What would be the effects on theoretical physics if neutrinos go faster than light? Earlier today, I saw this link on Facebook about neutrinos going faster than the speed of light, and of course, re-posted. Since then, a couple of my friends have gotten into a discussion about what this means (mostly about time-travel), but I don't really know what this really implies. This made me wonder...
What are the biggest and most immediate implications of this potential discovery?
Related: Superluminal neutrinos
 A: While interesting, even potentially enormous for physics, you can still bet on the sun coming up tomorrow.  One thing I like to point out to people who are enamored with the fact that science is constantly changing is that any new changes have to fit the old observations into them.  The article even mentions this specifically.
If it turns out that neutrinos have the potential to travel faster than light, the fact remains that general relativity does a fantastic job of explaining a wide variety of phenomena and it always will.
A: There is no chance that this observation reflects neutrino physics. The neutrinps from supernova 1987a arrive 3hrs before the light, due to blocking of the supernova light by matter. Let us double this to 6hrs to include some dubious measurements, and assume that all the 6hrs is due to the superluminal neutrion travel. Then the time difference for 400km vs. 168,000 light years is $2.5 \cdot 10^{-12} s$, and this is 4 orders of magnitude smaller than the measured deviation. This means that if neutrinos outrun light by this much, the neutrinos from the supernova would have come in about a year earlier than the light.
The distance measurement is tricky, because the light-path is not the same as the neutrino path--- the neutrinos go through the Earth. If you measure the distance by sending radar between towers, you have to deal with curvature corrections due to mountains inbetween, buildings etc, which can easily add 20m of path-length over 400km. So I assume that they measured the distance using GPS. But then you have the issue that you are relying on U.S. government assurances that the absolute GPS positions are reliable to 20m. Relative distances might be ok even when absolute distances are off over large distances.
I can't say more without seeing the measurement, but it is certain in the scientific sense of 5 sigma confidence that this is not a correct result, so it is probably best to classify this as an irresponsible publicity stunt.
AFTER SKIMMING THE PAPER: No error bound on the GPS absolute position
Their estimate of distance measurements is based on the excellent relative values for displacement given the GPS coordinates. They can detect cm shifts in the Earth's crust etc. But the whole point is that you need the relative distance between the two points, and they have absolutely no independent calibration of the error in the long distance measurement, and blow smoke and mirrors with how accurate the short distance measurements are.
Here is the reference they give for their absolute distance measurement; they did none of their own; http://www.iers.org/nn_11216/IERS/EN/IERSHome/home.html , and they did no error estimate on the values they get from this. This is no good.
I don't know any way to calibrate the absolute position independently which is more accurate than the neutrino beam, so the best interpretation of the paper is that they used the neutrino beam to measure the distance between the recieving and emitting point with better accuracy than the project above gives.
Satelite Abberation
Given that the Earth is rotating with a speed v of approximately 400m/s, there is an abberation in the apparent angular position of satellites which is of the order v/c, and is normally negligible. The magnitude of the abberation between two instantaneous measurements 700 km apart depends on the angular position of the satellite in the sky, and for a satellite at 20,000 km gives a difference in estimated position of about 20m, times a trigonometric factor which can reduce this by 10% to 1%.
I don't see an estimate of correction for angular abberation in the paper.
A: Before I answer, a couple caveats:


*

*As Adam said, the universe isn't going to start behaving any differently because we discovered something.

*Right now it seems much more likely (even by admission of the experimenters) that it's just a mistake somewhere in the analysis, not an actual case of superluminal motion.


Anyway: if the discovery turns out to be real, the effect on theoretical physics will be huge, basically because it has the potential to invalidate special relativity shows that special relativity is incomplete. That would have a "ripple effect" through the last century of progress in theoretical physics: almost every branch of theoretical physics for the past 70+ years uses relativity in one way or another, and many of the predictions that have emerged from those theories would have to be reexamined. (There are many other predictions based on relativity that we have directly tested, and those will continue to be perfectly valid regardless of what happens.)
To be specific, one of the key predictions that emerges out of the special theory of relativity is that "ordinary" (real-mass) particles cannot reach or exceed the speed of light. This is not just an arbitrary rule like a speed limit on a highway, either. Relativity is fundamentally based on a mathematical model of how objects move, the Lorentz group. Basically, when you go from sitting still to moving, your viewpoint on the universe changes in a way specified by a Lorentz transformation, or "boost," which basically entails mixing time and space a little bit. (Time dilation and length contraction, if you're familiar with them) We have verified to high precision that this is actually true, i.e. that the observed consequences of changing your velocity do match what the Lorentz boost predicts. However, there is no Lorentz boost that takes an object from moving slower than light to moving faster than light. If we were to discover a particle moving faster than light, we have a type of motion that can't be described by a Lorentz boosts, which means we have to start looking for something else (other than relativity) to describe it.
Now, having said that, there are a few (more) caveats. First, even if the detection is real, we have to ask ourselves whether we've really found a real-mass particle. The alternative is that we might have a particle with an imaginary mass, a true tachyon, which is consistent with relativity. Tachyons are theoretically inconvenient, though (well, that's putting it mildly). The main objection is that if we can interact with tachyons, we could use them to send messages back in time: if a tachyon travels between point A and point B, it's not well-defined whether it started from point A and went to point B or it started from B and went to point A. The two situations can be transformed into each other by a Lorentz boost, which means that depending on how you're moving, you could see one or the other. (That's not the case for normal motion.) This idea has been investigated in the past, but I'm not sure whether anything useful came of it, and I have my doubts that this is the case, anyway.
If we haven't found a tachyon, then perhaps we just have to accept that relativity is incomplete. This is called "Lorentz violation" in the lingo. People have done some research on Lorentz-violating theories, but it's always been sort of a fringe topic; the main intention has been to show that it leads to inconsistencies, thereby "proving" that the universe has to be Lorentz-invariant. If we have discovered superluminal motion, though, people will start looking much more closely at those theories, which means there's going to be a lot of work for theoretical physicists in the years to come.
A: Since Hawking already found quantum effects to get around the black-hole roadblock thrown up by unadulterated, un-quantum General Relativity, it is possible that there are similar quantum tunnelling ways to get around the notion that $c$ is a roadblock.  In fact, General Relativity already requires modification of the constancy of the speed of light in vacuo anyway, and the sky did not fall for that, so even if these experimenters are correct, which seems unlikely, this need not be any more revolutionary than what we already realised: there are fundamental problems with unifying Special Relativity with Quantum Mechanics and we may not have been finished yet.  Without being earthshaking, it would be great and exciting to have a measurable incompatibility between Special Relativity and Quantum Theory more accessible to experiment than the Planck scale.
A: If the  results of the OPERA experiment will be confirmed, that would probably mean that neutrinos are tachyons with negative mass square. As David wrote, tachyons are consistent with special relativity (SR), i.e. SR is not violated, but we still need to do something with potential causality violation.
But this situation is not new in physics. A thought experiment authored in 1935 by Einstein, Podolsky and Rosen (EPR) was intended to demonstrate that quantum theory does not comply with causality. It took about 50 years to establish that both causality and quantum theory are preserved in this thought experiment. By the way, it is not a thought experiment any more. The idea of EPR paradox is used in real experiments on quantum teleportation, as well as in newest information security systems.
I've found a book by Moscow State University professor Yakov Terletsky published in 1966 (unfortunately only in Russian). In this book Terletsky is making an attempt to explain existence of tachyons without violation of causality. Here is the link:
http://lib.mexmat.ru/books/8667
