Is Gauge-Free Electrodynamics the Solution to Gauge Ambiguities?

[Demystifier]

Today (a revised version of) the paper
http://lanl.arxiv.org/abs/0903.4340
attracted my attention.

Abstract:
We propose a reformulation of electrodynamics in terms of a {\it physical} vector potential entirely free of gauge ambiguities. Quantizing the theory leads to a propagator that is gauge invariant by construction in this reformulation, in contrast to the standard photon propagator. Coupling the theory to a charged Abelian Higgs field leads at the quantum level to a one loop effective potential which realizes the Coleman-Weinberg mechanism of mass generation, thus resolving the issue of its gauge dependence. We relate our results to recent work by Niemi et. al. and Faddeev, where similar strategies are adopted in a version of the electroweak theory. Other theories with linear Abelian gauge invariance, like the linearized spin 2 theory of gravity or the antisymmetric tensor field, which may all be reformulated in terms of physical vector or tensor potentials without gauge ambiguities, are also discussed briefly.

Any comments?

 

[Careful]

Today (a revised version of) the paper
http://lanl.arxiv.org/abs/0903.4340
attracted my attention.

Abstract:
We propose a reformulation of electrodynamics in terms of a {\it physical} vector potential entirely free of gauge ambiguities. Quantizing the theory leads to a propagator that is gauge invariant by construction in this reformulation, in contrast to the standard photon propagator. Coupling the theory to a charged Abelian Higgs field leads at the quantum level to a one loop effective potential which realizes the Coleman-Weinberg mechanism of mass generation, thus resolving the issue of its gauge dependence. We relate our results to recent work by Niemi et. al. and Faddeev, where similar strategies are adopted in a version of the electroweak theory. Other theories with linear Abelian gauge invariance, like the linearized spin 2 theory of gravity or the antisymmetric tensor field, which may all be reformulated in terms of physical vector or tensor potentials without gauge ambiguities, are also discussed briefly.

Any comments?

I just glanced for a brief moment because I have thought about this for a long while too (but then kind of dismissed the idea because all such formulations have to be nonlocal). Their formulation is nonlocal too because of the inverse of the d'alembertian on page two. It seems they have been sloppy there because you need a tiny imaginary mass to regulate such expression, no?

 

[Demystifier]

Their formulation is nonlocal too because of the inverse of the d'alembertian on page two.

That equation is indeed nonlocal, but that equation is NOT crucial for their theory. Indeed, their theory is NOT based on the transversal part of A^{\mu}. And even if it was, the theory would be local as long as one interprets the theory in its own terms, rather than establishing the connection between their and standard formulation.

 

[Careful]

That equation is indeed nonlocal, but that equation is NOT crucial for their theory. Indeed, their theory is NOT based on the transversal part of A^{\mu}. And even if it was, the theory would be local as long as one interprets the theory in its own terms, rather than establishing the connection between their and standard formulation.

Then, what they are doing must be fully equivalent to the standard formulation Actually, it is even worse than that, if you watch formula (5) at page 3, you see they do not even care to define the charge current. Of course when they would have spinorial matter, they would have to use the ordinary current, but then their formalism would give the wrong equations of motion in general since that current gets redefined by - \partial_{\mu} \Gamma. \Gamma is then fixed upon a residual gauge transformation by \partial_{\mu} J^{\mu} = \box \Gamma. But we know this current is conserved from the symmetries of the fermion action (and the coupling to the electromagnetic field), so \Gamma is pure gauge, \box \Gamma = 0. So classically, this theory is fully equivalent to the standard formulation (in the Lorentz gauge you also have this residual gauge transformation, but not in the current but on the side of the gauge field !) if and only if \Gamma = constant (and otherwise it gives the wrong results). Quantum mechanically however, we are in sh*t since gauge invariance of a massless spin one particle is actually required for ensuring Lorentz covariance at least on Hilbert spaces. So, this paper is not doing what it claims to do.

It is actually pretty easy to see that any reformulation of electrodynamics without gauge invariance has to be nonlocal (at least if the theory is fully equivalent to the old one classicaly) and Wilson loops or field strenghts integrated over two surfaces are essential ingredients of such formulation.

Careful

 

[Demystifier]

Well, my impression is that, essentially, they are doing electrodynamics in one particular gauge (Feynman gauge), in which locality and Lorentz invariance are manifest. If that is so, then it is trivial that their approach is equivalent to the standard one. But it is interesting to point out that nothing physically important is lost if one formulates electrodynamics in that form, without even mentioning gauge symmetry. It suggests that gauge symmetry is much less important than it is usually believed, which is why I find this paper interesting.

Actually, I always had an impression that A^{\mu} should be more fundamental than F^{\mu\nu} (see e.g. the Aharonov-Bohm effect), and this paper reinforces such a view.

What seems more problematic to me is whether gauge symmetry can also be removed in a similar way from NON-ABELIAN gauge theories (weak interactions and QCD).

 

[Demystifier]

Consider also an analogy. Consider the action of a free nonrelativistic particle with the trajectory x(t). The Lagrangian is
L=v^2/2
where v=dx/dt. Here x is analogous to A^{\mu} and v is analogous to F^{\mu\nu}. The Lagrangian is invariant under the "gauge" transformation
x(t) -> x(t) + x_0
which is nothing but the translation invariance. From this symmetry, in analogy with electrodynamics, one might conclude that velocity of a particle is more fundamental than the position. Yet, we know that position is more fundamental than velocity, though not absolute position, but relative position with respect to another particle or with respect to some "natural" physically chosen origin x=0.

In a similar sense, it seems sensible to me that A^{\mu} might be more fundamental than F^{\mu\nu}, but with respect to some fixed "natural" gauge. And I think this is exactly what the paper above tries to prove.

 

[Careful]

Well, my impression is that, essentially, they are doing electrodynamics in one particular gauge (Feynman gauge), in which locality and Lorentz invariance are manifest. If that is so, then it is trivial that their approach is equivalent to the standard one. But it is interesting to point out that nothing physically important is lost if one formulates electrodynamics in that form, without even mentioning gauge symmetry. It suggests that gauge symmetry is much less important than it is usually believed, which is why I find this paper interesting.

Well, I don't recall what the Feynman gauge is, but it is the Lorentz gauge. And no, as I pointed out, they do not retrieve ordinary electrodynamics, since they have residual *global* degrees of freedom. Your comment about gauge invariance not being so fundamental is not really adequate I believe. Gauge invariance, as I said, is mandatory for standard Hilbert space quantization. There whole discussion is on the classical level where indeed, the motivation for gauge invariance is rather poor.

Actually, I always had an impression that A^{\mu} should be more fundamental than F^{\mu\nu} (see e.g. the Aharonov-Bohm effect), and this paper reinforces such a view.

What seems more problematic to me is whether gauge symmetry can also be removed in a similar way from NON-ABELIAN gauge theories (weak interactions and QCD).

Sure, but if you go over to the magnetic lines/electric flux variables I gave you previously, everything is fully equivalent.

Careful

 

[Demystifier]

Gauge invariance, as I said, is mandatory for standard Hilbert space quantization.

Can you explain that? Or point to a reference?

I mean, canonical quantization is much easier in a completely fixed gauge (such as the Coulomb one), so I don't see why gauge invariance is important for Hilbert space quantization.

 

[Careful]

In a similar sense, it seems sensible to me that A^{\mu} might be more fundamental than F^{\mu\nu}, but with respect to some fixed "natural" gauge. And I think this is exactly what the paper above tries to prove.

But they fail in doing so - as I explained before (because the Lagrange multiplier creeps in the EM current). Moreover, as I said, you are fighting in the wrong court, it are by the laws of quantum mechanics that you should make your case.

 

[Careful]

Can you explain that? Or point to a reference?

I mean, canonical quantization is much easier in a completely fixed gauge (such as the Coulomb one), so I don't see why gauge invariance is important for Hilbert space quantization.

Hmm, because if you work in a gauge, Lorentz covariance gets ''hidden''. Lorentz invariance only exists if and only if one has gauge invariance (for massless spin one particles). Reference, see Weinberg - volume I, page 250.

 

[Demystifier]

Hmm, because if you work in a gauge, Lorentz covariance gets ''hidden''. Lorentz invariance only exists if and only if one has gauge invariance (for massless spin one particles). Reference, see Weinberg - volume I, page 250.

When you fix the gauge completely, like with the Coulomb gauge, then manifest Lorentz invariance is lost. However, this is true already at the classical level. I don't see how that problem becomes more difficult at the quantum (Hilbert space) level. Indeed, all practical calculations in QED can be done in the Coulomb gauge, and final physical results turn out to be Lorentz invariant.

 

[Careful]

When you fix the gauge completely, like with the Coulomb gauge, then manifest Lorentz invariance is lost. However, this is true already at the classical level. I don't see how that problem becomes more difficult at the quantum (Hilbert space) level. Indeed, all practical calculations in QED can be done in the Coulomb gauge, and final physical results turn out to be Lorentz invariant.

See the reference I have given you ! Lorentz covariance has a DIFFERENT meaning in quantum theories than in classical ones. At least you would have to give up Fock space quantization or Hilbert spaces in order to restore Lorentz covariance at the quantum level if you abandon gauge invariance.

 

[Demystifier]

See the reference I have given you ! Lorentz covariance has a DIFFERENT meaning in quantum theories than in classical ones. At least you would have to give up Fock space quantization or Hilbert spaces in order to restore Lorentz covariance at the quantum level if you abandon gauge invariance.

Sorry, but Weinberg says nothing like that on page 250. So I have no idea what you are talking about.

 

[Careful]

Sorry, but Weinberg says nothing like that on page 250. So I have no idea what you are talking about.

Yeh, that's right, you should have read to page 251 formula 5.9.31 and subsequent comments...

 

[Demystifier]

Thanks, I will need to think more carefully about (5.9.31) ...

 

[Careful]

Thanks, I will need to think more carefully about (5.9.31) ...

I don't say you cannot abandon gauge invariance at the quantum level, but then you will have to give up a very fundamental principle of QFT. So, all these discussions have to happen at the quantum level and this paper contributes nada, nil, in this respect.

Careful

 

[tom.stoer]

Isn't that just a reformulation of standard electrodynamics? I mean during standard gauge fixing one eliminates longitudinal and scalar photons; this is done in this paper using the "physical condition". So gauge symmetry is somehow bypassed, but eventually one arrives at the well-known Feynman gauge.

It would be interesting to try to "guess" the physical formulation of QED w/o going throgh all these messy gauge fixing procedures.

 

[Careful]

It would be interesting to try to "guess" the physical formulation of QED w/o going throgh all these messy gauge fixing procedures.

Yes, but my point was that such formulation has to be nonlocal and cannot depend upon the vector potential (unless you modify QFT), that's all.

 

[tom.stoer]

Yes, but my point was that such formulation has to be nonlocal and cannot depend upon the vector potential (unless you modify QFT), that's all.

What do you mean by that? If you look at the gauge fixed Hamiltonians for QED or QCD they are "non-local", that's a well-known fact.

 

[Careful]

What do you mean by that? If you look at the gauge fixed Hamiltonians for QED or QCD they are "non-local", that's a well-known fact.

Sure, but when dealing with the issue of Lorentz covariance, you do not start from the gauge fixed version, right. You start from the gauge invariant one. For example, the action in this paper gives (for constant Lagrange multiplier) well known classical electrodynamics in the Lorentz gauge, but if you would quantize this theory, Lorentz invariance would be broken (since the latter requires gauge invariance). Therefore, my comment is that this version will not give usual QED at the quantum level. In order to even write down an appropriate classical theory for electrodynamics which does not require gauge fixing, one needs to formulate it in terms of new nonlocal variables like the magnetic field lines (Wilson loops) and field strength integrations over two surfaces.

In the path integral language, this goes as follows ... you start with ordinary Maxwell, perform a gauge fixing which adds effectively a new part to the action (which depends upon an unphysical parameter) and one can construct the proper quantum measure like this. However, if you care to look at the kinetic term of the action in this paper, then you see the latter is not gauge invariant either (as it should). The distinction lies here: if you would take the effective action after breaking the gauge symmetry in the path integral formulation, your classical equations of motion wouldn't give ordinary Maxwell theory anymore (since there is effectively no constraint), but quantum mechanically this new term is completely irrelevant if we restrict to gauge invariant observables (classically it ain't). In this case, if you would integrate out the Lagrange multiplier, then this part of the action would just give you a gauge fixing delta function which simply is a gauge fixing for electromagnetism *if and only if* the kinetic term is corrected to the usual one. That's all I had to say (but then this paper is a trivial excercise).

 

[tom.stoer]

I see what you mean, but I think you cannot be sure about breaking of Lorentz invariance w/o explicit calculation.

Canonical quantization in non-Lorentz-invariant gauges (only those make sense to me in canonical quantization as otherwise you still have unphysical degrees of freedom, negative norm states and things like that) leads to a Hamiltonian (plus other generators of the Poincare algebra) which at first glance break Lorentz (Poincare) invariance. One exampe is the Coulomb potential which looks like an "instantaneous interaction". Explicitly checking the Poincare algebra shows that the algebra still closes and that the theory is fully Poincare invariant, even so this symmetry is "hidden" due to gauge fixing. Afaik this has been done in QED in (spatial) axial and Coulomb gauge. I am not sure if all generators have been regularized and if the algebra has been checked to be anomaly-free.

So I think w/o checking explicitly you cannot be sure.

What I am questioning is if this whole procedure provides any new insights. It is clear that for QED one can do something like that based on educated guesses. But I doubt that it will work for QCD (here guessing is much harder).

 

[Careful]

I see what you mean, but I think you cannot be sure about breaking of Lorentz invariance w/o explicit calculation.

I just extended my previous message which answers your question. If the authors do not adjust the kinetic term, they do not have QED (and break Lorentz invariance as well).