Spinor Info

In the last several years, much of the time when I was wearing my physicist’s hat I was working on a theory of modified gravity.

Modified gravity theories present an alternative to the hypothetical (but never observed) substance called “dark matter” that supposedly represents more than 80% of the matter content of the Universe. We need either dark matter or modified gravity to explain observations such as the anomalous (too rapid) rotation of spiral galaxies.

Crudely speaking, when we measure the gravitational influence of an object, we measure the product of two numbers: the gravitational constant G and the object’s mass, M. If the gravitational influence is stronger than expected, it can be either because G is bigger (which means modified gravity) or M is bigger (which means extra mass in the form of some unseen, i.e., “dark” matter).

In Einstein’s general theory of relativity, gravity is the curvature of spacetime. Objects that are influenced only by gravity are said to travel along “geodesics”; their trajectory is determined entirely by the geometry of spacetime. On the other hand, objects that are influenced by forces other than Einstein’s gravity have trajectories that deviate from geodesics.

Massless particles, such as photons of light, must travel on geodesics (specifically, “lightlike geodesics”.) Conversely, if an originally massless particle deviates from a lightlike geodesic, it will appear to have acquired mass (yes, photons of light, when they travel through a transparent substance that slows them down, such as water or glass, do appear to have an effective mass.)

Modified gravity theories can change the strength of gravity two ways. They can change the strength of Einstein’s “geometric” gravity (actually, it would be called “metric gravity”); or, they can introduce a non-geometric force in addition to metric gravity.

And herein lies the problem. One important observation is that galaxies bend light, and they bend light more than one would expect without introducing dark matter. If we wish to modify gravity to account for this, it must mean changing the strength of metric gravity.

If metric gravity is different in a galaxy, it would change the dynamics of solar systems in that galaxy. This can be compensated by introducing a non-geometric force that cancels out the increase. This works for slow-moving objects such as planets and moons (or spacecraft) in orbit around a sun. However, stars like our own Sun also bend light. This can be observed very precisely, and we know that our Sun bends light entirely in accordance with Einstein’s general relativity theory. This cannot be explained as the interplay of geometric curvature and a non-geometric force; photons cannot deviate from the lightlike geodesics that are determined in their entirety by geometry alone.

So we arrive at an apparent contradiction: metric gravity must be stronger than Einstein’s prediction in a galaxy to account for how galaxies bend light, but it cannot be stronger in solar systems in that galaxy (or at the very least, in the one solar system we know well, our own), otherwise it could not account for how suns bend light or radio beams.

I have come to the conclusion that it’s not galaxy rotation curves or cosmological structure formation but modeling the bending of light and being able to deal with this apparent paradox is the most important test that a modified gravity theory must pass in order to be considered viable.

I have been thinking about neutrinos today. No, not about faster-than-light neutrinos. I was skeptical about the sensational claim from the OPERA experiment last year, and my skepticism was well justified.

They may not be faster than light, but neutrinos are still very weird. Neutrinos of one flavor turn into another, a discovery that, to many a particle physicist, had to be almost as surprising as the possibility that neutrinos are superluminal.

The most straightfoward explanation for these neutrino oscillations is that neutrinos have mass. But herein lies a problem. We only ever observed left-handed neutrinos. This makes sense if neutrinos are massless particles that travel at the speed of light, since all observers agree on what left-handed means: the spin of the neutrino, projected along the direction of its motion, is always −1/2.

But now imagine neutrinos that are massive and travel slower than the speed of light. As a matter of fact, imagine a bunch of neutrinos fired by CERN in Geneva in the direction of Gran Sasso, Italy. It takes roughly 2 ms for them to arrive. Now if you can run very, very, very fast (say, you’re the Flash, the comic book superhero) you may be able to outrun the bunch. Looking back, you will see… a bunch of neutrinos with a velocity vector pointing backwards (they’re slower than you, which means they’ll appear to be moving backwards from your perspective) so projecting their spin along the direction of motion, you get +1/2. In other words, you’re observing right-handed neutrinos.

This is just weird. On the surface of it, it means that our fast-running Flash sees the laws of physics change! This is in deep contradiction with the laws of special relativity, Lorentz invariance and all that.

How we can interpret this situation depends on whether we believe that neutrinos are “Dirac” or “Majorana”. Neutrinos are fermions, and fermions are represented by spinor fields. A spinor field has four components: these correspond, in a sense, to a left-handed and a right-handed particle and their respective antiparticles. So if a particle only exists as a left-handed particle, only two of the four components remain; the other two (at least in the so-called Weyl representation) disappear, are “projected out”, to use a nasty colloquialism.

But we just said that if neutrinos are massive, it no longer makes sense of talking about strictly left-handed neutrinos; to the Flash, those neutrinos may appear right-handed. So both left- and right-handed neutrino states exist. Are they mathematically independent? Because if they are, neutrinos are represented by a full 4-component “Dirac” spinor. But there is a possibility that the components are not independent: in effect, this means that the neutrino is its own antiparticle. Such states can be represented by a two-component “Majorana” spinor.

The difference between these two types of neutrinos is not just theoretical. The neutrino carries something very real: the lepton number, in essence the “electronness” (without the electric charge) of an electron. If a neutrino is its own antiparticle, the two can annihilate one another, and two units of “electronness” vanish. Lepton number is not conserved.

If this is indeed the case, it can be observed. The so-called neutrinoless double beta decay is a hypothetical form of radioactive decay in which an isotope that is known to decay by emitting two electrons simultaneously (e.g., potassium-48 or uranium-238) does so without emitting the corresponding neutrinos (because these annihilate each other without going anywhere). Unfortunately, given that neutrinos don’t like to do much interacting to begin with, the probability of a neutrinoless decay occurring at any given time is very small. Still, it is observable in principle, and if observed, it would indicate unambiguously that neutrinos are Majorana spinors. (A prospect that may be appealing insofar as neutrinos are concerned, but I find it nonetheless deeply disturbing that such a fundamental property of a basic building block of matter may turn out to be ephemeral.)

Either way, I remain at a loss when I think about the handedness of neutrinos. If neutrinos are Dirac neutrinos, one may postulate right-handed neutrinos that do not interact the way left-handed neutrinos do (i.e., do not participate in the weak interaction, being so-called sterile neutrinos instead). Cool, but what about our friend, the Flash? Suppose he is observing the same thing we’re observing, a neutrino in the OPERA bunch interacting with something. But from his perspective, that neutrino is a right-handed neutrino that is not allowed to participate in such an interaction!

Or suppose that neutrinos are Majorana spinors, and right-handed neutrinos are simply much (VERY much) heavier, which is why they have not been observed yet (this is the so-called seesaw mechanism). The theory allows us to construct such as mass matrix, but once again having the Flash around leads to trouble: he will observe ordinary “light” neutrinos as right-handed ones!

Perhaps these are just apparent contradictions. In fact, I am pretty sure that that’s what they are, since all this follows from writing down a theory in the form of a Lagrangian density that is manifestly Lorentz (and Poincaré) invariant, hence the physics does not become broken for the Flash. It will just turn weird. But how weird is too weird?