Thread: Issue with Maxwell equations

Hello,
I`m a bit confused about a rather subtle issue with electromagnetism. We have two Maxwell equations in covariant formulation





My problem is that EM field (potential) can be "derived" assuming U(1) gauge symmetry of Dirac lagrangian. One simply defines covariant derivative with corresponding gauge field which is nothing else than electromagnetic four-potential. If one wants to write kinetic term for gauge field one gets



Using E-L equations one gets first (dynamical) Maxwell equation for potentials as equation of motion.

My issue is this...: How to get second equation? Is it some nontrivial information we have to insert by hand? As far as I know these equations serve only as constraints and if EM field satisfies them in one time then they are satisfied at all times. Is that right?

... and this: In case of Lorentz force if I understand it correctly it´s only possible four-vector up to constant ( ) therefore only possible candidate for electromagnetic force. Is this correct?

Thanks.

How to get second equation?

Good question, and I am not too sure what the answer needs to be. However, I would approach this in a different manner; you start with the 4-potential ( obtained in the way you mentioned ), and then get the field via exterior differentiation :



Poincare's Lemma then immediately implies



which is precisely equivalent to . Physically this really just means that in the absence of any sources the field can't have "boundaries". This seems intuitive to me and follows straight from basic exterior calculus, but it is quite possible that there is a deeper reason which I am not aware of. This is just my two cents' worth.

Ah yes nice. I had to revise my rusty knowledge of differential forms Ok, so first equation emerges simply as equation of motion for gauge field and second emerges simply through definition of F and dd = 0. Could you comment on Lorentz force problem? eg. is the form of Lorentz force also "derived" considering it is up to constant only possible covariant four-vector? I have a deeper reason why I am asking this just want to verify my assumptions.

Originally Posted by Gere

Ok, so first equation emerges simply as equation of motion for gauge field and second emerges simply through definition of F and dd = 0.

Yes, this would be my take on it. Remember though that I am only an amateur, so there might be other things at play here that I am not myself aware of.

is the form of Lorentz force also "derived" considering it is up to constant only possible covariant four-vector?

I don't know the answer to this question, though it is a very good one. I have always considered the Lorentz law an empirical result, that cannot be derived from any underlying principle - it is precisely the Lorentz force that allows us to define the components of the field tensor in terms of classical E and B fields.

I have a deeper reason why I am asking this just want to verify my assumptions.

Ok, then maybe someone could comment on this who is more of an expert on the matter than I am. What is your deeper reason ?

Originally Posted by Gere

Hello,
I`m a bit confused about a rather subtle issue with electromagnetism. We have two Maxwell equations in covariant formulation





My problem is that EM field (potential) can be "derived" assuming U(1) gauge symmetry of Dirac lagrangian. One simply defines covariant derivative with corresponding gauge field which is nothing else than electromagnetic four-potential. If one wants to write kinetic term for gauge field one gets



Using E-L equations one gets first (dynamical) Maxwell equation for potentials as equation of motion.

My issue is this...: How to get second equation? Is it some nontrivial information we have to insert by hand? As far as I know these equations serve only as constraints and if EM field satisfies them in one time then they are satisfied at all times. Is that right?

... and this: In case of Lorentz force if I understand it correctly it´s only possible four-vector up to constant ( ) therefore only possible candidate for electromagnetic force. Is this correct?

Thanks.

The answer to your questions can be found in paragraphs 30-33 , The Classical Theory of Fields by Landau and Lifshitz.

Originally Posted by xyzt

Originally Posted by Gere

Hello,
I`m a bit confused about a rather subtle issue with electromagnetism. We have two Maxwell equations in covariant formulation





My problem is that EM field (potential) can be "derived" assuming U(1) gauge symmetry of Dirac lagrangian. One simply defines covariant derivative with corresponding gauge field which is nothing else than electromagnetic four-potential. If one wants to write kinetic term for gauge field one gets



Using E-L equations one gets first (dynamical) Maxwell equation for potentials as equation of motion.

My issue is this...: How to get second equation? Is it some nontrivial information we have to insert by hand? As far as I know these equations serve only as constraints and if EM field satisfies them in one time then they are satisfied at all times. Is that right?

... and this: In case of Lorentz force if I understand it correctly it´s only possible four-vector up to constant ( ) therefore only possible candidate for electromagnetic force. Is this correct?

Thanks.

The answer to your questions can be found in paragraphs 30-33 , The Classical Theory of Fields by Landau and Lifshitz.

But I thought Lifshitz only did work with hydraulics.

Yes - but it's more instructive to run the argument backwards

Suppose the F-field, then the 2-form , is given by a section of the principal bundle on Minkowski spacetime, a manifold recall, whose structure group is a Lie group. Now define, ad hoc, so to speak, the 1-form , the vector potential. We will assume, again on the fly, that this is a connection on our bundle.

Then I can show, with some difficulty I grant you, that, by definition of the covariant derivative, is the curvature of . Notice that curvature is a Lie algebra-valued (-valued) 2-form by definition.

So, all is well, since we know that is a 2-form. And we also know that, by definition,

, the Bianchi identity .

But, if we already know that then it must be that , then this requires that our structure group has an algebra with all brackets zero. And if we further require our group to be unitary, then we must choose , where of course all Lie brackets vanish.

Thus have we derived the necessary (but not sufficient) structure group for "ordinary" electromagnetism.

My problem is this. In standard electroweak unification one gets 4 gauge fields by assuming SU(2)xU(1) invariance on lefthanded dublets and U(1) invariance on righthanded singlets. Using the same procedure for finding kinetic term in lagrangian for these fields one gets exactly the same form



only difference being non-Abelian nature of three fields corresponding to SU(2) which results in interaction terms between gauge fields.

Physical fields can be constructed from these using some simple assumptions, basicaly we get two charged vector bosons and two uncharged being photon and Z. Since kinetic term for Z is the same as kinetic term for photons we get same equation of motion (up to mass term) for Z field. I never paid much attention to this since this is only for gauge fields respectively potentials and it also represents only first Maxwell equation. However second ME is true simply by definition therefore Z field satisfies Maxwell equations almost exactly (up to mass term). I still disregarded this because I felt that electrodynamics as we know it is defined mainly by interactions between field and matter which is contained in definition of Lorentz force.

I long thought of Lorentz force as sort of empirical law through which we can define E and B eg. how they act on charge. However while reviewing my old notes from special relativity I found that Lorentz force is derived up to constant by requirement of covariance on force four-vector and is the only candidate for such object. It is not derived in mathematicaly super-rigorous way more like "what else could it be?" much like Einstein field equations.

Using very naive approach and "standard" definition of E and B I simply wrote down Maxwell equations for Z field from standard Proca equation and interaction terms with fermions this way



where m is mass of Z field and J in four-current made of flux of electron and neutrino fields. Role of electric charge here takes another coupling constant which is different for righthanded and lefthanded electrons and neutrinos and is function of electric charge and weak coupling constant.

So naively:



and



with standard notation for scalar and vector potential. Since m is macroscopicaly small I would tend to disregard it.

Therefore this naive way the similarity between EM field and Z seems much more literal than I would think before. I always though of Z as something like massive photon only not in such a literal way.
Of course there is massive distinction in that large mass of Z leads to Yukawa potential eg. this behaves like screened Coulomb potential like in plasma or metals but similarity seems interesting to me.

Any thoughs?, glaring mistakes?

Originally Posted by Guitarist

Yes - but it's more instructive to run the argument backwards

Suppose the F-field, then the 2-form , is given by a section of the principal bundle on Minkowski spacetime, a manifold recall, whose structure group is a Lie group. Now define, ad hoc, so to speak, the 1-form , the vector potential. We will assume, again on the fly, that this is a connection on our bundle.

Then I can show, with some difficulty I grant you, that, by definition of the covariant derivative, is the curvature of . Notice that curvature is a Lie algebra-valued (-valued) 2-form by definition.

So, all is well, since we know that is a 2-form. And we also know that, by definition,

, the Bianchi identity .

But, if we already know that then it must be that , then this requires that our structure group has an algebra with all brackets zero. And if we further require our group to be unitary, then we must choose , where of course all Lie brackets vanish.

Thus have we derived the necessary (but not sufficient) structure group for "ordinary" electromagnetism.

This looks interesting though I will admit complicated for me right now as I´m not very well acquinted with differential forms. I will look it up.