

But when you throw in an external electromagnetic potential you get, for instance, transitions up from some fields and transitions down from other fields. It doesn't just evolve to permanently be in that superposition. And that's because QM becomes QFT as you include the vacuum fields and other fields in a way where those fields themselves can change.

But the dBB theory just doesn't have that part of quantum mechanics because it doesn't have a QFT version. So it really just doesn't apply to the exact situation Dirac was discussing.