Not necessarily. (Disclaimer: Physics background but this is not my area of expertise; I am working from memory of courses I took >5 years ago). In electroweak unification, there are four underlying gauge fields, superpositions of which make up the photon, W bosons, and Z boson. You have to adjust the coefficients of the combinations very carefully to make the photon massless and the weak bosons heavy. You could adjust them slightly less carefully and have an extremely light, but not massless, photon, without touching the underlying gauge fields; then you can derive Maxwell and whatnot using the gauge fields instead of the physical particles, and presumably save SR as well.
Observe that the current experimental upper limit on the photon mass (well, I say current—I mean, the first result that comes up in Google; it’s from 2003, but not many people bother with experimental bounds on this sort of thing) is 7x10^{-19} eV, or what we call in teknikal fiziks jargon “ridiculously tiny”.
SR doesn’t depend on behaviour of gauge fields. Special relativity is necessary to have a meaningful definition of “particle” in field theory. The gauge fields have to have zero mass term because of gauge invariance, not Lorentz covariance. The mass is generated by interaction with Higgs particle, this is essentially a trick which lets you forget gauge invariance after the model is postulated. It doesn’t impose any requirements on SR either.
I was thinking of how Lorentz invariance was historically arrived at: From Maxwell’s equations. If the photon has mass, then presumably Maxwell does not exactly describe its behaviour (although with the current upper bound it will be a very good approximation); but the underlying massless gauge field may still follow Maxwell.
First we may clarify what is exactly meant by “following Maxwell”. For example in electrodynamics (weak interaction switched off) there is interaction between electron field and photons. Is this Maxwell? Classical Maxwell equations include the interaction of electromagnetic field and current and charge densities, but they don’t include equation of motion for the charges. Nevertheless, we can say that in quantum electrodynamics
photon obeys Maxwell, because the electrodynamics Lagrangian is identical to the classical Lagrangian which produces Maxwell equations (plus equations of motion for the charges)
photon doesn’t obey Maxwell, because due to quantum corrections there is an extremely weak photon self-interaction, which is absent in classical Maxwell.
See that the problem has nothing to do with masses (photons remain massless in QED), Glashow-Weinberg-Salam construction of electroweak gauge theory or Higgs boson. The apparent Maxwell violation (here, scattering of colliding light beams) arise because on quantum level one can’t prevent the electron part of the Lagrangian from influencing the outcome even if there are no electrons in the initial and final state. Whether or not is this viewed as Maxwell violation is rather choice of words. The electromagnetic field still obeys equations which are free Maxwell + interaction with non-photon fields, but there are effects which we don’t see in the classical case. Also, those violations of Maxwell are perfectly compatible with Lorentz covariance.
In the case of vector boson mass generation, one may again formulate it in two different ways:
the vector boson follows Maxwell, since it obeys equations which are free Maxwell + interaction with Higgs
it doesn’t follow Maxwell, because the interaction with Higgs manifests itself as effective mass
Again this is mere choice of words.
Now you mentioned the linear combinations of non-physical gauge fields which give rise to physical photon and weak interaction bosons. The way you put it it seems that the underlying fields, which correspond to U(1) and SU(2) gauge group generators, are massless and the mass arises somehow in the process of combining them together. This is not the case. The underlying fields all interact with Higgs and therefore are all massive. Even if the current neutrino affair lead to slight revision of photon masslessness, the underlying fields would be “effectively massive” by interaction with Higgs (I put “effectively massive” in quotes because it’s pretty weird to speak about effective properties of fields which are not measurable).
Of course, your overall point is true—there is no fundamental reason why photon couldn’t obtain a tiny mass by the Higgs mechanism. Photon masslessness isn’t a theoretical prediction of the SM.
Photons having mass would screw up the Standard Model too… right?
Not necessarily. (Disclaimer: Physics background but this is not my area of expertise; I am working from memory of courses I took >5 years ago). In electroweak unification, there are four underlying gauge fields, superpositions of which make up the photon, W bosons, and Z boson. You have to adjust the coefficients of the combinations very carefully to make the photon massless and the weak bosons heavy. You could adjust them slightly less carefully and have an extremely light, but not massless, photon, without touching the underlying gauge fields; then you can derive Maxwell and whatnot using the gauge fields instead of the physical particles, and presumably save SR as well.
Observe that the current experimental upper limit on the photon mass (well, I say current—I mean, the first result that comes up in Google; it’s from 2003, but not many people bother with experimental bounds on this sort of thing) is 7x10^{-19} eV, or what we call in teknikal fiziks jargon “ridiculously tiny”.
SR doesn’t depend on behaviour of gauge fields. Special relativity is necessary to have a meaningful definition of “particle” in field theory. The gauge fields have to have zero mass term because of gauge invariance, not Lorentz covariance. The mass is generated by interaction with Higgs particle, this is essentially a trick which lets you forget gauge invariance after the model is postulated. It doesn’t impose any requirements on SR either.
I was thinking of how Lorentz invariance was historically arrived at: From Maxwell’s equations. If the photon has mass, then presumably Maxwell does not exactly describe its behaviour (although with the current upper bound it will be a very good approximation); but the underlying massless gauge field may still follow Maxwell.
First we may clarify what is exactly meant by “following Maxwell”. For example in electrodynamics (weak interaction switched off) there is interaction between electron field and photons. Is this Maxwell? Classical Maxwell equations include the interaction of electromagnetic field and current and charge densities, but they don’t include equation of motion for the charges. Nevertheless, we can say that in quantum electrodynamics
photon obeys Maxwell, because the electrodynamics Lagrangian is identical to the classical Lagrangian which produces Maxwell equations (plus equations of motion for the charges)
photon doesn’t obey Maxwell, because due to quantum corrections there is an extremely weak photon self-interaction, which is absent in classical Maxwell.
See that the problem has nothing to do with masses (photons remain massless in QED), Glashow-Weinberg-Salam construction of electroweak gauge theory or Higgs boson. The apparent Maxwell violation (here, scattering of colliding light beams) arise because on quantum level one can’t prevent the electron part of the Lagrangian from influencing the outcome even if there are no electrons in the initial and final state. Whether or not is this viewed as Maxwell violation is rather choice of words. The electromagnetic field still obeys equations which are free Maxwell + interaction with non-photon fields, but there are effects which we don’t see in the classical case. Also, those violations of Maxwell are perfectly compatible with Lorentz covariance.
In the case of vector boson mass generation, one may again formulate it in two different ways:
the vector boson follows Maxwell, since it obeys equations which are free Maxwell + interaction with Higgs
it doesn’t follow Maxwell, because the interaction with Higgs manifests itself as effective mass
Again this is mere choice of words.
Now you mentioned the linear combinations of non-physical gauge fields which give rise to physical photon and weak interaction bosons. The way you put it it seems that the underlying fields, which correspond to U(1) and SU(2) gauge group generators, are massless and the mass arises somehow in the process of combining them together. This is not the case. The underlying fields all interact with Higgs and therefore are all massive. Even if the current neutrino affair lead to slight revision of photon masslessness, the underlying fields would be “effectively massive” by interaction with Higgs (I put “effectively massive” in quotes because it’s pretty weird to speak about effective properties of fields which are not measurable).
Of course, your overall point is true—there is no fundamental reason why photon couldn’t obtain a tiny mass by the Higgs mechanism. Photon masslessness isn’t a theoretical prediction of the SM.
Ok, I sit corrected. This is what happens when an experimentalist tries to remember his theory courses. :)