If by "quantum non-locality" you mean ``entanglement'', I do not think that, in principle,

complex Minkowski space has any role in that.

Complex Minkowski space could play a fundamental role at short distances, when quantum mechanics and general relativity merge. In fact, Born's Reciprocity Principle, which proposes an invariance of the laws of nature under the interchange of four position and four momenta, implies the existence of a minimal (Planck) length and, therefore, a ``granular'' structure of space-time (see e.g. http://arxiv.org/abs/1006.5958 for a discussion and references). You could also see here a kind of ``non-locality''.

Maxwell's equations are *classical* equations-so they cannot describe quantum phenomena, in general and entanglement in particular-they don't describe photons, but (classical) electromagnetic waves.

Minkowski space(time) is, also, classical.

If you want to describe *quantum* effects, you need to introduce quantities that depend on Planck's constant and look at the consequences this leads to.

It's essential to keep in mind that quantum mechanics is, usually, expressed in *phase space*, not *spacetime* and a lot of confusion results from mixing the two. Quantum mechanics in spacetime is described through the path integral formalism of Dirac and Feynman. But here all quantities are *classical* and quantum effects are described in a more indirect fashion.

Dear Manuel and Stam, thank you for your answers. Yes quantum effect cannot be derived from classical equations, nonetheless there is known correspondence at least on theory between the two. For instance, Ward and Volkmer show how Schrodinger equation can be derived from classical wave equation (see http://arxiv.org/pdf/physics/0610121.pdf). Of course there is missing planck constant in classical wave equation. Furthermore, others are able to find connection between Schrodinger equation and Newton law ( such as Edward Nelson). But i am not sure if there is nonlocality in classical wave equation.

Stam -- Please enlighten this non-expert in quantum theory: Is there no inverse canonical transformation to take quantum theory in phase space into space-time?

Dear Victor,

You ask on quantum non local correspondence with Maxell equations and these are at least two very different aspects.

Classical Electrodynamics is a local theory if we think that it cannot take into account the topological terms due to have the Maxwells equations written as local fields. In fact they are obtained from an action which only use the Lorentz scalar invariant associated to the fields and it forgets the pseudoscalar invariant E.B which is the one that could give topological non local information.

On the other hand Classical Electrodynamics can be quantized as it was done in QED quite sucessfully. And the introduction of topology in Electrodynamics is made through the Chern numbers (or other topological number) associted to the fields and in fact to invariants as the Lorentz invariant that was mentioned previously. Mathematically the best form to glue the local and local properties of Electrodynamics is to represent it in a principal fiber bundle with the characteristic classes associated.

Finally there is not a unique correspondence between a local theory and a global or non local one: they depend strongly on the boundary conditions of the functional space of their solutions. This is one of the great problems for trying to quantize the Einstein's gravity. I attach you one paper which shows it.

There does exist a transform from phase space to space-time, the Wigner transform. Of course it's not covariant since quantum mechanics is non-relativistic, but that's beside the point. This is an operator-valued transform in one-particle quantum mechanics and doesn't have anything to do with Maxwell's equations, however.

By definition, quantum effects are functions of Planck's constant. Schrödinger's equation describes quantum effects, *because* it contains Planck's constant. The quantity of interest, the wave function is a function on phase space, NOT space-time, whatever people may claim. To pass from phase space to space-time one either uses the Wigner transform of the wave function or the path integral formalism-Feynman's paper of 1948 in Rev. Mod. Phys. is the reference on the subject.

In the path integral formalism, Planck's constant enters in the phase factor, that weighs each path.

In Nelson's formalism, Planck's constant enters in the variance of the noise term.

It's a new constant, unrelated to anything else. Since it's dimensionful, we can choose *units* such that its numerical value, in those units, is equal to 1-but that doesn't mean it has disappeared from the physical consequences of the equations.

What's interesting is that one can obtain *classical* effects as limiting cases of *quantum* effects.

Thank you, Daniel and Stam. I will read your paper. Best wishes

Thank you, Stam.

You're welcome. Another point to keep in mind is that the generalization of the Wigner transform to systems of an arbitrary number of particles, i.e. a field theory, is hard to get. The way to describe therefore field theories, in a way consistent with their symmetries, classically and quantum mechanically, is the path integral formalism. The lattice regularization then leads to a formulation that is, also, mathematically well-defined. Here, however, most explicit calculations require numerical methods, but can be checked by mathematical analysis.

Of course all this holds for *fixed* spacetime background. How to describe quantum effects, when the space-time can fluctuate, too, is, still, an open issue.

I think that the question was very clear, although complex or impossible to answer as it stands: Can quantum nonlocality be explained using (complex) Maxwell equations?

Obviously, no; if we follow it literally. But we can extend a little bit the response for distinguishing quantum and nonlocality from classical theories as the Maxwell's Electrodynamics. I have attached a paper devoted to explain how there are not univoque answer when you pass from a classical to quantum theory ( best example and no resolved: Einsten's gravity).

Let me to tell you that the Hamiltonian formalism in Electrodynamics is not allowed univocally too because there are not a Legendre transformation independent of the gauge and also the electric field doesn't depends of a time derivative of the scalar potential. See for instance:

D.Baldomir and P.Hammond, Geometry of Electromagnetic Systems, Clarendon Press, 1996.

I attach you another reference more for trying to explain the difficulties and where they are for both concepts: quantum and nonlocality.

@Spiros. Thank you for your answer, but can you please be more specific on which paper that you cited?

@Daniel. Thank you for your comments, yes I agree with you that classical electrodynamics cannot describe quantum phenomena, but as Gersten and also Raymer and Smith have shown, it is possible to derive Maxwell wave function for photon. See for instance, http://arxiv.org/pdf/quant-ph/0604169.pdf. So it seems there can be correspondence at least in theory to describe quantum mechanics in terms of electrodynamics.

Furthermore, i read elsewhere a quote by a physicist who says that up to now there are two theories to describe electron, first using Maxwell theory, and second using quantum mechanics. Since experiment shows that electron can behave like particle or wave, then it seems that there should be coherent picture between quantum mechanics and clasical electrodynamics. If Maxwell theory cannot explain quantum mechanics, does it mean that we should better use nonlinear electrodynamics? Best wishes

Maxwell's electrodynamics doesn't know what is an electron and I do not know what is non linear electrodynamics. In quantum mechanics the electron may be described by a de Broglie matter wave which is very far of classical electrodynamics.

The Schrödinger equation, although we always speak about the wave function, it is not a wave equation at all. Meanwhile the photons or the electromagnetic waves follow a real wave equation equivalent to Maxwell's equations.

In QED you can combine them but separating clearly one of the other.

The paper that you recommend is clearly wrong, besides it doesn't prove anything and only write analogies for equations, what call Maxwell's equations in the flow of fig.1 are nothing more than Faraday and Ampere when the density of current is zero. Notice that in this case even the fields are not determined following the Helmholtz theorem.

Thank you, Daniel, for your answer. Yes we can use De Broglie matter wave to describe electron, but then we have a wave function. But the Schrodinger wave function only has probabilistic meaning. You are an expert, i hope you don,t mind if i ask one more question: is there electromagnetic picture of the quantum wave function? Thanks

No, the wave function even is without physical meaning. What has meaning is its square following Max Born and gives the density of probability only!

Ok, i understand what you mean that the wave function does not have a physical meaning. Then if we follow your argument that electron can be described by a wave function, but the wave function only has probabilistic interpretation, then it would mean that the electron itself is not a physical entity. Am i right? I think that is part of the reason why Einstein-Podolski-Rosen wrote in 1935 that the quantum mechanics is incomplete. Best wishes

The electron itself is not described by the wave function, what describes the wave funcion as an eigenfunction of the Hamiltonian y the eigenenergy or its momentum. Thus it is posible to know a wave length or a frequency of a real wave movement. This is the de Broglie wave of matter, broadly speaking.

In any case I recommend you to read a basic quantum mechanical textbook as:

Claude Cohen-Tannoudji, Bernard Diu, Frank Lalöe, Quantum Mechanics,

Vol. I and II.

That you can find in internet and it is very good and exhaustive, from my humble point of view.

Thank you, Daniel, for the reference, i will try to find it. Btw, there is new paper by Blackledge and Babajanov (2013) whihch discuss wave function solution based on correspondence between Helmholz and Schrodinger equations. See http://arrow.dit.ie/cgi/viewcontent.cgi?article=1080&context=engscheleart2&sei-redir=1&referer=http%3A%2F%2Fwww.google.com%2Furl%3Fsa%3Dt%26rct%3Dj%26q%3Dhelmholtz%2520schrodinger%2520correspondence%2520greenleaf%26source%3Dweb%26cd%3D3%26ved%3D0CDAQFjAC%26url%3Dhttp%253A%252F%252Farrow.dit.ie%252Fcgi%252Fviewcontent.cgi%253Farticle%253D1080%2526context%253Dengscheleart2%26ei%3DrvYQU_bkK4yCrgff5oGgBA%26usg%3DAFQjCNHTmlZruy9zTnrPZowKp5r1g-6rjA#search=%22helmholtz%20schrodinger%20correspondence%20greenleaf%22.

Another paper discussing Helmholtz equation is written by Greenleaf et al., but they only use it to solve invisibility cloaking problem. Best wishes

Dear all, thank you so much for all your answers and comments. Allow me to share a summary of this discussion in a pdf file. Hopefully this discussion may stimulate further investigation. best wishes

Recently with my work on the Synthetic Physics see researchgate Germano Resconi, I found that Maxwell equations give us the basic scheme for gravity and quantum mechanics, Now because in quantum mechanics we have no locality we argue that the scheme of classical Maxwell Equation give us the non locality wihtout to introduce complex number

I would caution all those who say that the wave function of the matter wave has no physical meaning, that this is one philosophical viewpoint but far from proven. In field theory, the electromagnetic wave and the matter wave are quantized in exactly the same way, with just the difference of commutation operator: photons and other bosons commute, while fermions anticommute. Otherwise the fields of both are treated exactly the same. It would then be hard to argue that they must be utterly different phenomena.

In classical physics, we use position (x, y, z) and momentum (Px, Py, Pz) to describe a particle, along with it's other characters, such as mass, etc.; in quantum physics, the position-momentum description is replaced by a more general description, namely, the wave function. The wave function has every physical meaning you would ask, beyond the simple probabilistic implications of where the particle is. For example, studying it's phase behaviours led to the Anderson Localization. It also covers other characters of a particle normally not important in classical physics, such as spin. Regarding Einstein-Podolski-Rose' 1935 paper, I would like to remind everyone in this discussion that it was simply to point out that the classical quantum mechanics interpretation of wave function can lead to a paradox, also known as EPR Paradox, against the causality implied by basic assumption in the special relativity.

@Resconi and @David. Thank you for your answers. So do you think that classical wave equation corresponds to wave function? Can you prove your argument? Because the standard quantum interpretation is that the wave function only has probabilistic interpretation.

@Jian-Yang. Thank you for your answer. So do you think that there is spin description too in classical wave equation? Please clarify. Thanks.

No.

The wave function in physics is a clear object: amplitude of probability, i.e. it has not physical units. Actually what is very important is that it corresponds to act as an state in front of a Hamiltonian operator for giving us the eigenvalues as the quantized energies associated to such statates and with discrete spectrum. This is what corresponds to the experimental values able to be reproduced and well measured.

On the other hand, the potential scalar of electrodynamics is also with a very good meaning: energy by unit of charge extended in the space surrounding electric charges, which is conservative if time is not involved. The units are volts.

Say, there are no possibility to interprete an amplitude of density of probability (without units) with a scalar potential (volts) which everyday is measured not only in laboratories but in normal life.

@Daniel. Thanks for your answer. Best wishes

@Johan. Thank you. So do you think that nonlocality can be explained using Maxwell equation (without Copenhagen interpretation) ? Best wishes

In the "Explanation" offered you do not throw away EPR / Bell's inequalities', but rather use Hadamard's principle about the truth to suggest that "local realism" is a wrong concept... Is this the bleeding edge of the theory? Or does it have predictions that would not only repackage QM but show where it is plain wrong?

Dear Johan,

After reading your argument:

What I clearly argued, and if you are willing to read the literature on my arguments, is that within the volume of a SINGLE EM wave, local realism does not exist since the EM-energy is continuously distribited mass energ (you must know about E=m*c^2).

Please, what is the mass of an EM? Do you think that the photons have rest mass? Do you think that an EM have a center-of-mass: what is the physical meaning? May you reduce the motion of an EM to a point where it is the center of mass? Wonderful, please don't waste your time writing here and make a good publication. That is a new PHYSICS

Dear Johan,

The energy of a photon given by Einstein is E=pc, where p is the linear momentum and c the velocity of light. No E=mc² because the mass at rest for a photon must be zero and you cannot reach by Loretz transformations a another mass different. If you do not want to use quantum mechanics, you can use the Poynting vector for defining the linear momentum p; but notice you never use the concept of mass there.

The center of mass of system is a trick for reducing many diferent bodies in motion to only one. But in this case is usefulness because the center of mass of photons cannot be followed in space and time, notice that a monochromatic wave is not localized in a point and in fact it must have also a non defined length (no wavelength obviously).

@Johan. Thank you for your answer and posting, your ideas seem very interesting. Allow me to ask some points:

- did you write a paper describing your idea on nonlocality explained by classical electrodynamics? I just found a paper by Orefice in researchgate where he discusses helmholtz entanglement. But it lacks derivation of nonlocality. So i think it will be useful if you can prove nonlocality from maxwell equations.

- It is interesting that you assume photon mass. That reminds me to Proca equations, which are an extension of Maxwell equation. Vigier has derived an estimate of photo mass.

- it would be very interesting if you can also explain in a paper about your idea to describe rtsc (room temperature superconductivity). I read Tajmar paper which describes superconductor electrodynamics using proca equations, but i am not sure if it can explain rtsc.

Your further comment will be appreciated. Best wishes

Quick question: in this complex EM what happens to the tensor F_mu_nu and the vector potential A_mu, and in particular, how does it affect the Aharonov-Bohm effect?

Johan, thank you for the url. I will read it. Concerning publications, yes i know that publishing non standard papers in iop or prl is almost impossible, but if you wish you can try non mainstream journals, like Annales de la Fondation Louis de Broglie. They accept novel ideas. I once tried myself to publish in prl but it has been rejected. Best wishes

@Asher. Thank you for your question, I will try to answer that question. Perhaps complex minkowski is not needed to explain quantum nonlocality, maxwell equations may suffice to do that. Regarding aharonov-bohm effect, there is a paper by Oscar Chavoya-Aceves in arxiv which describes that effect from classical electrodynamics. Best wishes

Oops.... looks like entered the wrong chatroom.

Dear Johan,

You are absolutely wrong and if you allow me one advice: don't write things that people can show as yours, future is very long and full of serious people making a hard work.

P=mc cannot be true because Einstein relativity doesn't allow it. m different of zero and moving at the velocity of light is forbidden as one student of first course in Physics knows.

Physics is not a free interpretation of the reality but only one which the natural laws permit

Dear Johan,

Let me see how mass tranforms m/ squrt (1- v²/c²), say if m different to zero then this is not defined because is m divided by zero (no defined operation in my kindergarden). When you put p=mc, you obviously assume that m is different to zero. Did you understand me?

Dear Victor,

When one opens a forum of questions I think that it is necessary to be responsible of what is said there and at least trying to understand the answers because we cannot squander the trust palaced in us, in other case this interesting place could be absolutely out of interest. At least I have tried to do it always.

For all that I have been obliged to give you a bad mark to the question and to form of drive it. I am sorry but it is very difficult to understand that one person who doesn't know that

E²=( mc²)² + (pc)²e

is the energy in special relativity or he answers about the Aharonov-Bohm in the form that he has done to another person.

Without e sorry.

Dear Johan,

For ending, if you put m=0 in the last formula, i.e. the rest mass of the photon, then you obtain the one that I told you E=pc. Perhaps you are right and I am very ignorant but never have distinguished between dynamic mass or rest mass. Rest mass is the one with velocity zero respct to the observer, but this is not related with "dynamic mass" in my humble knowledge.

@Johan and @Daniel. I think both of you agree that photon has no rest mass. I have tried to check another forum on this, see for instance: http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/photon_mass.html. I think the standard physics says that photon cannot have rest mass. However, few physicist like the late Prof. Vigier suggest that photon has rest mass. But that is another issue.

@Daniel. Thank you for your comment, but i think a question like this in researchgate is valid at least because of two reasons: a. It is a sincere question ( because somebody does not know the answer, and wants to know it), b. it is intended to attract genuine answers from scientific colleagues. However, this is not a moderated forum that is why i cannot edit or become reponsible for any comments by others. Nevertheless, so far i always find that non mainstream ideas as well as mainstream ideas are very interesting to ponder, although in physics the ultimate test is with experiment. Best wishes

The argument about zero photon mass is tightly tied up with gauge invariance. Yet, we allow gauge symmetries to break (e.g. electroweak theory). Gauge invariance is asked for by Maxwell equations -- that is -- if you solve ME you get the vector potential A as a reduction or a common platform for E & B. The substance of the fields are now the photons - quanta of A - and the invariance is with respect to a phase (U(1)). For Maxwell, 'A' was in fact, a statement about locality - he didn't like the formulation offered by the flux. The interesting point about all this is that in QM the 'mathematical' nature of the vector potential becomes a reality through the AB effect - but thereby extenuates locality. Nobody likes the non-local manifesto of QM but we have learnt to live with it quite efficiently - that is - there are no unsolved paradoxes, only conceptual difficulties. Aharonov suggests that if we move to modular variables of the canonical operators then together with 'total uncertainty' we recover some intuition about QM's interference. It's worth noting that physics is an empirical science where dogmas are tested in the lab. The dogma of 'local realism' has been proved to be wrong yet QM holds because causality is rescued with uncertainty. It is a mater of taste should someone chose to 'live' in a multi-verse and involve conscience into the physical realm but when it comes to the lab experiments tell the story.

In the Synthetic Physics researchgate Germano Resconi I discover that Maxwell equation can be a prototype of a family of theories like Maxwell equation where we include quantum mechanics and gravitation as theories of the same family but with non locality. So Maxwell was the father of a set of theories beyond electromagnetic phenomena

@Asher and @Resconi. Thank you for your answers. Regarding multiverse hypothesis, i think it would be more interesting if it yields some testable predictions. However, i read that many leading physicists seem to accept multiverse/many world intepretation without question. Best wishes

In my paper I have no hypothesis but modells generate with maxwell equations. In the paper of Synthetic Physics you can see a lot of testable predictions. Thank you for your replay. Please read the paper and give me your important contribution

Germano

In my paper on the Synthetic Physics as you can see I have a lot of testable predictions. In my work we have no hypothesis or many world interpretations but we have a synthesis of many different theories in one meta theory that is built like Maxwell equations. If you have time to read the paper I will be very gratefull to have your important remarks

@Resconi. Ok, i will try to read your paper soon. Best wishes

@Resconi. I just read your paper at a glance, and allow me to make a few comments. While i am no familiar with all matrices, i think your maxwell description is interesting. However, i don,t know why then you choose to connect qm and fisher entropy, which lead you to write a new wave mechanics with pure probabilistic meaning. Won,t it be much more interesting if you can connect your maxwell description to the wave mechanics? Therefore you will find an electromagnetic/deterministic interpretation of wave function. for an example of deterministic qm, see 't Hooft.

Of course, that is up to you. But if read the comments by Einstein, De Broglie and Schrodinger, all of them don,t like the probabilistic interpretation of qm. Furthermore, if you read the original paper by Schrodinger (1926), then you will find that he prefers to interpret his wave function from vibration. In this regard, i plan to write a paper soon on this vibration interpretation of qm. So if you like, perhaps we can write paper together on this issue. Best wishes

I think Asher is perfectly right.

We can say and quote many things. But the problem is: whether one believes that

the equations represent tha materialities involved.

If you are a non-believer: the Church will kill you.

If you believe: no one would care.

With all due respect, one has to clear up the fundamentals first. If we do not have them, experiments at least will help us dismiss the rubish.

Best regards to you all

@Orlando. Thank you for your answer. Best wishes

Thank you, Arno. Do you have a paper describing how you can get there? Best wishes

Victor,

You wrote,

"...if you read the original paper by Schrodinger (1926), then you will find that he prefers to interpret his wave function from vibration."

This is misleading (though it refers to the misleading title 'Quantization as Eigenvalue Problem'). On p. 513 of the 2nd Commun. Schrödinger requires, that it should exist special mathematical solution methods for the stationary Schrödinger equation, which account for the non(!)-classical nature of the problem of quantization. - Such a method is given by the recursion formulas, which uniquely solve that equation without (!) using boundary conditions.

Best wishes,

Peter

Thank you, Peter, for your answer. Best wishes

Johan,

The method is described in the following publications.

- P. Enders & D. Suisky, Quantization as selection problem, Int. J. Theor. Phys. 44 (2005) 161-194;

- P. Enders, Von der klassischen Physik zur Quantenphysik, Springer 2006;

- http://cdn.intechopen.com/pdfs/41542/InTech-Quantization_as_selection_rather_than_eigenvalue_problem.pdf