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### But what if they are faster than light?

This week the OPERA experiment released a statement about their famous “faster than light” neutrino measurement. In September scientists announced that they had measured the speed of neutrinos traveling from CERN to Gran Sasso and they found that they arrived slightly sooner than they should do according to special relativity. There was a plethora of scientific papers, all kinds of rumors and speculation, and most physicists simply refused to believe that anything had traveled faster than light. After months of diligent study, OPERA announced that they may have tracked down two sources of experimental error, and they are doing their best to investigate the situation.

But until we get the results of OPERA’s proposed studies we can’t say for sure that their measurement is right or wrong. Suppose that they reduce the lead time of the neutrinos from 60ns to 40ns. That would still be a problem for special relativity! So let’s investigate how we can get faster than light neutrinos in special relativity, before we no longer have the luxury of an exciting result to play with.

The OPERA detector (OPERA Collaboration)

Special relativity was developed over a hundred years ago to describe how electromagnetic objects act. The electromagnetic interaction is transferred with electromagnetic waves and these waves were known to travel extremely quickly, and they seemed to travel at the same speed with respect to all objects, no matter how those objects were moving. What Einstein did was to say that the constancy of the speed of light was a fundamental law of nature. Taking this to its logical conclusion meant that the fastest speed possible was the speed of light. We can call the fastest possible speed $$s$$ and the speed of light $$c$$. Einstein then says $$c=s$$. And that’s how things stood for over a century. But since 1905 we’ve discovered a whole range of new particles that could cast doubt on this conclusion.

When we introduce quantum mechanics to our model of the universe we have to take interference of different states into account. This means that if more than one interaction can explain a phenomenon then we need to sum the probabilities for all these interactions, and this means we can expect some strange effects. A famous example of this is the neutral kaon system. There two lightest neutral kaons are called $$K^0$$ and $$\bar{K}^0$$ and the quark contents of these mesons are $$d\bar{s}$$ and $$s\bar{d}$$ respectively. Now from the “outside” these mesons look the same as each other. They’ve got the same mass, they decay to the same particles and they’re made in equal numbers in high energy processes. Since they look identical they interfere with each other, and this gives us clues about why we have more matter than antimatter in the universe.

Since we see interference all over the place in the Standard Model it makes sense to ask if we see interference with a photon. It turns out that that we do! The shape of the Z mass peak is slightly asymmetric because of interference between virtual Z bosons and virtual photons. There are plenty of other particles that the photon can interfere with, including the $$J/\psi$$ meson, and the $$\rho$$ meson. In fact, any neutral vector meson with no net flavor will do. Einstein didn’t know about any of these particles, and even if he did he never really accepted the conclusions of quantum mechanics, so it’s no surprise that his theory would require that the speed of light is the fastest speed (that is, $$c=s$$.) But if the photon interferes with other particles then it’s possible that the speed of light is slightly lower than the fastest possible speed ($$c<s$$). Admittedly, the difference in speed would have to be very small!

In terms of quantum mechanics we would have something like this:
$|light>_{Einstein} = |\gamma>$
$|light>_{reality} = a_\gamma |\gamma> + a_{J/\psi} |J/\psi> + a_Z |Z> + \ldots$

As you can see there are a lot of terms in this second equation! The contributions would be tiny because of the large difference in mass between the massive particles and the photon. Even so, it could be enough to make sure that the speed of light is ever so slightly slower than the fastest possible speed.

At this point we need to make a few remarks about what this small change in speed would mean for experiments. It would not change our measurements of the speed of light, since the speed of light is still extremely fast and no experiment has ever showed a deviation from this extremely fast speed. Unless somebody comes up with an ingenious experiment to show that the difference between the speed of light and the fastest possible speed is non-zero we would probably never notice any variation in the speed of light. It’s a bit unfortunate that since 1983 it’s been technically impossible to measure the speed of light, since it is used in the definition of our unit of length.

Now we know that photons can interfere with other particles it makes sense to ask the same question about neutrinos. Do they interfere with anything? Yes, they can interfere, so of course they do! They mix with neutrinos of other flavors, but beyond that there are not many options. They can interfere with a W boson and a lepton, but there is a huge penalty to pay in the mass difference. The wavefunction looks something like this:
$|\nu_e>(t) = a(t)_{\nu_e}|\nu_e> + a(t)_{\nu_{\mu}}|\nu_\mu> + a(t)_{\nu_{\tau}}|\nu_\tau> + a(t)_{We}|We>$
(I’ve had to add a time dependence due to neutrino mixing, but it’s essentially no more complicated than what we had for the photon.)

That means that the photon could get slowed down slightly by the interference with other particles (including particles in the vacuum) and that neutrinos could get slowed down more slightly by their interference terms with other particles. And that way we could get neutrinos traveling faster than the speed of light and special relativity could remain intact. (In this description of the universe we can do what used to seem impossible, we can boost into the rest frame of a photon. What would it mean to do that? Well I suppose it would mean that in this frame the photon would have to be an off-shell massive particle at rest.)

The SN 1987 supernova, a rich source of slower than light electron neutrinos (Hubble, ESA/NASA)

Now I’ll sit back and see people smarter than I am pick holes in the argument. That’s okay, this isn’t intended to be a serious post, just a bit of fun! There are probably predictions of all kinds of weird effects such as shock waves and time travel that have never been observed. And there are plenty of bits I’ve missed out such as the muon neutrinos traveling faster than electron neutrinos. It’s not often we get an excuse to exercise our analytic muscles on ideas like this though, so I think we should make the most of it and enjoy playing about with relativity.