The principle of least action/time, and geodesics of spacetime

[Alexander]

Again incorrect. Coulomb law is more fundamental than Maxwell equations. It follows from Heizenberg uncertainty principle, translational symmetry of space and 3-dimensionality of space.

 

[pmb]

Originally posted by Alexander
Again incorrect. Coulomb law is more fundamental than Maxwell equations. It follows from Heizenberg uncertainty principle, translational symmetry of space and 3-dimensionality of space.

You missed my point. Coulomb's law **IS** one of Maxwell's equations!

Please provide a proof, or a referance to a proof, of this claim -- "It follows from Heizenberg uncertainty principle, translational symmetry of space and 3-dimensionality of space."

Pmb

 

[Alexander]

Coulomb law is NOT one of Maxwell equations. Gauss law is, which is derived from Coulomb law and from definition of electric field.

Inverse square law for some of natural forces (like gravity and electric force) follows mathematically from interaction of massles virtual particles in 3-d space, providing that space is translationary symmetric (=momentum conservation). See QED texts for derivation of inverse square law.

Existence of virtual particles follows from Heizenberg uncertainty principle (which in turns follows from wave nature of all particles).

 

[pmb]

Alexander wrote

Coulomb law is NOT one of Maxwell equations. Gauss law is, which is derived from Coulomb law and from definition of electric field.

Thank you. Yes. That is correct. I meant to say that Coulomb's law is equivalent to one of Maxwell's equations namely Gauss's law.

Anyway - Your claim was "Coulomb law is more fundamental than Maxwell equations" which is pretty far from being the truth.

Let's forget about this claim that Coulomb's law is derivable from quantum principles as you suggest - that is not relavent to your comment to which I was originally referring to. Nameley your claim that Maxwell's equations are derived from relativity. Or to quote you directly

you can't refer to Maxwell equation(s) trying to prove constancy of speed of light, because Maxwell equations are CONSEQUENCE of relativity, because they are DERIVED from relativity (namely, from existence of electric charge, Lorents transformations of coordinates (which gives rize to magnetic component), and 3-dimensionality of space).

Please derive Coulomb's law from
(1) existence of electric charge
(2) Lorents transformations of coordinates
(3) 3-dimensionality of space

I question the derivation from quantum principle since they axioms of QED might be based on Maxwell's equations and since I'm not formally familiar with QED I have no wish to discuss it here.

Did you really mean that it Maxwell's equations are derivable from quantum principles *plus* special relativity?

Pete

 

[Alexander]

Yes. Maxwell equations are classical limit (h--->0) of QED equations of interactions of charged particles with virtual massles bosons (virtual photons in QED). The result of this interaction is exchange by momentum (in translationary symmetric space momentum shall conserve).

There is no "force" in quantum interactions - there is only exchange by momentum, energy, spin, charge and other conserved quantities. Force is a classic concept standing for "average of momentum change rate", F = <dp/dt>. The momentum of virtual photon is what we call "force", so to speak - in this case "Coulomb force", and a bunch of virtual bosons is what we call "field" ("electric field" if those bosons are virtual photons). Due to Heizenberg uncertainty principle (HUP)if two particles (say, electrons) are sharing same virtual boson (that is where interaction of electrons or other distant from each other particles comes from), then the momentum of this boson is inversely proportional to the distance between particles.

Without going into QED the Coulomb law can not be derived. Here is a rough sketch of how inverse square "forces" and inverse square "fields" originate from HUP of exchange by virtual massles boson. If two electrons are separated by distance r, then to "share" or "exchange" by same virtual photon (which is moving with speed c) the photon must exist for at least t~r/c time. The longer the virtual photon is around, the less energy it can have according to HUP: E~h/t, thus the less momentum it can carry p=E/c~h/ct. Recalling that classic "force" is nothing else but the avarage rate of momentum exchange F~p/t, you get inverse square law: F~p/t=(h/ct)/t =h/ct²=h/(c(r/c)²)=hc/r²

By the way, hc is indeed close to e²/4piepsilon (which is a factor in classic Coulomb law of interaction of two electrons), and the difference (called fine constant factor) arises in transition grom QED to classic limit due to screening of actual charge of electron (which is about sqrt(137) more than its classic limit 1.6x10^-19) by polarized virtual pairs around it.

 

[pmb]

Alexander

I'm not sure why you went into all that. When I said that I wasn't formally familiar with QED that only meant that I couldn't go into a grad level quantum field theory class and pass a test on the subject. I didn't mean don't understand/know it. I'm just not interested in it at the present time as I suggested.

Thanks for the effort though.

Pete

 

[Integral]

Seems like it is time for some moderator intervention in this thread. It really seems that it is starting to circle with no hope of final resolution.

It is clear that historically relativity was NOT necessary to derive the constancy of the speed of light. It is also clear that Einstein was able to POSTULATE the constancy of the speed of light due to the work of Maxwell. That fact that Maxwell’s derivation of the speed of light as a constant created a 30 year period of turmoil in Classical Physics is sufficient proof for me that relativity is not NECESSARY to discuss the constancy of the speed of light.

This does not mean that given a new and greater understanding of the universe that the constancy can be seen to play a fundamental role which can be viewed independently of Maxwell’s Equations. Since I am not familiar with a development of relativity that does not rely on the constancy of the speed of light, it is not clear how one can use relativity to prove c constant.

If Alexander or pmb wishes to start a thread dedicated to that topic, the discussion can continue.