added much later: It turned out that the OPERA results were flawed.

I leave the text below and the comments as testimony of my confusion about all that when the surprising observation seemed somehow possible.

-----

I wonder if neutrinos should travel slightly faster than photons according to standard physics, i.e. quantum field theory as we know it (*).

The reason would be the Scharnhorst effect.

If c denotes the 'bare' speed of light then real photons should actually travel a tiny bit slower than c, due to the interaction with virtual particles in a 'real vacuum': c(physical) < c. The bulk of the effect comes from interactions with virtual electrons and positrons (see e.g. this paper for more details).

But neutrinos interact only weakly with those and therefore the velocity of neutrinos should be closer to c; In other words neutrinos would be slightly faster than photons.

I am not sure if the OPERA experiment detected anything real, but if the results are indeed confirmed one should take a second look at the Scharnhorst effect imho; This time considering the difference between neutrinos and photons instead of looking at photons between Casimir plates.

with credit to A.M., a friend of mine and yet another quant interested in physics, who reminded me of this effect, which was discussed already in the 1990s. But if this is all b.s. the embarrassment is of course fully mine.

added later: A very simplified calculation shows that the Scharnhorst effect could have the right order of magnitude!

According to the Scharnhorst paper, equ. 10, the increase of the speed of photons between Casimir plates is to first order approximately

1 + 0.01*alpha

^{2}/(mL)

^{4}.

As we move the plates closer and reduce L (in a thought experiment) we eliminate more of the interaction with the virtual particles and the speed of photons gets closer to the 'bare' speed of light c.

However, we cannot reduce L below 1/m (in this approximation) and thus the Scharnhorst formula

gives a maximum correction

c = c(bare) = c(physical)*( 1 + 0.01*alpha

^{2}).

If we assume that neutrinos travel at a speed close to the 'bare' speed c, then they would travel faster than photons in the 'real vacuum' by a factor of approximately 1 + 10

^{-6}.

This is pretty much what OPERA measured.

(*) added even later: Heather Logan thinks this explanation cannot work and I changed the sentence to reflect that this is not 'standard' opinion; After all she is a physics professor and my last lesson in QED was about 20 years ago. On the other hand I don't see where the argument goes wrong. Obviously I appreciate any input.

added one day later: After some more thinking and reading I can now better formulate this idea as follows:

i) In QED the Ward identities ensure that after quantization of the electromagnetic field the longitudinal polarization of the photon vanishes.

ii) This implies (by comparison with classical fields?) that the photon mass remains zero and the velocity of photons is the light speed c.

iii) On the other hand, the Scharnhorst effect suggests that photons travel at a 'dressed' speed d and one can increase (in principle) d between Casimir plates (see all the above).

iv) The standard interpretation is that d = c to keep QED simple, but one has then to explain away the acausality from photons travelling at increased speed between Casimir plates.

v) In light of the OPERA experiment, the proposal is to assume d < c and (because the Ward identities hold) for all possible experiments the longitudinal polarization of the physical photon still vanishes, so that it would behave like a massless particle, although it travels slightly slower than the 'true' speed of light c. (By the way, I am not the first thinking along such lines.)

vi) The neutrino would travel very close to the 'true' speed of light c and thus slightly faster than photons. The estimate above suggests that the effect could be in the right ballpark,

see the comments for more

## 11 comments:

An interesting idea. But if the explanation is that the "true" speed of light is larger than the speed of photons, it should appear at other particles, too. Protons in the LHC are 0,036 ppm slower than the speed of light (Tevatron: 0,3ppm), so they should be faster than the speed of photons if that effect is so large. That would result in Cherenkov radiation, which is not observed. I do not know how precise the speed of the particles is measured, that would give another test.

How about this?

Neutrinos travelling through the crust is like particles travelling through very narrow Sharnhost plates and thus slighty faster, but in vacuum there is no effect.

So you can explain OPERA and SN1987A in one swoop.

If Heather is right (and I guess she is the more I think about it) and the OPERA result is confirmed (and I guess it may not) then I will consider your explanation next 8-)

Maybe rock does speed up neutrinos (due to distorting the physical vacuum a la Scharnhorst) rather than slow them down.

And it would also explain the SN1987A result (no rocks).

I've been thinking along similar lines. You might be interested in this paper: http://arxiv.org/abs/hep-th/9408016 (Speed of Light in Non-Trivial Vacua), where the conclusion is that the measured speed of propagating photons depends on the energy density, such that a positive energy density causes a slowdown, a negative energy density (as in Casimir plates relatively to the usual QED vacuum) a speedup of light. So maybe we have only been observing the 'dressed' value of the speed of light so far...

>> maybe we have only been observing the 'dressed' value of the speed of light so far...

Yes. I have updated the main text to make it clearer what this could mean.

It seems to me that giving up the simplifying assumption (or actually quasi-renormalization) d=c could be the least painful way to save the appearances if the OPERA result turns out to be correct.

@Anonymous: Perhaps one can combine the Mikheyev–Smirnov–Wolfenstein effect with the Schranhorst effect to explain why neutrinos travel faster than light in a rock.

This would explain *everything*.

The vacuum remains Lorentz-invariant and SN1987A would have no effect.

The OPERA neutrinos would be slightly faster than light (and the effect would be on the order estimated in my text).

Interesting! I wasn't aware of this... I was about to give up on the Scharnhorst effect explanation, as it seems impossible to get the right energy scaling behaviour to be consistent with both SN1987A and OPERA, but maybe this needs to be looked at again.

Incidentally, I'm not so sure one should worry much about the Ward identities etc. -- one could always adopt a 'Lorentzian' approach to SR (see Bell's 'How To Teach Special Relativity'), in which, loosely, electromagnetic theory appears relativistic independently of any 'true' background geometry because electromagnetic interactions are what define our rods and clocks.

Something analogous is done in condensed matter, where obviously there is a preferred, Newtonian, reference frame -- the rest frame of some crystal lattice, say --, and nevertheless, an 'inside observer' would see a special relativistic 'spacetime' if there is some maximum speed at which lattice excitations propagate, since all he has access to in order to probe this are, well, lattice excitations. QED built upon these foundations would be just as Lorentz invariant.

>> consistent with both SN1987A and OPERA

well, SN1987A was mostly lower energy e-neutrinos while OPERA was mostly higher energy mu-neutrinos.

If neutrinos have a positive mass, then the high energy OPERA neutrinos could travel close to the 'bare' speed c (and thus faster than light).

But the SN1987A neutrinos would be slower and perhaps in this case just about the same speed as photons traveling at the dressed light speed.

almost any explanation which assumes that both OPERA and SN1987A are correct would have to use such a coincidence.

However, the "rock speeds up neutrinos" explanation would easily explain both but on the other would have the trouble to explain why this does not lead to acausality (if neutrinos in rock would travel faster than the relativity speed of light).

>> Ward identities

yes, one would try to turn Heather's argument around:

photons move at the dressed speed d<c, but the Ward identities hold, which explains why photons appear as massless to us.

>> well, SN1987A was mostly lower energy e-neutrinos while OPERA was mostly higher energy mu-neutrinos.

Yes, but the functional dependence of the neutrino speed on energy can't be what you'd expect from SR with the constraints imposed by SN1987A and OPERA, both of whom see little to no variation in speed over the energy regimes they probe. There's already a paper out discussing this (including limits from other neutrino experiments): http://arxiv.org/abs/1109.5172

Perhaps MSW can provide the right kicker...

the same idea with much more detail here: http://arxiv.org/abs/1110.0132

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