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Killtech

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- 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.

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.

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