John wrote:

Any spinning massive object with its angular momentum pointing in the direction of its velocity will look like its angular momentum is pointing in the opposite direction of its velocity if you run faster than it.

As said I have an uneasyness with this picture, in particular this picture assumes somewhat that there exists a “neutrino charge” which behaves according to some kind of (hidden) Maxwells equations but even if one would accept this assumption for the moment then as the Wikipedia article says if an observer is faster the particle appears to move backwards, that would actually mean that the helicity would stay the same, because angular momentum (and therefore magnetic moment) AND relative velocity switch sign. But then in electrodynamics electrical and magnetic fields change when changing the frame and I have no idea what this does to the whole picture. And then one gets already for the electron an anomalous magnetic moment, if one takes quantum effects into acount, who knows what this is for the neutrino?

I doubt there’s a paper on this; there would only be a paper if someone discovered it wasn’t true.

Why? I don’t know how difficult it is to measure a neutrino’s spin or hellicity, but such an experiment could eventually say something about wether a neutrino needs to have a mass or not.

]]>Are there experiments, where this has been tested?

Any spinning massive object with its angular momentum pointing in the direction of its velocity will look like its angular momentum is pointing in the opposite direction of its velocity if you run faster than it. It’s easy to test this for large objects, a bit harder for small objects… but someone must have done it for electrons. I doubt there’s a paper on this; there would only be a paper if someone discovered it *wasn’t* true. This would be an immensely shocking discovery!

Wikipedia wrote:

“it is possible for an observer to change to a reference frame that overtakes the spinning particle, in which case the particle will then appear to move backwards, and its helicity (which may be thought of as ‘apparent chirality’) will be reversed.”

Are there experiments, where this has been tested? Do you know? That is I understand the mental picture that if you overtake a rotating particle and look onto it, then the rotation direction will appear reversed and thus the magnetic moment. Apriori one could wonder that since electrical and magnetical fields change roles depending on frames that something strange might happen, but lets assume that at least theoretically this is still in line with the above picture – but then one may still wonder about the fact that spin might not always be explained with charges rotating around one axis.

]]>This is clearing up a lot of my misconceptions, thank you! That makes a lot more sense why neutrinos would oscillate and not just have a chance of being observed in the wrong flavor.

If I’m understanding you correctly, this means that the electron neutrino, muon neutrino, and tau neutrino are *not* particles in the sense of corresponding to representations of , but “small neutrino,” “medium neutrino,” and “large neutrino” are. However, I’m guessing it’s still useful to use the flavor basis for their direct sum, because that’s how the same vector space breaks up as a representation of a different symmetry group. (Would that be the from the weak force?)

Okay, so a representation corresponding to electrons of both handednesses would be an irreducible representation of , but as a representation of the smaller group , it breaks up as a direct sum of two smaller irreps, and you can use the operators to project onto them. That fits with what I’ve been trying to glean from the wikipedia and stackexchange pages.

How nice that the double cover of the identity component of P is the right group to study—then we can just look at representations of its Lie algebra instead!

]]>Owen wrote:

I was about to ask whether there’s an example where we decompose a single sum of identical irreducible representations in two different ways (so that they can’t have a common refinement), but is that perhaps what neutrino oscillation is all about? If “neutrinos” form a representation that can be divided up into identical subrepresentations either by flavor or by mass , is the point that these aren’t necessarily the same decomposition? So if the sun creates a neutrino in a “small” mass eigenstate, it could be a superposition of different flavors?

It’s certainly true that the ‘mass basis’ for neutrinos is different than the ‘flavor basis’. And it’s certainly true that neutrino oscillations are caused by this fact.

However, ‘mass’ is part of the specification of an irreducible representation of the Poincaré group (or more precisely, the double cover of its connected component). So, the three representations are *not isomorphic*. We can sum them up and get

and we can then break this sum down into a direct sum of three vector spaces

However, these new summands are *not subrepresentations*. And that’s exactly *why* there can be neutrino oscillations!

If they were subrepresentations, they’d be invariant under time translation. So, if you started with a vector in corresponding to an electron neutrino, it would *stay* in

But it *doesn’t*. That’s why electron neutrinos can turn into neutrinos of other flavors, if you just wait.

Owen wrote:

I notice that in your example of two different decompositions of a representation into subrepresentations (via charge or handedness) there’s a common refinement into four subrepresentations—right-handed electrons, left-handed electrons, right-handed positrons, and left-handed positrons—so the whole representation looks something like and we were decomposing it into the first time and the second time.

Hmm. Good point. This contradicts things I believe, but it’s also a logical consequence of things I’ve said! So, I’ll have to debug my understanding.

What I *wanted* was for the electron to be described by a representation , the positron to be described by another copy of for the electron/positron to be described by the direct sum and for chirality to give another way of decomposing this direct sum into two copies of which correspond to the ‘right-handed electron/positron’ and the ‘left-handed electron positron’.

All this is internally self-consistent, and I was trying to use it as an example of this observation of yours:

What if an irreducible representation V is repeated in the decomposition: ? Are there two types of particles that are identical in every way? (Or infinitely many, because there are infinitely many ways to include into ?)

However, it’s not consistent with my claim that the chirality operator comes from a central element of the double cover of the Poincaré group. This would make the representation for a right-handed particle where chirality takes the value +1, *not isomorphic* to the representation for the left-handed electron/positron.

Okay, I see the problem now. I was being sloppy in my use of the term ‘Poincaré group’. The original Poincaré group is the isometry group of Minkowski spacetime. It has four connected components! There is also a smaller group, the connected component containing the identity, which deserves to be called .

Either way, we need to use a double cover. So, we have to decide whether a ‘type of particle’ is an irreducible representation of or of . I was taking an inconsistent attitude on this question! As we’ll see, the right attitude is to use

Why does this matter? Here’s why:

The element I’m calling which gives the chirality operator, lies in but not the subgroup . It commutes with everything in but not with everything in

For example, reflection in any space direction gives an element of that does not commute with In fact, acting on any representation, this element will map chirality eigenvectors with chirality +1 to chirality eigenvectors with chirality -1, and vice versa.

In plain English: “reflection interchanges right-handed particles and left-handed particles.”

I need to get some work done, but with should let me straighten out my story. For example, any sort of ‘right-handed particle’ or ‘left-handed particle’ will correspond to a representation of but not the bigger group

Any theory of physics that’s symmetrical under reflections will give representations of . However, the theory that describes our world is not symmetrical under reflections, so the correct group to use is

]]>Thank you again! Phrased in terms of reps and irreps the idea of “types of particles” makes so much more sense.

I notice that in your example of two different decompositions of a representation into subrepresentations (via charge or handedness) there’s a common refinement into four subrepresentations—right-handed electrons, left-handed electrons, right-handed positrons, and left-handed positrons—so the whole representation looks something like and we were decomposing it into the first time and the second time.

I was about to ask whether there’s an example where we decompose a single sum of identical irreducible representations in two different ways (so that they can’t have a common refinement), but is that perhaps what neutrino oscillation is all about? If “neutrinos” form a representation that can be divided up into identical subrepresentations either by flavor or by mass , is the point that these aren’t necessarily the same decomposition? So if the sun creates a neutrino in a “small” mass eigenstate, it could be a superposition of different flavors? And Fermilab is doing the same thing but with the decompositions by mass and handedness?

]]>]]>I believe there are several things that need to be pointed out regarding the MiniBooNE results, which are all reasons for neutrino phenomenologists not being nearly as excited about this as you might expect. First and foremost of these is the OPERA exclusion from the money plot, which measures the exact same channel as MiniBooNE with the same L/E in an accelerator experiment. The exclusion limits in the plot are actually all from other experiments examining the same channel and so the oscillation interpretation of the best-fit is directly excluded by other experiments. On top of this, something that is not shown in the figure are the exclusion limits that would come from other oscillation channels just from the assumption of probabilities adding up to one. This is where atmospheric and reactor neutrinos come in and this essentially rules out the entire fit region.

The second thing to note is that the fit is a rather bad one. Although the chi2/dof is seemingly fine, the signal is concentrated in a few bins at low energy where the background is the largest and likely most poorly understood (the first MiniBooNE papers did not include this region as a signal region exactly because of this). The rise with energy is simply put just too steep to be due to oscillations.

Third, the people doing global fits with sterile neutrinos typically disregard this region of MiniBooNE data because of what I just mentioned. Regardless of what they do with steriles, they end up with a fit that does not explain the MiniBooNE excess at low energies. Some find a good fit with steriles and LSND results, but that result would typically be compatible with a MiniBooNE excess at higher energies, as was initially expected. MiniBooNE itself does not provide much statistical weight to those fits.

With the global picture in mind, it seems very unlikely to me that the appropriate resolution of this discrepancy is sterile neutrinos. It is very possible that something is afoot, be it new physics or a background that is poorly understood, but that something is probably not sterile neutrinos. The best way of testing this in the future is using experiments with multiple detectors to decrease the reliance on Monte Carlo simulations of backgrounds and fluxes. The Fermilab short-baseline program should do pretty well in this regard.

Also, regarding the right-handed neutrinos in the canonical seesaw, they are typically not Dirac. We introduce an additional right-handed SM singlet to build Dirac mass terms via the Yukawa couplings to the left-handed lepton doublet in the same way we give up-type quarks masses. The big difference is that the right-handed neutrinos are SM singlets and therefore allow for a heavy Majorana mass term between them. The resulting neutrinos (both light and heavy) are Majorana particles. Of course, you have no a priori handle on the scale of the right-right Majorana mass and if it is very very small you end up just with light pseudo-Dirac neutrinos.

10:19 AM, June 02, 2018