Can the hydrodynamic formulation of QM be understood purely classically?

[Killtech]

TL;DR Summary

It seems extended hydrodynamics allows to put QM in terms of classically understandable quantities. Going through some example cases it seems there is a new unavoidable decoherence effect which produces familiar QM-behavior.

I was wondering this question for quite some time because this view seems to work surprisingly well and also coincides with how the state space must be setup in any attempt model QM within classical probability theory.

So starting with the hydrodynamic formulation from https://www.researchgate.net/publication/277307706_Hydrodynamic_Formulation_of_Quantum_Electrodynamics. First off this formulation extends classical QM by allowing interactions with EM-field to go in both ways but isn’t full QED yet. So it’s somewhere in between.

Anyhow we have a system of PDEs (PIDEs): Maxwell + Continuity Eq. + Madelung + Quantization Condition. Furthermore the interaction between EM-fields and the charge fields (density and current) is the same as for a purely classical charge and current distributions and looks absolutely nothing like one would expect from a stochastic interaction. Thus one can interpret them as physical distributions from the mathematical role they in-fact play. Lastly ##\rho_{el}## is proportional to ##\rho## so one could just reformulate the equations to drop ##\rho## altogether.

So that’s a system of PDEs where only classical quantities are involved and where nothing violates classical physics per se: Maxwell defines no equation for the current so this spot is open to be filled by Madelung + Quantization and the remaining equations are classical to begin with. This only throws away the idea of point like charged particles from classics by replacing it with fields. Given that such charged point particles are a source of a lot of problems in the classical theory (e.g. self-interaction via its own EM-field) it’s fair to look at this option. The only issue is that ##\rho## is normalized from its probability origin – a restriction which cannot hold for ##\rho_{el}## classically in general. This means it the equations have to be understood as merely a proxy of something more complex. Also within this picture the role mass is left out as the system is already complete (however mass is implicitly embedded in Madelung).

Now I was checking how this view makes classical sense of some well know examples. And for example it renders the H-atom behavior perfectly understandable from purely classical point of view. Solving the equations in full is of course very complicated but one can do a few proxies: taking the regular solutions of Dirac-H-atom (which only neglects the coupling of the EM-field to ##\rho_{el}## of the electron) and transforming them to hydrodynamic quantities should be a good proxy given how well experimentally founded it is. Now looking at what effects the full equations would have on the solutions: For an energy eigenstate ##|\psi_{nlm}>## the charge ##\rho## and current ##j## are static thus so are the resulting ##E## ##B## fields they induce. However a superposition of two different energy states has an additional mixing term ##2<\psi_{nlm}|\psi_{n’l’m’}>## in ##\rho## which oscillates with a frequency proportional to the difference in energy levels. Now for a classical charge distribution this means it will radiate off energy via EM-waves emission of that frequency and energy conservation would imply the H-eigenstate with the lesser energy would survive. This is the observed behavior and ironically makes it classically perfectly reasonable by the very same mechanic classical physics fails with Bohr’s old model due to Larmor decay (where again the point like charge causes the problem).

More in QM terms this behavior looks just like the Rabi oscillation in reverse: there is no external field supplied but instead the superposition itself becomes the source of it to which it loses energy instead of absorbing it. I think the decay of such superposition could be understood as an unavoidable decoherence effect where the EM-field coupling takes the role of the environment. Interestingly this effect would apply in general to any superposition of energy eigenstates even when those are spatially very far away. In those cases the almost-nonlocal behavior of ##\rho## comes into full play as it can move at unrestricted speed and simply becomes faster the greater the distance. So this looks like it would take away the job for the wave function collapse in most scenarios such that the classical view can avoid having to discuss special measurement mechanics.

In any case I wonder where this view fails/creates any inconsistencies because as of now stressing it through example produces rather well know behavior than anything obscure. Okay, the only weird thing is that charge can travel far faster than light - so only Newtons gravity force can beat it within classical physics. Even so I don't understand why the hydrodynamic formulation isn't more popular as it makes most of QM quite intuitive - well, at least on the interpretation level as it is fair to say it's not analytic calculation friendly.

 

[PeterDonis]

this formulation extends classical QM by allowing interactions with EM-field to go in both ways

What do you mean by "classical QM"? That looks like an oxymoron, since "classical" usually means "not quantum".

 

[Killtech]

What do you mean by "classical QM"? That looks like an oxymoron, since "classical" usually means "not quantum".

basic QM as in considering Schrödinger or Dirac equation as a separate equation only, e.g. where EM-fields affect the equation but it does not affect them back.

 

[PeterDonis]

a system of PDEs where only classical quantities are involved

No, it isn't. The probability density ##\rho##, the momentum density ##\bf{p}##, and the energy density ##E## are all derived from the original Dirac spinors, so they aren't classical.

 

[Killtech]

No, it isn't. The probability density ##\rho##, the momentum density ##\bf{p}##, and the energy density ##E## are all derived from the original Dirac spinors, so they aren't classical.

in the equations they act as-if classical. from the equations alone you cannot tell that they are supposedly probabilistic in origin or from some wird charged gas someone tries to model. or even if they were of statistical nature you would expect them to look differently. this is exactly what also classic probability theory points to: you cannot even model them properly as anything else than real fields - they store the required information needed to make a correct time evolution and a classical field just stores the same level of information.

In any case you don't run into any immediate problems if you take them as classical. Or do you see where is could cause problems?

 

[PeterDonis]

in the equations they act as-if classical

I'm not sure what you mean by "act as-if classical". According to the paper, the equations make the same predictions as standard QED. If that statement is correct, the equations are not "as-if classical", since they make different predictions from classical electrodynamics.

 

[PeterDonis]

from the equations alone you cannot tell that they are supposedly probabilistic in origin

Sure, if you just look at the equations in isolation and don't look at how they're derived or what assumptions had to be made to derive them or what the quantities in them actually mean. That doesn't strike me as a valid procedure.

 

[PeterDonis]

even if they were of statistical nature you would expect them to look differently

How?

 

[PeterDonis]

for example it renders the H-atom behavior perfectly understandable from purely classical point of view

No, it doesn't, because you have to make non-classical assumptions to get the behavior. In classical electrodynamics the H-atom can't even exist in the first place: the electron radiates and spirals into the proton. You have to forbid this solution by hand (in the paper, this is done by imposing the Bohr-Sommerfeld quantization condition by hand). That's not classical; that's quantum.

 

[Killtech]

No, it doesn't, because you have to make non-classical assumptions to get the behavior. In classical electrodynamics the H-atom can't even exist in the first place: the electron radiates and spirals into the proton. You have to forbid this solution by hand (in the paper, this is done by imposing the Bohr-Sommerfeld quantization condition by hand). That's not classical; that's quantum.

I wouldn't say that. You indeed add assumptions but those assumptions don't cause any conflicts within classical framework but however allow to make for proper predictions.

On your classical H-atom: this is only true if the electron remains a point like charge. if it could change its form (or rather its charge distribution) to for example a ring filling the entire orbit you cannot get it to radiate energy any more. don't forget that the base assumption you are making that a same sign charge could be hold into a point like form by a magical infinite force (to overcome its own ##\frac 1 r## repulsion) is exotic to begin with (and with Loretz-Abraham . so instead assuming that the particle might in fact be some kind of soliton solution to some unknown fundamental field equation that somehow finds a way to balance internal forces to keep it together (in which case it would likely have a nature of chaging form depending on its enviorment) is in comparison perhaps less problematic - especially given that the original assumption fails whereas it#s not so clear for the other.

 

[PeterDonis]

those assumptions don't cause any conflicts within classical framework

They most certainly do.

On your classical H-atom: this is only true if the electron remains a point like charge. if it could change its form (or rather its charge distribution) to for example a ring filling the entire orbit you cannot get it to radiate energy any more.

This is personal speculation and is off limits here at PF. Thread closed.

(If you seriously think it's not personal speculation, then you can PM me a reference and the moderators will consider it.)