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Killtech

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- 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?

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?