**Update:** On Saturday, I added a blog entry discussing possible mistakes in the Opera paper.

**Update:** On Thursday, AP, BBC (who renamed Opera to Orion: fixed) and Reuters (and all other media) finally leaked the news: the neutrinos are claimed to have arrived 60 nanoseconds before the light (18 meters over *c*).

Because this is said to be a 6-sigma signal, their total error margin of the timing should be 10 nanoseconds (3 meters over *c*); recall that the distance is 732 km. I leave it to the reader to decide whether this accuracy is plausible given the messy birth and detection of the particles. One nanosecond is the duration of one cycle of your iPhone microprocessor, among other things. Ten nanoseconds is 40% of the lifetime of the charged pion or 80% of the lifetime of the charged kaon. I can kind of imagine that they're doing something really silly, like imagining that each pion or kaon lives at least for the lifetime and then it dies. But some of them decay immediately; this error could erase most of the 60-nanosecond discrepancy.

Talk on Friday: watch at 4 p.m. Prague Summer Time; Preprint is finally out (click!)Anna has suggested that their GPS-based timing device may have neglected that the electromagnetic waves are moving slower than

See #MundaneNeutrinoExplanations for some ongoing Twitter fun such as:

Bartender says: "We don't serve neutrinos here." Neutrino enters a Gran Sasso tavern. :-)

*c*through the atmosphere: if the collaboration did an error in this subtlety, they get an error of exactly the same magnitude to explain the "signal". The index of refraction of the air is 1.0003, so light needs to penetrate a 10-km layer of the atmosphere as it would need to get through 10.003 km of the vacuum which would exactly produce the 3-meter delay. Make the atmosphere a bit thicker because the satellites are not right above your head; add the delays from both directions and you may already produce those 18 meters of error (or most of it).

*Text below was originally posted on September 19th*

**Experimental claims on neutrinos that are faster than light**

Tommaso Dorigo is spreading a six-sigma (6.1, to be more precise) rumor involving neutrinos and his countrymates in an experiment melodically named Opera (well, the result is actually mostly due to French intruders from Lyon, but that's a detail):

A Six-Sigma Signal Of Superluminal Neutrinos From Opera! (Dorigo)The first link points to a Yahoo cache because Dorigo had to scrap his article: as an employee of IFNF in Italy that co-funds Opera, this is a top-secret classified material for him. On Friday afternoon, I returned the original Dorigo link: it works again.

Can neutrinos be superluminal? Ask OPERA (Gibbs)

Supernovas and neutrinos (Strassler)

When we visited Milan, a pigeon managed to deposit some products of its metabolism directly into the pocket of my shorts. This procedure was apparently achieved from a nearly impossible (vertical?) angle – much like if your world line in the spacetime were spacelike instead of timelike.

This experience taught me to be careful about angles measured in Italian towns, especially the towns that are famous for an opera. That lesson will turn out to be useful momentarily – in the case of the spacetime surrounding the Gran Sasso mountain.

*A theater that is prettier than La Scala.*

And by the way, we were so disappointed by the exteriors of the famous La Scala, the local opera house, that I decided to post a picture of our local J. K. Tyl's Theater in Pilsen which all Pilsner tourists preferred over La Scala. ;-)

Make no mistake about it, this trash talk is a preparation for the soccer match, FC Viktoria Pilsen vs AC Milan, in the UEFA Champions League. We will show them the door much like Napoli did a few days ago ;-) before we will do the same thing with Barcelona, a Catalan village where they thought they could play soccer before they met any players from Pilsen. :-) Unfortunately, BATE Borisov (Belarus) played a tie against Pilsen in the first match of our group; Milan tied with Barcelona in our group, too.

**Back to neutrinos**

Fine. What Dorigo says is simple. They measured how much time it takes for the tau neutrinos to cause a reaction in Gran Sasso tavern in central Italy. The (currently) tau neutrino \(\nu_\tau\) came together with a mostly muon-neutrino beam originally created by decays of pions \(\pi\) and kaons \(K\) into muons \(\mu\) and muon neutrinos \(\nu_\mu\) at CERN, more precisely in SpS which is a pre-accelerator of the LHC.

They tried to be careful about the timing, measured the delay, calculated the speed. And their resulting speed of the neutrinos is

\[ v = c + 6 \sigma. \] It is greater than the speed of light – by an excess of speed that is six times larger than their total error margin. If this claim were true, the probability that this excess happens by chance is lower than one part in half a billion. Much like the pigeon's feces near the opera house in Milan, the Opera collaboration claims that the neutrinos are moving along impossible angles in the spacetime.

*The Opera experiment*

Of course, the claim is false although I can't tell you exactly why because I wasn't told any further details about the rumor, either. They have surely screwed the calculation of some timing in the multi-step process.

The claims about a superluminal neutrino have a long history. All those claims in the past seem to be impossible although people are usually unaware of the new evidence that identified the errors in the superluminal claims.

Supernova observations are the most reliable source to tell us that neutrinos are moving slower than the speed of light – because their mass is actually positive. For high energy neutrinos, the deviation of the speed from the speed of light is smaller and the neutrinos arrive simultaneously with the light produced during the supernova explosion. But high-energy neutrinos that are faster than light seem to be incompatible with some observations of the supernovae.

You have to assume that neutrinos have a different speed when they move through the Italian rocks. They're perhaps faster in rocks than in the vacuum, Dorigo will tell you. But rocks are almost transparent for neutrinos, according to everything we know about neutrinos today. It's hard to imagine how the rocks could matter. Moreover, local interactions of the rocks with the neutrinos, if they were relevant, would almost certainly slow the neutrinos down instead of speeding them up.

You may see my answer on theoretical physics stack exchange with a list of theoretical papers that discussed superluminal/tachyonic neutrinos...

**Opera vs violinist Einstein**

More importantly, superluminal neutrinos – or any other particles – contradict special relativity. If you could shoot superluminal neutrinos, they would move along spacelike paths in the spacetime and the Lorentz symmetry would allow you to boost the SpS a little bit so that the neutrinos would be moving backwards in time (there is no qualitative difference between spacelike paths going to the future and those going to the past). By boosting the SpS accelerator, you could create neutrinos that would move backwards in time and you could kill your grandfather before he had his first sex with your grandmother, thus rendering your own existence needed for the homicide inconsistent with the result of the homicide.

This shouldn't happen. Your grandfather in the past has the right to protect his record of having sex with your grandmother – much like the Italian prime minister has the right to defend his record that he was able to do eight women in a single night (he couldn't manage more, he says, even though eleven women, one for each dimension of the spacetime in M-theory, were waiting in the line). So neutrinos should better be slower than light, otherwise the pride of Berlusconi who just left a court (in Milan, of course) would be in trouble. And the pride matters more than his job because he's only a prime minister in his spare time, we were told. :-)

*CERN and Gran Sasso are 732 km i.e. 2.44 light milliseconds away from each other.*

More than ten years ago, around the year 2000 or so (when I was a Rutgers grad student), there were claims that the radioactive decay of the tritium atoms implied that neutrinos should have been superluminal. The statistical significance of the claims was apparently very low. I don't know what exactly happened with that claim but because no one has talked about this revolution in recent years, I guess that it slowly evaporated. This was the first time when people tried to quantitatively analyze the beta-decay of atoms and not just the bare nuclei so I am sure that they did something wrong, or they were just unlucky with the statistics.

**Tachyons and spin**

In bosonic string theory, the 26-dimensional ancestor of modern (super)string theory that was randomly discovered in the late 1960s, one has a troublesome particle called the tachyon. It is a scalar particle associated with the field \(T(x^\mu)\) whose squared mass is negative:

\[ m_T^2 = -\frac{1}{\alpha'} = -\frac{T_{\rm string\,\,tension}}{2\pi} \] for the open string tachyon and 4 times larger for the closed string tachyon. At the level of classical particles, the negative squared mass looks like faster-than-light propagation of particles that allows you to inconsistently change your grandfather's bedroom record.

At the level of field theory, a negative squared mass means that the potential (density of potential energy to be more accurate),

\[ V(T) = \frac{m_T^2}{2} T^2, \] acquires a maximum rather than minimum at \(T=0\). That's very bad because this value of the tachyon field becomes unstable. The field wants to roll down and find a true minimum. If it doesn't find a true minimum, it will continue to roll indefinitely. Quantum mechanically, this increasingly fast rolling on the slope looks like a spontaneous creation of tachyonic particles or their pairs with both positive and negative energies.

Superstring theory eliminates tachyons and guarantees that the theory contains fermions as well. But we have never seen a string theory with a tachyon which is not a scalar particle. Is it a coincidence? Can't we have a tachyon with spin? A Dirac tachyon? Or a vector tachyon?

The answer is that it is no coincidence. In string theory, tachyons have to be scalar particles. The reason is that to give the tachyon a nonzero spin, you must excite it by some stringy oscillators that are connected with large spacetime coordinates. But these oscillations raise the squared mass of the resulting particle and because of that, the particle's squared mass can no longer be negative. In bosonic string theory, you may only obtain spin by \(\alpha_{-n}^\mu\) which increases \(N\) by one (minimal amount for \(n=1\)); in supersymmetric string theory, one may also have \(\psi_{-n}^\mu\) where \(n\) may be as low as \(n=-1/2\) in the antiperiodic (NS) sector. However, the ground state energy is less tachyonic in the RNS superstring case and this is enough to bring you to the massless level with just a single excitation, too.

Of course, those calculations only apply to the free string theory – without any interactions. (But the result is still true even in you consider arbitrary orbifolds etc., even in the twisted sectors where the ground state energy is less tachyonic or not tachyonic at all.) Can you have Dirac particles when you turn on interactions? Can you have spin-1/2 fermions which are tachyonic?

**Effective field theory vs spinning tachyons**

Well, the answer is that this is impossible even at the level of effective field theory. Recall that the kinetic term for a Dirac particle is

\[ {\mathcal L}_{\rm kin} = \bar \psi \gamma^\mu \partial_\mu \psi \] and is independent of the mass. The mass term

\[ {\mathcal L}_{\rm mass} = -m \bar\psi \psi \] has a coefficient of \(m\). Note that the mass is not squared in this case (unlike the case of the scalar tachyonic particle). Consequently, you can't make the world lines spacelike because that would mean an imaginary value of \(m\) and the action would fail to be real (or the Hamiltonian wouldn't be Hermitian).

In other words, you know that the Klein-Gordon equation may be derived from the free Dirac equation if you apply the Dirac operator twice. That's why the squared mass in the Klein-Gordon equation has to be the square of a real number: it can't be negative.

See my somewhat technical report about neutrino oscillations and Majorana vs Dirac character of the neutrinos (nu.pdf) that I presented in the late 1990s.Similar field-theory arguments may be used to show that gauge bosons associated with spontaneously broken gauge symmetries can't be tachyonic, either: in \(m_A^2 = g^2 v^2\), both factors on the right hand side (the squared coupling and the squared vev) are positive. One may probably generalize these arguments to arbitrary spins: the only allowed spin of a tachyonic particle in a Lorentz-invariant theory is zero. Such scalar tachyonic particles make the theory unstable.

So there are various reasons why I think that the claim will go away. They will review the accuracy with which they may determine the timing and the signal will no longer be six sigma (but I can't even tell you whether the wrong timing affects the decays at CERN or the detection in Gran Sasso). Superluminal particles are the ultimate extraordinary claim, it seems to contradict many other observations that have eliminated such a possibility, so some truly extraordinary evidence is surely necessary.

Statistically speaking, six standard deviations is a strong signal but it is still "of order one". For such a new experiment that is measuring something totally unusual, you must always wonder why the signal is just 6 times larger than their error margin. Why are they comparable? Why isn't the accuracy 50 times better than needed? Times that may be measured by clocks in Italy span many possible orders of magnitude. From this viewpoint, "6 sigma" is still too close to "1 sigma" and the big claim is likely to be due to an underestimate of some systematic errors.

Let me say it in other words. When there are claims about deviations from the currently valid theory that violate its basic properties, we shouldn't just wait for some statistically significant result that says "something unusual is going on". We should wait for a big enough accuracy so that the experimenters can tell us what the deviation is and how it phenomenologically depends on certain quantities. For example, they should first say that the speed is \(v=c+a/E_\nu\) and the coefficient \(a\) should be measured accurately enough and by various places and perhaps by independent methods.

There are no tachyons in Nature. Well, to be more accurate, the Higgs boson would be a tachyon if we could spend our lives near the maximum of the Higgs potential, \(H=0\). However, this point is unstable so the Higgs would quickly roll down – and did roll down – to a value where the energy density is lower, to a true minimum. That's why the electroweak symmetry is spontaneously broken – assuming that God gives us His particle. And when we expand around this point, there are no tachyons left.

So what Opera has produced is just a Phantom of the Opera:

*Phantom of the Opera, by Mr Martin Chodúr, the 21-year-old recent winner of the Czech and Slovak American Idol, and the 13-year-old recent winner of the Czech and Slovak Talentmania, Miss Patricia Janečková. I assure dear Polish visitors that our beloved shared star Ewa Farna can also sing opera but Ewakuacja may still win. ;-)*

Just to be sure, a phantom is a ghost, usually of a deceased person, and – as the singers suggest – it exists inside your mind only.

If you're interested in physics, and physics is the hottest topic in Poland since the times of Maria Salomea Skłodowska :-), try other articles on this blog. For example, my ex-colleague and one of world's top physicists Lisa Randall of Harvard (extra dimensions etc.) guest blogged over here today (Tuesday) about her new book, Knocking on Heaven's Door.

Neutrinos are being mentioned or discussed in 125 articles on this blog.

## snail feedback (36) :

You make all these claims about Berlusconi, but your post appeared 3 days before the news was released. So who's really travelling in time?

You say that you got the "rumour" in advance from a cache at Yahoo. A company that doesn't even exist any more! Well, barely.

I really dislike when news agencies read some speculations on some blogs and interpret it as scientific results..

I find your post very informative. However, what I am interested in knowing is how the newspapers interviewed the scientists. Because it looks as if the scientists are positive that the results are true.

I'm just wondering if the reporters actually interviewed the scientists, but cherry-picked the juicy bits and edited the responses to fit an exciting story..

Do you know anything about the response of the scientists to these news articles?

It's just that so many incorrect rumors are spread this way, by reporters incorrectly reporting some hypothesis or speculations to be factual..

I'm not going to try to follow your physics as it's just been too many years. However, one thought that occurred to me was what if there was some strange form of quantum entanglement between the two labs? This would allow for FTL signal transmission.

Loki

Dear Tómas,

don't be too dogmatic about the waiting. The result was leaked by Tommaso Dorigo 3 days ago and those who know some sociology realize that he's damn close to the source so it was pretty much guaranteed that the very same information would be released officially within days. I didn't hesitate to write about it because it was no rumor: it was just the official release that was sped up a little bit.

I am confident that most physicists think that the media react properly in this particular case. In fact, most of the articles also quote top theoretical physicists such as John Ellis who explain that this is an extraordinary claim that requires extraordinary evidence - why it contradicts all things we knew about the speed limit at least since 1905. Everyone closer to the theory is convinced that the result will go away; the experimenters clearly believe that they did everything carefully and their big claim is right. Well, they claim it because they don't appreciate how powerful relativity is relatively to the fallible humans such as themselves, but I can't rigorously prove that they're wrong at this point.

Dear Leigh, people believe all kinds of things (Poe's law) so I will better avoid jokes here. ;-) Let me swear that I haven't killed my grandfather when he was a little asexual boy and I am not aware of any violations of the cosmic speed limit that I would ever perpetrate.

The news was leaked 3 days ago by Dorigo who is very close to the actual authors of the paper so we knew it would be right. Yahoo is not doing too well but it still exists and just recently changed its CEO. If I remember well, it was some woman and they hope that her exit could save them which I am not quite sure about. Google didn't cache the article.

All the best

Lubos

The neutrino does have one special feature that might allow it to be tachyonic without the vacuum breaking down, it left hand chirality. Producing a back to back, neutrino antineutrino pair from empty space, is made impossible by angular momentum, that would result one unit of Angular momentum being created from nothing. So for handed weyl (not dirac) neutrino, a tachyon mass might not be forbidden. An alternative is that light doesn't actually travel at relavistic maximum speed due to being slowed slightly by vacuum effects, ones that don't slow neutrinos. However despite either of these let outs. I still think it likely that the result is purely an experimental error.

Interesting thoughts but: No, BDO, if you just imagine two neutrino energy-momentum vectors that are exactly inverse to each other, one of them has the opposite sign of energy than the other and it goes backwards in time, so it looks like an antiparticle spinning in the opposite direction (in frames where the energy is positive) and it is therefore allowed by angular momentum conservation: the situation will look like a continuous world line of a single particle in a fixed frame, anyway.

What is really relevant is a decay to vacuum with more than 2 external lines. For example, take a Z boson and two neutrinos. The Z-boson may compensate your violation of spin by one and the momenta may also add to zero if you allow negative-energy spacelike energy-momenta for the neutrinos. So this instability would still be there.

I'm not going to contemplate a negative FitzGerald contraction this week, or next week either.

Or maybe never. I probably never will, as long as I am not forcibly retired for senility.

I don't think Ricci or Levi-Civita would be overly impressed with this opera (opera buffa)?

Dear Lubos,

I never relished your comments as much as I have today; most appropriate set of comments by you. Hats off.

A Mitra

What if light in vacuum is going a bit slower than the "speed of light" and neutrinos know the actual speed limit better?

Thanks, Abhas Mitra!

Péter, a very good point. I wanted to make the same one.

Obviously, if we're forced to admit that neutrinos are faster than light, yours is the preferred interpretation: relativity still has the maximum speed and it's the neutrino speed, while light is slower.

A slower light is OK, doesn't violate the limitations of causality. However, it's still problematic for many reasons. Lorentz invariance still implies that the only way how a photon may be slower than the maximum speed is for its mass to be positive.

A positive photon mass would lead to an energy-dependent speed of light which is not measured. Moreover, a massive photon would mean that electromagnetism is a short-range force - with range depending on the mass. Unless the mass would be so tiny that it wouldn't be enough to explain those 18 meters collected by the neutrinos, I don't think you can construct any model of slower-than-max-speed photons that agrees with all the observations we know (such as long-range property of the geomagnetic field etc.).

Usually we say that, according to relativity, nothing can move faster

than light. But, in fact, according to relativity, the only

requirement is that there is a speed limit, c. This speed may be the

same one that is observed for the propagation of light in vacuum, if

Maxwell's equations are correct (massless photon). Even so, there may

be corrections due to interactions of photons with matter, or e +

e-pairs from the vacuum. Perhaps neutrinos, even though massive, are

even more effectively faster than the fastest ever observed photon.

This is jsut a conceptual matter of principle. Probably the speed of

light is exactly the same as the speed limit. In this case, I would

say that the experiment has some problem of unknown systematic error.

Usually we say that, according to relativity, nothing can move faster

than light. But, in fact, according to relativity, the only

requirement is that there is a speed limit, c. This speed may be the

same one that is observed for the propagation of light in vacuum, if

Maxwell's equations are correct (massless photon). Even so, there may

be corrections due to interactions of photons with matter, or e +

e-pairs from the vacuum. Perhaps neutrinos, even though massive, are

even more effectively faster than the fastest ever observed photon.

This is jsut a conceptual matter of principle. Probably the speed of

light is exactly the same as the speed limit. In this case, I would

say that the experiment has some problem of unknown systematic error.

I would think that if the result was true, it would be evident before - such as indication of solar prominences before they were observed optically.

Anyway I believe this is nothing but an incorrect analysis of error of response time and the true result is the same speed to within the limitations of the measurement.

(Everybody's brother in law has been out there proving Einstein "wrong" about this, that, and the other thing since about 1939, when Einstein became somewhat of a celebrity)

I laughed so soft when I read this.

Thanks for articulating some of your time travels on this subject. :)

You make all these claims about Berlusconi, but your post appeared 3 days before the news was released. So who's really travelling in time?

No, BDO, if you just imagine two neutrino energy-momentum vectors that are exactly inverse to each other, one of them has the opposite sign of energy than the other and it goes backwards in time, so it looks like an antiparticle spinning in the opposite direction (in frames where the energy is positive) and it is therefore allowed by angular momentum conservation: the situation will look like a continuous world line of a single particle in a fixed frame, anyway.

What is really relevant is a decay to vacuum with more than 2 external lines. For example, take a Z boson and two neutrinos. The Z-boson may compensate your violation of spin by one and the momenta may also add to zero if you allow negative-energy spacelike energy-momenta for the neutrinos. So this instability would still be there....

and so it goes

a piano player momentum

momenta monumenta

Many already mentioned a controversy between the superluminal fraction observed in the Opera experiment (2x10^-5) and what has been observed in the 1987 supernova. If that fraction is applied to the supernova, the time delay must be about a year, but actually it was just 2 hours. Interesting, that the controversy could be resolved if one suggest that the neutrino, when it "virtually" goes through the electron, somehow manages make it instantly. It is interesting, because the required cross section of such virtual interaction (let call it "swap") have to be about the typical neutrino-electron cross section 10^-48 m^2. With that assumption, we would have that the neutrinos in the supernova become superluminal just when they go through the matter, as well as across the rocks in the Opera experiment. In vacuum their velocity is equal to the speed of light. What remains with that model, that one should prove that the causality didn’t suffer from the swap.

Regards, Daniel Usikov. 9/23/11

Luboš Motl said...Lorentz invariance still implies that the only way how a photon may be slower than the maximum speed is for its mass to be positive.

A positive photon mass would lead to an energy-dependent speed of light which is not measured. Moreover, a massive photon would mean that electromagnetism is a short-range force - with range depending on the mass.

Not necessarily. Physical vacuum is a complex medium with all kinds of possibly unknown ingredients (like e.g. "dark matter" and who-knows-what).

Mass of quasi particles called "free photons" in vacuum may depend on their energy in a way that makes this particular medium dispersionless, at least in the frequency range it was tested so far.

Dear Lubos,

Your arguments for impossibility of time-like fermions are based on considering Dirac particle lagrangian. What if the neutrinos are Majorana fermions, not Dirac? Majorana equation is equivalent to Dirac equation with imaginary mass term.

Dear Peter, could you please write your prescription for the speed of a photon as a function of its energy E? What you write is just some vague "there are ghosts and dragons behind every corner" and "everything goes" and it doesn't answer anything.

I am telling you that any function v(E) you write for the speed of light will either be in blatant contradiction with known and measured properties of light, or it will be unhelpful to demystify faster-than-light neutrinos.

The only reason why photons would be dispersionless is that their frequency-independent speed is actually a universal constant, the maximum speed. If a particle's speed is not Nature's maximum speed, there's no reason why this speed should be energy-independent, so generally it won't.

Dear Murod, it's not true in any useful sense that the Majorana equation is a Dirac equation with an "i". Instead, the Majorana equation is identical to the Dirac equation but the Dirac spinor has to satisfy a reality condition, the Majorana condition, and this condition requires "i" to be written down in any basis.

But there's no sense in which the mass allows you to insert a previously nonexistent "i". If you write the allowed Dirac "m psibar psi" and you substitute the Majorana psi-proportional prescription for "psibar", you get the Majorana mass term with the allowed real mass. In both cases, the masses are real and the trajectories have to be timelike, nearly null for small masses, and not spacelike, and in both cases, this condition follows from the reality of the action.

Best wishes

Lubos

Dear Lubos,

I understand your point. But let me indicate a "useful sense that the Majorana equation is a Dirac equation with an "i"".

Dirac eqn can be written in the following form (where η and ξ are two-component spinor and co-spinor)

i∂η=mξ, i∂ξ=mη

Let's replace m with a matrix A acting on spinor/co-spinor, and require that η and ξ are eigenvectors of A:

i∂η = Aξ = mξ

i∂ξ= Aη = ¯m η

If A is a diagonal matrix A=(m,0,0,-m), we obtain Dirac eqn. The eigenvalue of A in this case is a real number m.

Majorana eqn can be obtained if A is off-diagonal matrix A = (0, im, -im, 0). The eigenvectors in this case satisfy reality condition, and the eigenvalue of matrix A is im.

Let me also note that introduction of the time-like particles helps to resolve "quarks confinement" problem if we consider proton as a time-like particle moving along the curved line. Time-like particle can be at three (or more) different places at the same time, so that we will always detect three (or more) "quarks" instead of one particle. Moreover, the number of "quarks" we detect will depend on the velocity of proton, as it actually observed by experiment. The higher is the velocity, the more "quarks" we see inside the proton.

Dear Murod, you write, among other strange things:

"Majorana eqn can be obtained if A is off-diagonal matrix A = (0, im, -im, 0). The eigenvectors in this case satisfy reality condition, and the eigenvalue of matrix A is im."

I apologize but as the freshmen who got A in their algebra course know, the eigenvalues of the matrix A = ((0,im),(–im,0)) which is clearly Hermitian, a multiple of sigma_y, are obviously and demonstrably +m and –m (both signs have the same degeneracy) which means that they are real rather than im. So your statement, in the form you wrote, is just wrong.

You may have meant that something else is im, like the matrix elements in a particular basis, but it's not surprising that matrix elements may be real or imaginary in different bases. But things like eigenvalues of an operator are obviously independent of the basis, and so is the rest mass of a particle. Whatever magic you will do, you can't present a valid proof that confuses massive and tachyonic particles because they are physically different.

Your comments about the "confinement problem" make no sense to me, sorry. First of all, there is no confinement "problem". Second of all, time-like character has nothing to do with a particle's being "at three points simultaneously". Moreover, quarks in a proton are not one particle at three point but three particles, each being at one point, which is something completely different.

When I came here from a link at 'Times' magazine, I thought that I'll enter an old mans blog... You know a bit like Einstein and such. Although I enjoy your way of writing, of course I cannot check if your statements are correct however... what I really would like to tell you is that you REALLY need a web designer.

My eyes got burned out. Since you have an interesting blog, make a bit more reader-friendly plz.

regards from Brno.

Luboš Motl said...The only reason why photons would be dispersionless is that their frequency-independent speed is actually a universal constant, the maximum speed. If a particle's speed is not Nature's maximum speed, there's no reason why this speed should be energy-independent, so generally it won't.

You are a bit rigid in your interpretation of theory, aren't you? Slower-than-the-ultimate-speed-limit photons in vacuum need not be

absolutelydispersionless, their dispersion only has to beundetectablewithin error bounds in the frequency range investigated so far.BTW, if the Opera measurement is correct (big if), it may tell us something about the interaction of photons and neutrinos. When neutrinos travel faster than the actual speed of light in a medium, they are supposed to induce some weak Cherenkov radiation (in spite of being neutral) and lose energy. It may explain the apparent contradiction between the terrestrial measurement and the SN1987a case.

732 km may not be enough to slow neutrinos down, even through solid rocks, but 160,000 light years is a different matter. Over astronomical distances even the slightest Cherenkov effect would slow them down to the local speed of light in the medium, which explains the near synchronicity.

One would even like to detonate several A-bombs all over the solar system at different distances from Earth to test it experimentally for good :)

Dear Peter, I am not that rigid. What you suggest contradicts all rules how we think about physics, and even things like the Michelson-Morley experiment.

You need the speed of light to be the same 0.9999912323 regardless of energy, for energies that go from fractions of eV to multiples of MeV and beyond because the speed has been measured for all of them. It's just implausible: Gell-Mann's totalitarian principle says that if an effect (such as dependence of the speed on the energy) is not forbidden by symmetries, it will occur.

It's really incompatible with the relativity even using the larger limiting speed. It's incompatible with precision Michelson-Morley, too: if the limiting speed were higher, the speed of photons would depend on the speed of sources and speeds of observers etc., and I claim that to a detectable extent today.

It's just vastly more likely that a grad student hacked the Opera collaboration and modified some numbers, or that one of thousands of possible errors (including wrong calibration of GPS that affects all of its users equally) has occurred.

I am not sure what definition of Dirac's gamma matrices or the differential operator \partial_{\mu} you use, but usually you need an 'i' in front of the differential operator in the kinetic term of Dirac lagrangian for the hamiltonian to be hermitian. I am just writing this since you seem very quick in writing sarcastic remarks to everyone who makes an error, so make sure you don't make your own.

Dear Michele, I have never written a full form of the Dirac equation which is a simple way to show that your accusation that I may have been imperfect in some "i" is a self-evident lie.

You may be confused by the formalism, remembering that you have an "i" somewhere, but you don't understand the physics behind these mathematical symbols and how the physics and mathematics are related.

The physics of the "superluminal fermions" is that the Dirac Lagrangian and any other related spinor Lagrangian - e.g. Majorana Lagrangian - guarantees through the reality of the action that the world line of the particle will be timelike, not spacelike, so you can't get superluminal particles.

Any opposite conclusion that someone gets about the spin-1/2 particles is an artifact of mistakes and bad mastery of the mathematical symbol and their meanings, and I reserve the right to continue to write sarcastic if not humiliating remarks about the people who aren't willing or able to learn and check this simple, formalism-independent insight.

Lubos, you really did not get my point. I understand the physics behind the formalism very well, and I agree with everything that you wrote in this blog. Your points are correct and expressed elegantly. But you don't need to make sarcastic remarks about an entire country because you don't agree with the findings of a group of scientists, and you surely don't need to be rude to other readers who can't handle basic linear algebra. I just picked on a silly 'i' to remind you that yes, that formula is imperfect (no matter how you try to turn it around), and that, to some extent, we all are.

While I agree that it is quite probable that there is some error in the experiment, I disagree about the consequences if it is correct. There is no big danger for causality, and no reason to be afraid for the life of your grandfather, because there is a quite straightforward causal solution - a preferred frame.

Just because there is a speed limit, it does not mean that every one obeys it...

Dear Berenyi Peter, I disapproved your newest attempted comment. You simply shouldn't try to post 4-kB long texts full of complete crackpottery, especially not as slow comments under heavily read articles such as this one.

People don't come here to read neverending delusions of morons like you because they can see it at millions of other places of the Internet.

What was the actual speed of the Neutrinos traveling at or claimed to be traveling at?

Why would a particle appear to be traveling at roughly the speed of light plus roughly pi?

I hope this helps.

I have seen this before,it was a spreadsheet error, I noticed that this happened when a person was demonstrating a simple experiment of measuring the speed of light in a microwave (do you know the one?)

c := speed = (distance) x (1/time) = (frequency) x (wavelength)

For some reason it was roughly off by pi 3.141 after some head scratching we worked it out that somewhere in the conversion between inches and kilometers (or it could have been centimeters to miles) the spreadsheet had made an error,the details are a bit sketchy because it was two years ago but it stands out as being to similar just to be a coincidence.

It would be funny if they were using the same spreadsheet!

When you say that Reuters "leaked" the news, what do you mean? :-)

i understand not because the speed of light is constant and because is the speed-limit to the universe. i think that it is due not the proper nature of the universe,but due to symmetry breaking to stronger CP,it is PT symmetry breakdown.then is possible that the speed of light is greater than the speed of light in our continuos spacetime,appear the violation of invariance of lorentz,both the lorentz's transformations are covariant with noncommutative matrix

due the increasing of mass to be linear with the incresing of speed implis that the spin for particles and antiparticles are asymmetrics.the rotational invariance is broken,but not globally.then the STR implies the existence of antiparticles as asymmetrics counterpartners of particles,being its originated by the breakdown of cp or deeply the breakdown of pt.then there are negative mass,or tachyons having imaginary mass.the mass is imaginary when quantified by others reference frames-as antiparticles supposely travelling back in time or with negative mass.but there supersymmetries that implies others continuities of spacetimes.

don't exist nothing that prevent in the lorentz's transformations the superluminal signals associated

to the right-left handed rotational system invariance partially broken ( CP + PT).then the relative motions

to speeds near the speed of light,that implies in the violation of cp vinculated in the deformation of spacetime( time dilatation and contraction of space)

i think that the isotropy of speed of light due the PT symmetry breaking is crucial to explain the relativistics effects about space and time,as well as understand that

the speed of light is generated constant due the violation of pt that explain simultaneouly the continuity and discreteness in the nature.the stronger violation obtained to CP and in the asymmetrics relations of matter and antimatter implies that curvatures of spacetime and its respectives metrics are derivatives of non-euclidean 4-dimensional manifolds are generated

by anti self dual and self dual -that are given by torsion tensor fields that torse the particles to one direction and antiparticles to opposed direction.

then the rotational invariance( spins-invariance of lorentz) is partially broken and restaured in CPT

the antimatter doesn't exist in the universe,the antiparticles are quantic vacuum locally energy due

the asymmetry of the matter in the relativistic speeds

the antiparticles are products of the 4-dimensional manifolds that gives symmetry of spacetime,with broken rotational invariance to left-right handed systems

then appear the superluminal neutrinos due the

left-right handed asymmetry.that gives metrics of the curvatures of spacetime generated by the violation of cp in the universe

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