Schrödinger's original interpretation

[Killtech]

TL;DR Summary

The original interpretation may be mathematically the hardest to analyze, yet it gives a glimpse of promise to resolve the long outstanding mysteries QM. Maybe Schrödinger was right all along?

I wasn't sure if i should call it an "interpretation" and post it in this forum or not as it isn't entirely clear if it makes any different predictions within its region of it validity.

Anyhow, the original idea of Schrödinger that made him come up with his equations is very different from the popular interpretations of QM, yet of all the interpretation is is most remarkable in how it reconciles QM with classical physics, fixes the most major inconsistencies in classic EDyn and makes almost all of QM seemingly weird behavior actually rather intuitive.

So, Schrödinger gave up the most abstract and contradictive assumption of classical physics: the existence of point like charged particles. Instead he tried to interpret $\Psi$ as a classical continuous charge distribution. Sadly he failed to make the interpretation work in time and Born beat him to it - but completely changed the interpretation and brought point like particles back on the menu. But the fix also heals Schrödingers interpretation if $\rho = \Psi^\dagger \Psi$ is interpreted as a charge density instead - which directly leads to the semi-classical non linear Schrödinger-Maxwell equations (of course there are more general formulations including spin and relativity like Maxwell-Dirac but they are even harder to deal with). A detailed depiction can be found here.

Now, Schrödinger-Maxwell has only become of interest not so long ago as it's power is hidden behind it's non linear nature which makes is hard to handle. Because of that only approximative solutions and numerical results are know so far, but they are disturbingly well agreeing with observations. There is also the issue that these equations describes a case more general then classic QM due to the full two-way coupling to the EM-fields but yet without modelling charge quantization so not as general as QED which makes it inherently hard to compare with either in terms of an interpretation.

But what makes it so interesting, is how it is able to explain quantum behavior within classical concepts. Starting with the case of hydrogen atom one realizes that energy eigenstates are the very only states with a stationary classical charge and current. Any superposition of any two such states produces an oscillatory term in the charge distribution of the form $\sin((E_n-E_m)t) \langle \Psi_n | \Psi_m \rangle$. Now using the identical classical reasoning that made Bohr's atomic model instable, Schrödinger's model instead predicts the emission of light with discrete frequencies whenever it is disturbed from an equilibrium energy eigenstate. The classically emitted wavelength of the light incidentally perfectly fits the observation. Energy conservation argument then yields the decay to the lower state. (What's weird though is that such terms are still present even if the eigenstates are locally separated, for example solutions for two spatially distant finite potential wells). Finally simulations of Schrödinger-Maxwell yield that such an emission would happen very swiftly well in range of what is actually observed. So it is capable to explain and even model spontaneous emissions which classical QM cannot. Furthermore going beyond hydrogen, this interpretation makes it immediately obvious why the Hartree-Fock method should work very well for many electron atoms.

What i find most interesting about it is that is shows that properties like quantization aren't unique to QM at all but something that comes quite natural with non linearity. In fact most quantum behavior is very reminiscent of non-linear equations. For example soliton solutions which are waves that behave like particles are known for almost 200 years and they can have such funny abilities like being normally stable but capable to break into multiple new solitons in certain conditions. They make you also question why someone would ever consider tunneling a weird behavior. As for spin, well stable vortexes are actually a special type of solitons. So when we just forget about any interpretation for a second and just look at what what experiments show us, nothing of it is actually that special or particularly new (which made me wonder what is even the problem to handle it with classical probability theory). It's merely the interpretation that tries to describe something in terms of properties it doesn't have that make it appear weird and require things like quantum probability. Schrödinger's cat does not contest the predictions of QM, it merely points out its inappropriate interpretation by Copenhagen. Because if $\Psi$ is interpreted some transformation of a physical entity (as with this interpretation), then there is simply no conflict whatsoever with classic probability. Again, it's the idea of a charged point particle that causes all the trouble.

That said, it's really noteworthy how Schrödinger interpretation actually heals classical physics. For the start, Schrödinger equation can be equivalently rewritten into two real equations for the charge (continuity equation) and current (Madelung). As such it conveniently completes Maxwell equations as they only lack a fundamental equation for the current (Note that Madelung is more alike what you expect a current equation for a charged gas to look like and quite different from a Kolmogorov forward equation of an actual particle). By doing so it solves a long standing contradiction of classical physics: the self interaction of a classic electron which produces unphysical behavior known by Lorentz-Abraham solution. Furthermore there is the issue that charge of same sign exerts a repulsing force and compressing it into a single point requires an infinite amount of energy. Thus classically an electron is an unstable energy bomb that can destroy the entire universe if no magical force is added to hold it together. But Schrödinger interpretation amusingly considers it to do just that (minus destroy the universe) when it is free: it just disperses over the entirety of space which makes its wave properties more prominent until it starts interacting with something. So the electron is basically modeled as a shape shifter.

Lastly there is the question of comparison with QED. Now, from the theory of stochastic processes we know that we can rewrite the time evolution of any process (including any non-linear deterministic one) into a linear form via a Markov kernel. There is a certain irony in that this is actually the very same formalism as in QM. The Markov kernel times $-i$ works the same as a Hamilton operator. However linearizing problems like that comes at the expense of usually increasing the dimensionality of the problem to infinity. Now QFT just look like that had happened to it, specifically because due to particle indistinguishability (of same type) the resulting space of possible observations lacks that dimensionality. So if the process could be reversed there should exist a finite dimensional non linear equations that represent the identical time evolution. Note that Maxwell-Schrödinger can already be obtained as a limit from QED but naturally this implies many simplifications such that the charge quantization gets lost in the process.

Schrödingers interpretation perhaps makes only one big prediction that strongly differs to all other QM interpretations: there is no strict limitation to what can be measured. Heisenbergs inequality still holds but it's interpretation changes (uncertainty becomes a mean dispersion when the distribution is assumed physical rather then probabilistic) to something akin to an ideal gas equation: trying to compress a gas to a point will make its pressure diverge causing currents (impulse) within to range from plus to minus infinity - which is for one not even remotely surprising result and for the other not a restriction for measurements. And there are many successful implementations of weak measurement techniques which severely question the bold and universal assumptions about measurement Copenhagen makes. That said, within this interpretation weak measurements are much closer to the idea of a measurement then the strong interaction that QM calls "measurement".

So overall, the experimental results don't back up Copenhagen all that well - which makes me question if restriction of observables to linear operators instead of general functionals or if it just applies to very special interactions. The more we know, the more everything seems to lean back to where it started. Maybe Schrödinger was right all along?

 

[Sunil]

The starting objection against this is that the wave function is defined on the configuration space. Except in the one-particle case this is not the three-dimensional space.

 

[vanhees71]

Also it's disproven by observation. Measuring the position of a single electron, e.g., by putting a photo plate in its way, leads to a single spot not a smeared distribution as predicted by the Schrödinger equation when interpreting it as a classical field description. That's why Born came up with his probability interpretation, and it's the only interpretation which works for all observed facts yet.

 

[Killtech]

The starting objection against this is that the wave function is defined on the configuration space. Except in the one-particle case this is not the three-dimensional space.

The Madelung equation answers this. Schrödinger models the potential of the velocity field which allows a reduction of the current to a scalar. It is however general enough to model all currents that are curl free (similar as it suffices to model the electric field by its potential in the curl free case). The inclusion of curly currents is the addition of spin (how unexpected) and done by related equations. Curly currents/spin are apparently not the primary aspect that that gives the hydrogen spectrum its shape.

Also it's disproven by observation. Measuring the position of a single electron, e.g., by putting a photo plate in its way, leads to a single spot not a smeared distribution as predicted by the Schrödinger equation when interpreting it as a classical field description. That's why Born came up with his probability interpretation, and it's the only interpretation which works for all observed facts yet.

Except that what you described there also disagrees with Schrödinger-Maxwell already simply because the equations models a type of wave function collapse (which pure QM doesn't). It is seen in the hydrogen model as spontaneous emission, but it also present in solutions for multiple finite potential walls - which is the simplest approximation of a photo plate.

It's merely the stand-alone Schödinger equation that predicts a smeared outcome. But the classical coupling to Maxwell renders such an outcome unstable. The interesting thing is that the form of the quantum potential enables superluminar interaction such that spatial separation of the distribution is almost entirely ignored. The resulting collapse is therefore almost non local.

 

[Sunil]

The Madelung equation answers this.

Not really. You wrote "he tried to interpret Ψ as a classical continuous charge distribution". What could be the meaning of a charge distribution over a configuration space?

In the Madelung variables, we have, of course, a preserved flow in configuration space. This makes sense as a probability flow, like the classical Liouville equation. But an interpretation of a continuous charge distribution?

 

[Killtech]

Not really. You wrote "he tried to interpret Ψ as a classical continuous charge distribution". What could be the meaning of a charge distribution over a configuration space?

In the Madelung variables, we have, of course, a preserved flow in configuration space. This makes sense as a probability flow, like the classical Liouville equation. But an interpretation of a continuous charge distribution?

Did he define in on the configuration space or is it just your (or Born's) interpretation? Anyhow, rewritten into the Madelung form which expressed by the velocity vector field rather then its potential is the Euler equations. So the two equations can be interpreted as a time evolution equation of a classical scalar field and its current. So not in the configuration space anymore.

But fair enough, Maxwell-Schrödinger does not allow charge to flow in a circle. Therefore within Schrödingers interpretation the inclusion of spin is motivated purely on classical arguments already which leads to the Maxwell-Pauli and/or Maxwell-Dirac.

 

[Sunil]

Did he define in on the configuration space or is it just your (or Born's) interpretation? Anyhow, rewritten into the Madelung form which expressed by the velocity vector field rather then its potential is the Euler equations. So the two equations can be interpreted as a time evolution equation of a classical scalar field and its current. So not in the configuration space anymore.

Who is he? Madelung? The wave function being defined on the configuration space is straightforward, the standard mathematical apparatus of QT. You cannot describe any superpositional state of two particles without using at least the 6-dimensional configuration space, with two 3-dimensional wave functions you can describe only two independent particles. Rewriting $\psi(q)$ gives $\sqrt{\rho(q)}\exp(i \phi(q))$ and does not change the configuration space. And even if you use velocities instead of the potential $\phi(q))$, you gain only $v^i(q)=\frac{\partial\phi(q)}{\partial q^i}$ with the i running over the dimension of the configuration space.

 

[Killtech]

Who is he? Madelung? The wave function being defined on the configuration space is straightforward, the standard mathematical apparatus of QT. You cannot describe any superpositional state of two particles without using at least the 6-dimensional configuration space, with two 3-dimensional wave functions you can describe only two independent particles. Rewriting $\psi(q)$ gives $\sqrt{\rho(q)}\exp(i \phi(q))$ and does not change the configuration space. And even if you use velocities instead of the potential $\phi(q))$, you gain only $v^i(q)=\frac{\partial\phi(q)}{\partial q^i}$ with the i running over the dimension of the configuration space.

By "he" i meant Schrödinger. Maybe you look up his original attempts to see that his original idea isn't at all what the apparatus of QM repurposed it to be.

And look how you are trapped in your very classical interpretation of particles - since those are restricted to a 6 dim config space only. But Schrödinger didn't even try to describe a particle as such but merely it's empirical behavior that is reminiscent of a fluid in some cases. Forget about the particle idea entirely and rather try to understand that non-linear fluids can create particle like solutions at times (which is knows for some time from classical physics and mathematics). Just understand that that we don't need particles at all to come up with something that makes the same predictions as QM.

A fluid only needs a description in 4 dimensions, 1 for it's density, 3 for its current. If the fluid is restricted from being able to flow in a curl the velocity has a potential. This restriction allows the reduction of its equation to to merely 2 dimensions - like in Schrödingers case.

About the question of having multiple particles... well from non linear equations we know that soliton solutions can be simply combined. One, two, .. n solitons are described within the same single fluid - same as the Maxwell equations don't have a limit of how many waves you can fit in there. so the dimensionality doesn't grow if you add more particle-like solitons. The thing is that a classical field/fluid already has an infinite amount of degrees of freedom whereas a particles has the finite 6 spanning its configuration space. adding more particles requires to upsize the configuration space. On the other hand a field contains infinite information already (a basis decomposition for a field has countable infinite amount of independent members with each coefficient representing one information) so there is enough dimension to fit in any number of particle like objects, so no need to grow it.

Alternatively one could model two solitons by two sperate fluids growing the configuration space in dimension. However if both fluids have the same properties and there is no way to distinguish between them, then you just add a ton symmetries on top of the new dimensions. Those symmetries just say that you have unless baggage information (because that information allows to distinguish between the two which you do not want). removing such information is possible and goes along with reduction of dimensionality (and the process in quite complicated). In practice it just means that you can model the same thing either way.

 

[vanhees71]

Except that what you described there also disagrees with Schrödinger-Maxwell already simply because the equations models a type of wave function collapse (which pure QM doesn't). It is seen in the hydrogen model as spontaneous emission, but it also present in solutions for multiple finite potential walls - which is the simplest approximation of a photo plate.

It's merely the stand-alone Schödinger equation that predicts a smeared outcome. But the classical coupling to Maxwell renders such an outcome unstable. The interesting thing is that the form of the quantum potential enables superluminar interaction such that spatial separation of the distribution is almost entirely ignored. The resulting collapse is therefore almost non local.

There is no collapse in the standard Q(F)T. Some physicists see this as a deficit and implement a collapse, guessing new theories. At this time I don't see any hint that this is a needed extension of QT. Spontaneous emission is well-described in standard QED, and it's the most simple phenomenon showing that you have to quantize the electromagnetic field too and not only the "matter".

Of course if you approximate the matter by the non-relativistic Schrödinger equation (or the corresponding 2nd-quantized QFT formalism) the microcausality condition cannot be fulfilled and thus there's faster-than-light signal propagation. In fully relativistic QED with the matter also described by local relativistic quantum fields, there's no such thing. AFAIK, there's however no relativistically consistent Bohmian reinterpretation that obeys these causality constraints. That's why I don't consider the Bohmian interpretation as a convincing alternative to the minimal interpretation, which describes all known phenomena correctly.

 

[Killtech]

There is no collapse in the standard Q(F)T. Some physicists see this as a deficit and implement a collapse, guessing new theories. At this time I don't see any hint that this is a needed extension of QT. Spontaneous emission is well-described in standard QED, and it's the most simple phenomenon showing that you have to quantize the electromagnetic field too and not only the "matter".

The spontaneous emission in classical QM in usually explained using some hand-waving explanation and applying the formalism of measurement. So any implementation that reproduces that behavior within an extended theory can be understood as a collapse mechanic in QM.

Of course if you approximate the matter by the non-relativistic Schrödinger equation (or the corresponding 2nd-quantized QFT formalism) the microcausality condition cannot be fulfilled and thus there's faster-than-light signal propagation. In fully relativistic QED with the matter also described by local relativistic quantum fields, there's no such thing. AFAIK, there's however no relativistically consistent Bohmian reinterpretation that obeys these causality constraints. That's why I don't consider the Bohmian interpretation as a convincing alternative to the minimal interpretation, which describes all known phenomena correctly.

Well, the thing is that the above mentioned procedure does work with Maxwell-Dirac in a similar fashion. The Dirac equation is 8 dimensional (real dims) and therefore fits in the equivalent of two classical fields and their currents. The one interpreted as probability is limited in velocity, the other one originally interpreted as a charge because it could not be guaranteed to be positive definite is not subject to such restriction. So Dirac actually allows some of the internal information in the wave function to be superluminar and coupling that to Maxwell maintains a collapse mechanic.

Interpreted with Schrödingers view, it seems Dirac limits only the mass field to the speed of light but not charge. On that observation one might note that classically a superluminar charge current flowing in a curl exerts a net attractive force on itself because the magnetic force outweights the electric repulsion. So classically a superluminar spin allows a charged gas to hold together on its own... might be an indication why all charged particles have a spin.

 

[vanhees71]

The spontaneous emission in classical QM in usually explained using some hand-waving explanation and applying the formalism of measurement. So any implementation that reproduces that behavior within an extended theory can be understood as a collapse mechanic in QM.

Spontaneous emission is explained in QED. I don't know, what you mean by "classical QM". QM is just the non-relativistic approximation but it's QT not classical physics.

Well, the thing is that the above mentioned procedure does work with Maxwell-Dirac in a similar fashion. The Dirac equation is 8 dimensional (real dims) and therefore fits in the equivalent of two classical fields and their currents. The one interpreted as probability is limited in velocity, the other one originally interpreted as a charge because it could not be guaranteed to be positive definite is not subject to such restriction. So Dirac actually allows some of the internal information in the wave function to be superluminar and coupling that to Maxwell maintains a collapse mechanic.

Interpreted with Schrödingers view, it seems Dirac limits only the mass field to the speed of light but not charge. On that observation one might note that classically a superluminar charge current flowing in a curl exerts a net attractive force on itself because the magnetic force outweights the electric repulsion. So classically a superluminar spin allows a charged gas to hold together on its own... might be an indication why all charged particles have a spin.

Indeed, one of the reasons for the first-quantization one-particle interpretation of relativistic wave equations (Klein-Gordon, Dirac, etc.) to be inconsistent indeed is the lack of causality. You have to use field quantization (or equivalently Dirac's hole theory) and the interpretation of the negative-frequency modes as antiparticle modes of positive energy to get a consistent causal description via the so and only so possible local quantum fields.

 

[Sunil]

By "he" i meant Schrödinger. Maybe you look up his original attempts to see that his original idea isn't at all what the apparatus of QM repurposed it to be.

Why should I care about Schrödinger's initial attempts to develop quantum theory if I have not that much interest in history? A theory which has only a wave function $\psi(x), x\in\mathbb{R}^3$ is of no interest for me as a physical theory, given that it is obviously unable to describe entanglement.

And look how you are trapped in your very classical interpretation of particles - since those are restricted to a 6 dim config space only. But Schrödinger didn't even try to describe a particle as such but merely it's empirical behavior that is reminiscent of a fluid in some cases.

I'm not at all restricted to dimension 6, this was simply the smallest dimension for the configuration space. That's why I wrote "at least".

Forget about the particle idea entirely and rather try to understand that non-linear fluids can create particle like solutions at times (which is knows for some time from classical physics and mathematics). Just understand that that we don't need particles at all to come up with something that makes the same predictions as QM.

Particles are indeed irrelevant. We can do field theory. But to obtain the same predictions as QM you need not only something particle-like, but you also need entanglement. And that is the part where you have no chance.

A fluid only needs a description in 4 dimensions, 1 for it's density, 3 for its current. If the fluid is restricted from being able to flow in a curl the velocity has a potential. This restriction allows the reduction of its equation to to merely 2 dimensions - like in Schrödingers case.

This dimension is about something completely different.

Standard quantum theory works in $\mathcal{L}^2(Q,\mathbb{C})$. Your fluid theory works in $\mathcal{L}^2(\mathbb{R}^3,\mathbb{C})$. The choice of Q depends on the problem you consider, similar to the classical case where in the Lagrange formalism you also have the same configuration space Q. You have no chance to get all quantum effects using Schrödinger's one-particle theory in $\mathcal{L}^2(\mathbb{R}^3,\mathbb{C})$, in whatever interpretation and with modifying the Schrödinger equation in whatever way. You will fail, and it is known where you will fail, namely in the description of entanglement.

 

[Killtech]

Spontaneous emission is explained in QED. I don't know, what you mean by "classical QM". QM is just the non-relativistic approximation but it's QT not classical physics.

I know that. What I indented to say is if we look solely at spontaneous emission from classical QM (i.e. the stuff that students are taught in their first lecture about QT) then in this framework some try to explain it in the way that an energy in a higher energy eigenstate is slightly perturbed such that it gets into a superposition state with another energy state and then it behaves like at some point later it's energy is being measured which causes it to take a pure eigenstate again.

This is of course not a real explanation, but it shows that the process has a certain analogy with an energy measurement.

QED of course explains it in an entirely different fashion. But if you try to compare it to the primitive explanation attempt in QM, it still does formally implement something that fits QTs description of an energy measurement operation.

Indeed, one of the reasons for the first-quantization one-particle interpretation of relativistic wave equations (Klein-Gordon, Dirac, etc.) to be inconsistent indeed is the lack of causality. You have to use field quantization (or equivalently Dirac's hole theory) and the interpretation of the negative-frequency modes as antiparticle modes of positive energy to get a consistent causal description via the so and only so possible local quantum fields.

Well, I find causality is a hard to grasp concept for time symmetric systems with full two way interaction. The separation of cause and event feels completely artificial to me in that case. You take a solution of a system and tag one thing a cause and something else an effect, you reverse the time and the solution stays valid yet cause and effects are reversed. The concept is not well defined for systems that fulfil detailed balance. I prefer the word interaction. The word "inter" points out no direction of action and in reality it is just a question how we define the time coordinate. Why should we studently care about coordinate choices? More physically speaking changing Einsteins synchronization convention to something else changes causality but why should something that isn't physically observable be relevant for us?

 

[Killtech]

Why should I care about Schrödinger's initial attempts to develop quantum theory if I have not that much interest in history? A theory which has only a wave function $\psi(x), x\in\mathbb{R}^3$ is of no interest for me as a physical theory, given that it is obviously unable to describe entanglement.

Why do you think that, mathematically speaking? What is the basis of entanglement expressed in terms of information? it is that some information encoded in a system remain correlated in a special way. This is something that is also true for fluids without inner friction (which kind of works like decoherence for information). Take for example a superfluid at rest and make a disturbance such that a pair of counter rotating vortexes is created. Now because there is no friction, these vortexes will persist indefinitely and keep their orientation and because they were created as a pair their orientation will remain correlated regardless how far they venture apart until maybe another external disturbance of the fluid destroys it. So superfluidity allows recreating preservation of information needed for entanglement. Now the remaining part of engagement is the forced extraction by "measurement" transforming the originally deterministic information into a probability outcome - a type of collapse mechanic can fit that role, and apparently Schrödinger-Maxwell involuntarily has one.

Mathematically it is not relevant how the information is encoded into your state description. It just is relevant that it is guaranteed not to get lost in the time evolution.

But you are right, in that Schrödinger-Maxwell ist just a first simple proxy not able to single-handedly reproduce every single quantum effect known on its own. You need other more general equations for that, like some of which i have named here - same story as it is in QT. The interpretation is what guides towards those more general equations and motivates the things to look out for.

 

[Sunil]

Why do you think that, mathematically speaking?

I see that the mathematical structure you use is quite different from the one used in standard quantum mechanics. Therefore I would need more than a vague hope, namely a proof, to accept their equivalence.

So superfluidity allows recreating preservation of information needed for entanglement.

This was some vague qualitative consideration. By the way, superfluidity is a quantum effect. So what you have used here was a quantum effect to explain another quantum effect. Moreover with your words you explain only correlations, which could be quite classical correlations. But correlations are not sufficient to explain violations of the Bell inequalities.

Mathematically it is not relevant how the information is encoded into your state description. It just is relevant that it is guaranteed not to get lost in the time evolution.

There are other relevant things. Like the notion of locality. In de Broglie-Bohm theory we have all we need to identify locality, namely a deterministic theory, a continuous trajectory $q(t)\in Q$ with a velocity used in the equations. This would be a good base to prove the Bell inequalities if $Q\cong \mathbb{R}^3$. Namely, the theory is local in Q. Thus, with $Q\cong \mathbb{R}^3$ it would be local in $\mathbb{R}^3$. This is, of course, not yet Einstein-local. The dBB velocities are known to exceed c around zeros of the wave function. But these are rotations around the zeros which have no effects on the density, thus, are quite harmless. So it seems not unreasonable to hope for something like a Bell theorem for $Q\cong \mathbb{R}^3$.

But in another Q, say $(x,y)\in Q \cong \mathbb{R}^6$, a change in x immediately has consequences on the velocity of y because the wave function which is relevant for it depends on q, that means, on x too. Locality in Q alone gives you nothing.

But you are right, in that Schrödinger-Maxwell ist just a first simple proxy not able to single-handedly reproduce every single quantum effect known on its own. You need other more general equations for that, like some of which i have named here - same story as it is in QT.

That means, you have not more than a vague hope. Not yet a base to question established QT.

The interpretation is what guides towards those more general equations and motivates the things to look out for.

This is a point I agree with. Except that you have no interpretation of QM here. You have only an interpretation of a single particular case of QM, namely $Q\cong \mathbb{R}^3$.

 

[PeterDonis]

A fluid only needs a description in 4 dimensions, 1 for it's density, 3 for its current.

No, the density and current are not 4 dimensions. They are 4 functions of space and time. But, as has already been pointed out, that's not enough information to make correct predictions about things like experiments on entangled systems. A fluid model of this type cannot produce violations of the Bell inequalities.

You labeled this thread as "I" level, but the paper you referenced, and indeed this topic in general, is "A" level (and I will be editing the thread level accordingly). From your posts I am not sure you have the requisite background to understand either the claims your reference is actually making, or their limitations. Also, you have only given one reference, which means you are only seeing one side of the story. You should look for other papers critiquing the one you referenced.

 

[PeterDonis]

Moderator's note: Thread level changed to "A".

 

[vanhees71]

Standard quantum theory works in $\mathcal{L}^2(Q,\mathbb{C})$. Your fluid theory works in $\mathcal{L}^2(\mathbb{R}^3,\mathbb{C})$. The choice of Q depends on the problem you consider, similar to the classical case where in the Lagrange formalism you also have the same configuration space Q. You have no chance to get all quantum effects using Schrödinger's one-particle theory in $\mathcal{L}^2(\mathbb{R}^3,\mathbb{C})$, in whatever interpretation and with modifying the Schrödinger equation in whatever way. You will fail, and it is known where you will fail, namely in the description of entanglement.

Even for single particles there can be entanglement. E.g., for a particle like a neutron with spin and an associated magnetic moment going through a Stern-Gerlach magnet, it's momentum is entangled with the spin-component in the direction of this magnetic field. This can be well described in non-relativistic QT, using the Pauli equation, which is just the one-particle Schrödinger equation of a particle with spin 1/2 interacting with the electromagnetic field.

 

[Sunil]

Even for single particles there can be entanglement. E.g., for a particle like a neutron with spin and an associated magnetic moment going through a Stern-Gerlach magnet, it's momentum is entangled with the spin-component in the direction of this magnetic field. This can be well described in non-relativistic QT, using the Pauli equation, which is just the one-particle Schrödinger equation of a particle with spin 1/2 interacting with the electromagnetic field.

But you need already $\mathcal{L}^2(\mathbb{R}^3\times \mathbb{Z}_2,\mathbb{C})$ for this, $\mathcal{L}^2(\mathbb{R}^3,\mathbb{C})$ would not be enough.

Moreover, I'm not sure if this entanglement is sufficient to construct a violation of the Bell inequalities, given that this would require different measurements localized at different positions.

 

[vanhees71]

Good question. Quickly googling, I found this (open access):

https://www.nature.com/articles/s41598-021-85508-8

I've still to read it in detail though.

 

[Killtech]

I see that the mathematical structure you use is quite different from the one used in standard quantum mechanics. Therefore I would need more than a vague hope, namely a proof, to accept their equivalence.

But, as has already been pointed out, that's not enough information to make correct predictions about things like experiments on entangled systems. A fluid model of this type cannot produce violations of the Bell inequalities.

All right, fair enough. Let's take the example of CHSH as a simple example and just check what framework we need to build some empirical recreation of that - for example to simulate it on a computer. Then we can look what it means to implement the requirements within a fluid description. But note, that the only thing special about entanglement is that it can be maintained non-locally.

The information we need to save somehow is the spin orientation $S_p$ as a vector - in QT this corresponds to the axis along which the spin is deterministic when measured (i.e. the state is pure). either we store $S_p$ globally or locally but then we need two copies for the two objects which we need to keep synchronized. When measured by a detector set along $S_m$ we say that $S_p$ is forced to realign parallel or antiparallel which we determine by a random weighted coin flip depending on the cosine angle between $S_m$ and $S_p$ such that the probability distributions always match with QT prediction. in the local implementation we still need to synch up with the other copy of $S_p$ (that's really the non-trivial part) and we are done.

The synching is obviously non-local as it is instant - let's schedule the discussions that "instant" is not well defined with a relative concept of simultaneity for now. Just note that the resulting probability distributions don't change if the order of measurement is changed so it's something that only weak measurements could run into an issue with, not CHSH itself.

Anyhow, with a fluid we can easily save two vectors via two vortices, but we cannot implement "instant synching" with a locally interacting fluid equation obviously. Let's however keep in mind that we could employ a separate collapse mechanic, same as one QT interpretation does, though we just need standard Kolmogorov probability theory in the case we interpret the wave function as describing a physical fluid.

Alternatively, looking how Schrödinger-Maxwell itself implements a similar collapse, we notice that it doesn't happen instantaneously but it always takes some time. So if we were just to relax Bell's/Neumann's prerequisites a little bit by assuming measurement interaction does not happen perfectly instantaneously but is a very fast process that settles for a final value, we can look for a fluid equation that capable of that.

Now, the type of equations like Maxwell-Schrödinger with classical EM-coupling produce a local yet superluminar interaction like that. The description is in all instances time-symmetric. Interestingly this combined with the locality make the process entirely unaffected by relativity of simultaneity and clock synchronization conventions. Also note the original paper describes the dynamic of the system with the terminology of attractors for a reason as non linear equations have a habit of being sensitive to initial condition thus creating chaos and pseudo-randomness.

 

[akhmeteli]

Just a couple of relevant references.

Sebens, C.T. Electron Charge Density: A Clue from Quantum Chemistry for Quantum Foundations. Foundations of Physics 2021,51:75, 1–39.
Barut, A.O. Combining Relativity and Quantum Mechanics: Schrödinger’s Interpretation of ψ. Foundations of Physics 1988,18, 95–105.