Dear Arbab,

I think that the most that you could recover from just the Dirac equation is Faraday's law of induction and Gauss' law of magnetism, by inspecting the Bianchi identity on the U(1) connection. This has nothing to do with wave/particle duality.

Best wishes,

David

I agree with David that you could recover from just the Dirac equation is Faraday's law of induction and Gauss' law of magnetism

Yes, it is, I read a paper a couple of years ago. This paper did it brilliantly!

Dear David

Thank you for your answer.

Dear Golden

Thank you for your positive reply. I would be grateful if you let me know about that.

Best regards.

No, Maxwell's equations describe a massless, neutral,  spin-1 field, Dirac's equation a spin-1/2 field, whose charge isn't defined a priori.  These are field equations in spacetime.  The two fields have totally different properties: the first one transforms as a 4-vector, the second as a bispinor, under Lorentz transformations  in four spacetime dimensions.  This is all textbook material-cf. the first chapters of Itzykson and Zuber's book, ``Quantum field theory'', for example. (They can transform into one another under supersymmetry transformations, but then the spinors are Majorana, not Dirac. Cf., for instance Weinberg's vol. III, ``The quantum theory of fields''). 

As Stam Nicolis said, you don't derive Maxwell's equations from Dirac's; they describe two different things. Appropriately combined together, they yield quantum electrodynamics.

Moreover, Maxwell's equations don't even need to be derived in the first place. Once you postulate the existence of a thrice differentiable vector field, a metric, and define a massless current, the equations emerge as geometric identities.

The connection between the two is in the form of an interaction term: i.e., what Maxwell's equations "see" as currents and charges are, in fact, the charged fermions that obey the Dirac equation. Conversely, potential in the Dirac equation corresponds to the electromagnetic field that obeys Maxwell's equations.

This becomes evident when you study the appearance of the QED Lagrangian and compare it to the free field Lagrangians of electrodynamics and fermions, respectively.

Yes it is oossible...

Yes Sir! Second way is right. Maxwell' equations may be basic for all known phenomena in microworld. The h and a (1/137) before needed to involve there for it (then these will become Maxwell's quantized equations), then you can deduce (principally) all known correct-working relations from there. There need well enough math skills only. Some ideas you can find from my works - if to spend time.

http://vixra.org/author/george_kirakosyan 

Maxwell's equations describe _classical_ electromagnetic (EM) fields, i.e., the classical limit of photons in relativistic quantum field theory.

Dirac's equation, by contrast, describes a relativistic electron (generally fermions) rather than EM fields. I recall you can include the interaction of electrons with the EM fields in a semiclassical way, though.

Maxwell equations can be written in a Dirac-like form (see Laport and Uhlenbeck, Phys. Rev. 37, 1380 (1931)). This relationship follows from van der Waerden 2-spinor formalism. Moreover, it leads to  Riemann-Silberstein representation for electromagnetic field and Majorana-Oppenheimer quantum electrodynamics.

Dear Arbab,

You pose a very interesting question and you have received many replies that give, I think, good answers regarding the (different) nature of both systems spinors and photons, and so the impossibility to deduce one from the other (although the paper by Laport and Uhlenbeck cited by Vadim shows a very interesting game involving the equations for spinors and photons).

Along similar lines, it may be of your interest a discussion posed by Feynman many years ago and reported in a paper by Dyson, "Feynman's proof of the Maxwell equations" (http://scitation.aip.org/content/aapt/journal/ajp/58/3/10.1119/1.16188). In this work Feynman approaches the problem in quantum mechanics of finding the most general system consistent with canonical quantum commutators. You can view it as looking for the most general potential that can be consistently quantized with standard canonical commutators between coordinates and momenta. There is one appealing feature of the approach which is that the question is posed at the level of the equations of motion, without demanding any Lagrangian nor Hamiltonian. The answer is that the most general potential with these characteristics is the coupling of the point-like particle to the electromagnetic potential (the Lorentz force), and, if I remember correctly, it is even possible to obtain the dynamical equations for it. (A deeper insight into the problem reveals that imposing a set of standard commutators is equivalent to demanding the existence of the Lagrangian, thus the form of the Lorentz force's potential as output of the exercise.)

With this exercise in mind we may wonder if it's possible to perform a similar consistency computation involving Dirac's equation AND a set of given quantum commutators for fields and fields-momenta, and check that the possible consistent couplings are actually those given in QED Lagrangian.

Dear Arbab,

Dirac's equation was derived using "quantum logic".

Maxwell's equations were derived using "classical logic".

These two mindsets represent two separate independent mathematical approaches. Thus, to obtain one equation from the other equation would require adjustments in one or the other's logic; but if this is done, then one of the parent equations would be taken out of its realm.

Please see "Putting Quantum Mechanics in Perspective" on this site, which gives a more exhaustive view on this subject.

Regards,

Dan S. Correnti

Hi 

If I understand what you are asking, in a certain sense, U'd like to understand the mechanism that the Nature uses to allow a Spin1/2 particle to interact with a Spin1, being different and nevertheless with and without mass.

I know that there are two important factors that make the thing:

on one hand there is the gauge-invariance and on the other the fact that m, even if small, has a completely different meaning than m = 0.

So, what can be said is that, if you write the equations of dirac imposing solution E_x, E_y, B_x, B_y for propagation kz (i.e. the four 1d equation for the 4spinor replaced by those), then u get the solutions of Maxwell and can satisfy Dirac if you replace the normal derivative with a covariant one, for which the term "m" acts in a certain sense as a kind of curvature of space.

Answer is Yeess!

Exact answer is: Two massless Dirac equation are same with Maxwell equation with more scalar and pseudoscalar field,  but with different transformation properties to the L group.

Similarly in the case of massive Dirac field and Proca-Yukawa field.

See for example:

J. Brana and K. Ljolje, A costruction of free Maxwell's field from an eight component Dirac's field, FIZIKA, 6,(1974), p 117-126

J. Brana and K. Ljolje, Some advantages of spinor notation of electromagnetic field, FIZIKA, 7, (1975), p 1-12

I think this can be done, and some of the answers suggest this too.

To me, photons and electrons obey the same "general laws", and yes they are different in many regards-- electrons are Fermions and photons Bosons, so they have differnt statistical properties... photons would like to enter the same state as other photons (yielding the laser), and fermions can never do this, hence atomic structure, etc.

However, as far as QED goes, the wave particle "duality" still imposes its will on these objects, so they both have  things like diffraction, interference, EPR behaviors, etc.

No problem.. As far as the construction of Maxwell from charges, likely one of the

above references discusses these details..

Dear Arbab, you have asked an interesting question. As indicated by some experts in this thread, physically Dirac equation is different from Maxwell equations. Nonetheless, mathematically it can be shown that matrix Maxwell equations corespond neatly to Dirac equation, one famous example is Duffin-Kemmer-Petiau. Then what does it mean? To me such a correspondence seems to indicate that there is neat relation between quanum and electromagnetic picture, just like the exact correspondence between Poisson bracket and commutator bracket. Feynman's derivation of Maxwell equations is also using commutator bracket.

So my humble answer is that i agree with Francis, Varlamov, Brana, and others who think that there should be correspondence between Maxwell and Dirac equation, at least under matrix or quaternion unit. Alexander Gersten have shown that quantum picture should be accompanied by electromagnetic picture, and sometime ago i also derived Maxwell equations from Quaternion Dirac equation. See http://vixra.org/abs/1003.0010

http://vixra.org/abs/1003.0010

Maxwell's equations describe the electromagnetic field generated by electric or magnetic variable sources and therefore they describe an event of pure energy by deterministic laws. From Maxwell's equations  the "photon equations" can be deduced.

Dirac's equation describes instead the motion of leptonic particles that have mass. Besides the same equation is interpreted generally in non-deterministic shape.

The dualism wave-corpuscle can cause wrong conclusions if it isn't interpreted correctly. The two particles, electron and photon, in actuality are two different particles: electron is a massive leptonic particle to which it is possible to associate the De Broglie's wavelength or the Compton wavelength according to physical situations. Photon is a boson electromagnetic nanowave of pure energy to which it is possible to associate an equivalent mass, but the two particles are different physical entities. Besides leptonic particles can have different speeds with respect to a reference frame while photons have one only speed with respect to a reference frame.

@Fritz Wilhelm Bopp :  to me it seems that Dirac Eq is a subclass of KleinGordon, from which Maxwell is obvious case with m=0

Maxwell's equations represent the mathematical synthesis in vector shape of laws of electromagnetism: Gauss-Poisson's law, Faraday-Neumann-Lenz's law, Ampere-Maxwell's laws, Lorentz's law. The Dirac equation describes instead the motion of leptonic massive particles. If then one likes to generalize mathematical equations he must know it can be maked in different ways and above all he not must lose the physical meaning of things.

I agree then there is a great difference between statistics and probability.

@ Akira Kanda: U wrote

#Well Photons are stumbling block for physics. According to the UP, photons are supposed to have no trajectories. Is it not the case that MM used trajectories of photons to get the Constancy of Speed of Light result? Instead of putting thumb down why can't you just start a discussion if you have objection? #

Here starts my discussion:   :)

If I say: the photon does not move, it is at the same time (but in a broader view of the dimensionality of the world) either in the source (or whatever we call ) than in the destination (in what we call >, receiver): the photon is emitted when already exists simultaneously its destination. Then the propagation-velocity c arises as a measure of energy contained .... in somewhere (but I dont know);   

for you is  this sentence logically consistent with the established rules?

@ Akira Kanda: KleiGordon+Dirac+Maxwell just used to say that it is not (as it has been said) coz simply Dirac equation is a refinement of a subset of the KleinGordon equation, and Maxwell simply a particular case in which m=0 ;) 

@ Akira Kanda:  ;)  but  Feynman's ideas resolve in a elegant manner the following trick: who/what decides, during the electron-electron scattering, that a photon is to be emitted just at that time?  Maybe the meaning hidden here is that the electron, even before knowing that there is another electron with which it must interact .... already knows all about? 

of course... all is always based on conjectures, but some of these are interwoven strongly with experimental results that impact other phenomena... --> Physics is a measure of the phenomenological reality, but always supported by the interpretation of the phenomenon itself, which is a child of the same mechanism ... therefore a continuous loop....    it is always a Perspective, the evolution is a change in the perspective. But remaining to the e- / e- scattering... how can an electron "know" about the partner, if they are coming from a IN channel? i.e. they are decoupled for hypothesis

Dear Fulcoli, I think you would have to specify "an electron knows (?)" . Thanks.

@Daniele Sasso: Yes. If we have a charged particle Q in a "empty" space, and observe a motion of Q we can argue for the presence of a distant and different charged-particle, say Q', that interacts with Q, and does it through an entity called Field. But this happens in the classical perspective, in the modern one (QFT) we know about the existence of a media-boson like the photon, responsible of the interaction: the interaction resolve the following paradigm:

a body (a cherged particle, for example) remains in its "inertial" motion-status [GR speaking: following its geodesic] till is actioned by a external "force". Whilst The inertial status doesnt require to be propagated to whole universe becoz Inertia is already known, the Modified Status (for that particle) must be propagated, as information, through the whole universe in order to avoid a misalignment. When there is a change in the status, then we notice the presence of (radiation, in a classical view) or (boson, in a modern QF view). What is a boson? As Okun said: the boson is the perturbation of the integration Path for the Quantum-Action S.

Then my words: stands for the fact that, if there is an interaction between 2 charged-particles through a photon, say now, then we must define, almost 2 things:

who/what decides for this emission, and

why (it) decides precisely now!

In this view I like to follow Feynman intuition.... 2 e- in IN channel arent really decoupled....

Dear Fulcoli, the classic term "field" and the modern term "boson" in that physical context express the same concept: the photon. In the order of the contemporary Theory of Reference Frames there is also a different way for expressing the same concept:  electromagnetic nanowave that is related to an electromagnetic nanofield.

When a charged particle (let us say an electron and not any charged body) is into an inertial state nothing happens. When the initial inertial state of the electron is modified, you raise two questions relative to the emission of radiation: 1. who/what decides for this emission,  2.  why that emission happens at a precise time.

Questions that you raise are right if one wants to understand physics of considered events and that the post-modern quantum physics in all its versions doesn' t allow because it is based on an absolute indetermination of microphysical events. As per my researches in Relativity and in Particle Physics I can suggest a further answer :

1. The emission of radiation or boson happens actuality through two equal gamma nanowaves in different times and this emission is caused by the force of electromagnetic field that produces the acceleration.

2. The emission happens in two different times and precisely when the speed of accelerated electron reaches two precise values of speed.

@Daniele Sasso: why  answer 1: 2 gamma? R U referring to smtng hidden in ur words? And for 2nd answer: 2 precise values of speed (in which sense)? [ pls wrote 2 me at [email protected]  ]

Dear V. , there is nothing of mystery or of hidden in my words. They are deduced from my research. I will send you a few papers of mine about your questions. Please, I would want to know if you will receive them. Thanks.

Dear Arbab I. Arbab,

to understand field theory, one has to work hard, shut up and calculate, as you certainly know. Maybe you already know or will be interested in the fact that Maxwell's equation can be cast in a straightforward manner into Dirac form. This has been observed more than one hundred years ago by Riemann - for an introduction, see https://www.researchgate.net/publication/232905988_Complex_representation_theory_of_the_electromagnetic_field .

Sincerely yours,

Andreas Aste

Article Complex representation theory of the electromagnetic field

No.

Quantum mechanics introduces the symmetry group U(1) (the complex unit circle) to basic physics. 

Special relativity extends the Galilean group of 3D rotations and time translations to the Lorentz group, or the Poincare group.

The Klein-Gordon wave equation combines two ingredients: basic quantum mechanics and special relativity. The symmetry group of gthe Klein-Gordon equation is The Lorentz or Poincare group x U(1).

Dirac's wave mechanics combines three ingredients: basic quantum mechanics, special relativity, and the spin group (which is the double-covering of the Lorentz group). The symmetry group of Dirac's theory is U(1) x spin(1,3).

Maxwell's theory of electromagnetism introduces curvature to the U(1) symmetry of quantum mechanics. Without quantum mechanics and its minimal electromagnetic coupling, it is not obvious from classical electrodynamics that the electromagnetic field is a curvature tensor on a U(1) bundle.

Each of these theories -- quantum mechanics, special relativity, Dirac's theory, and Maxwell's electromagnetic theory -- introduces a different element to the structural symmetry groups of nature. In my opinion, the different structural groups represent different elements of nature at a level that we don't yet fully understand. Even if you could get the equations of Dirac and Maxwell into some common form, these theories are based in different elements in physics.

dear Akira,

I tried to understand what you were saying. Not sure i got it all, but it is interesting.

One statement i did object to, if my interpretation is right. You said Maxwell

has nothing to do with Relativity.  However, in his equations, one can set the charges,currents to zero, to get a wave equation, and the result is EM waves that

travel at the speed of light.. So, this relates to relativity..

best wishes, ken

Two very different things, Maxwell has to do with Classical electromagnetic

theory, and Dirac the Quantum behaviour of the electron. It is possible though

to show that Dirac obeys Lorenz invariance, and so does electromagnetism.

A common property, but not one derived from the other.

No they cannot be derived from one another. On the contrary to study the motion of a relativistic electron moving in an external electromagnetic field (φ,⃗A) , one has to extend the simplest form of Dirac equation by bringing in the concept of minimal coupling.  To this end one has the prescription: E → E - eφ and ⃗P → ⃗P - e⃗A. Hamiltonian contains new terms after this substitution is carried out, and one can clearly identify a piece arising out of the interaction between the magnetic moment of the electron and the external magnetic field.

Maxwell's Electromagnetic Field equations, have no aspect of quantum mechanics

in it, as far as I know. So, there is NO quantum unit within it. Planck and Einstein

had to add the "quantum:" aspect to obtain the photoelectric effect

and other quantum aspects would then fall out of QM, with very clever juggling,

including of course the Schrodinger equation... Etc. but no, i do not

think either can fall out of the other. Of course, Dirac's is more fundamental,

so one might be able to scale it to Hbar = 0, and get some effects like that, but

that is not so (hbar NOT equal 0)

It may be interesting to note that Maxwell's equations are -- up to unitary congruence -- a special case of the Dirac system (for mass zero) given appropriate constraints of the data. The details can be found in our paper "On a Connection between the Maxwell System, the Extended Maxwell System, the Dirac Operator and Gravito-Electromagnetism."