# How do electrons couple to gauge field?

a proca field

I think a mass term for the electromagnetic field breaks the gauge invariance of the theory. You are following a route which is technically simpler, but obscures the essential fact that the 4-potential is a gauge field. Namely, you can always set the mass of the photons to zero in the end of all the calculations and obtain the same result without ever discussing gauge invariance. But, then, your coupling term, the very topic of this thread, is inserted 'by hand' and does not have an obvious meaning. Also, by the Ward identities, I think the photon cannot acquire mass in any order of perturbation theory.

I think a mass term for the electromagnetic field breaks the gauge invariance of the theory. You are following a route which is technically simpler, but obscures the essential fact that the 4-potential is a gauge field. Namely, you can always set the mass of the photons to zero in the end of all the calculations and obtain the same result without ever discussing gauge invariance. But, then, your coupling term, the very topic of this thread, is inserted 'by hand' and does not have an obvious meaning. Also, by the Ward identities, I think the photon cannot acquire mass in any order of perturbation theory.

The mass of the photon field is zero by definition, not because you set it to zero due to some gauge symmetry. The gauge transformation of the photon field is the addition of the gradient of some local function, which doesn't change Maxwell's equations. Furthermore, requiring invariance of full QED Lagrangian (with coupling to spinor fields) under that transformation leads to an imposed local gauge symmetry of the spinor. The mass of the photon doesn't play a role in this, simply because it is absent due to the relativistic origins of QED.

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The mass of the photon field is zero by definition, not because you set it to zero due to some gauge symmetry.

ORLY? What definition is that? The vacuum polarization is nothing more but a self-energy of the photon. Ward-Takahashi identities, which are a consequence of gauge invariance protect the photon propagator from acquiring non-zero mass.

The gauge transformation of the photon field is the addition of the gradient of some local function, which doesn't change Maxwell's equations.

This is true in Classical Electrodynamics. However, In Quantum Theory, it has far more reaching consequences, because it also changes the phase of the quantum fields.

Furthermore, requiring invariance of full QED Lagrangian (with coupling to spinor fields) under that transformation leads to an imposed local gauge symmetry of the spinor.

I guess this is in accordance with my previous remark.

The mass of the photon doesn't play a role in this, simply because it is absent due to the relativistic origins of QED.

$$-\frac{1}{2} m^{2}_{\mathrm{photon}} \, A^{\mu} \, A_{\mu}$$
is gauge invariant?

$$-\frac{1}{2} m^{2}_{\mathrm{photon}} \, A^{\mu} \, A_{\mu}$$
is gauge invariant?

I never claimed that it was gauge invariant. I claimed that it wasn't there, which is a true statement for QED.

Cool, I was talking about something else.

Furthermore, I don't see how the Ward-Takahashi identities have any impact on the form of the original QED Lagrangian.

Furthermore, I don't see how the Ward-Takahashi identities have any impact on the form of the original QED Lagrangian.

You should familiarize yourself with renormalization.

You should familiarize yourself with renormalization.

I'm familiar with renormalization, that's why I wrote "original" instead of "renormalized" Lagrangian.

I'm familiar with renormalization, that's why I wrote "original" instead of "renormalized" Lagrangian.

If you really understood the gist of Renormalization, you would know that the "original" Lagrangian is not connected to reality.

I understand that. Apparently we were both confused with what the other one meant! ;)

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Actually, my comments started as a reply to OP's https://www.physicsforums.com/showpost.php?p=3491614&postcount=25" where he mentions "Proca Lagrangian". I don't know what your intention was or whatever it was that you had in mind.

I see, now everything is clear to me, I didn't realize you were talking about Proca theory. Sorry for the confusion!

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