I think it becomes a question of causal structure, this paper suggest that, assuming the Scharnhorst effect leads to signal velocities greater than c, then the ‘Scharnhorst photons’ have spacelike worldlines in Minkowski spacetime, i.e. they are like classical tachyons. As the paper points out, relativity can handle tachyons, however if you allow tachyons the relationship between causality and causal structure is weakened. In order to show that, under their presumptions, the Scharnhorst effect doesn’t lead to causality violations they define an effective metric on the region of Minkowski spacetime described by the Casimir vacuum that preserves the link between causality and causal structure and show that it obeys a fairly strong causality condition. So it seems that you can still describe the Casimir vacuum using a Minkowski spacetime background as the underlying theory is Lorentz-invariant, but as the Casimir vacuum itself isn’t Lorentz invariant, in some ways Minkowski spacetime isn’t necessarily the best background to describe it.
Some of the issues for these kind of quantum gravity theories are similar due to non-background independence. In this approach you have
spacetime A + quantum field ≈ spacetime B
where spacetime A is the background spacetime and spacetime B is a spacetime obeying the Einstein field equations.
But the causal structure will be that of spacetime A rather than spacetime B.
I was using c in the sense of the parameter appearing in the Lorentz transformations, and ‘speed of light’ otherwise. The constant c is simply a consequence of the spacetime geometry, which we change due to introducing the Casimir plates (though I haven’t read the paper Asymptotically Fat provides, which perhaps offers a different perspective; but even there, it appears it’s not that light in the Casimir vacuum travels at a speed closer to its ‘true’ speed, but that it becomes tachyonic when described in Minkowsky space). But the speed light travels at in the Minkowsky vacuum is necessarily c. The only way it couldn’t, would be for it to have a nonvanishing mass, which would mean a frequency-dependent speed distribution, etc. It’s just a fact of geometry, and that fact is built into relativistic QFT from the outset (via the assumption that spacelike separated fields commute), which hence can’t produce a different prediction.
I don’t think it’s really consistent to interpret the Scharnhorst effect as questioning c = ls (in the Minkowsky vacuum). Rather, it says ls > c in the Casimir vacuum, due to the energy density between the plates being effectively negative.
Interesting, thanks for pointing out the paper, I’ll have a look later. However, I would have thought that tachyons are far more problematic in QFT than they might be in SR, because you can then always lower the total energy by tachyon emission, thus leading to an unstable vacuum. And in SR, any tachyon should have a non-zero, though imaginary, mass, no? So a tachyon of zero mass should always propagate along lightlike worldlines. So, does there emerge some effective imaginary mass in such a case?
For this test in particular (photon mass), you can see the current limits here (PDF). The upper limit is around 27 orders of magnitude below the proton mass and comes from spacecraft-based studies of how the solar wind responds to the ambient magnetic fields. (A massive photon would modify the magnetic effects.)
Those are both classical theories. It shouldn’t be surprising that there could be small quantum effects. I believe the most accurate measurements of C come from actually measuring the speed of light. To say C(invariant) = C(speed of light), you’d need highly accurate independent measurements of both.
This reminded me that this question came up before, in the Can neutrinos travel Faster Than Light thread. I asked there how accurately Cinvariant had been measured, without assuming it was equal to Clight, but I never felt like I got a good answer. Pasta pointed me at this Wikipedia page, but it wasn’t clear which measurements were really independent. (I suspect that page has been modified in the mean time, and it’s even less clear now.)
Then when you say
you’re just assuming C = speed of light. Certainly they are very close to equal. But you can’t rule out a very small effect. The arxiv paper that Asympotically fat linked to agrees (C = the invarient speed, c = speed of light):
I foundsomeone in that Can neutrinos travel Faster Than Light thread who disagrees with you:
I think in this context I would describe special relativity as providing the symmetry of the background to both classical and quantum theories. I mentioned Maxwell’s equations specifically because of the lack of an explicit assumption that c is the invariant speed, however ‘c’ (as the speed of a photon in a Minkowski vacuum) can be back-derived from the quantum equivalent QED. The precise meaning of “a speed of a particle” is altered due to the differences between classical physics where particles always have well-defined trajectories and quantum physics where it is not necessary to assume that they do.
It’s clear though that in QED “the speed of a photon in a Minkowski vacuum” = “the invariant speed” and this comes from the assumption of Lorentz invariance. This is hardly surprising as the assumption of Lorentz invariance of classical electromagnetism is the same as the assumption that there must exist an invariant speed which is equal to the speed of light in a vacuum.
Of course you’re not wrong that the existence of an invariant speed is independent of light travelling at that speed, though if light did not travel at that speed then it’s speed would be variable so there would be no constant light speed to measure. Therefore a direct measurement of the speed of light compared to a direct measurement of the invariant speed can only falsify that light travels at the invariant speed or in other words in the absence of 100% precise measurements you cannot prove 100% that the speed of light is invariant. That said the sum total of the evidence puts an incredibly small upper bound on the mass of a photon and the deviation from an invariant light speed.
But if that’s the case (which is of course logically possible), then the only way to be consistent with QED is for the photon not to be a mass zero particle in Minkowski space.
Yes, my interpretation of the effect back then was somewhat muddled. I was speculating that the changed ‘vacuum’ the neutrinos see while travelling through matter could enable them to propagate faster than c (in this, I wasn’t alone), and used that faster speed as a baseline; I no longer think that made good sense.
Any particle’s speed, even a massive one’s, will be a good approximation of c provided that the particle’s energy is much greater than its mass. For the current experimental bounds on the photon mass, and for photon energies typically used in experiments, the speed of light is at least an excellent approximation to c.