Where were we? Ah yes, spontaneous symmetry breaking. When some field takes on a nonzero value even in empty space, and that field is affected by some symmetry transformation, the resulting symmetry is said to be "spontaneously broken," and becomes hard for us to see directly. The classic example is the electroweak symmetry of the Standard Model, which is purportedly broken by a Higgs field that we have yet to directly detect. The fields that get expectation values and spontaneously break symmetries are generally taken to be "scalar" fields -- that is, they are single functions of spacetime, not something more complicated like a vector field. If a vector field did get a nonzero expectation value, it would have to point somewhere, thereby picking out a preferred direction in spacetime. That means that Lorentz invariance -- the physical symmetry corresponding to rotations and changes of velocity -- would be broken. Lorentz invariance is a cornerstone of relativity (and thus of all of modern physics), so breaking it is often thought to be bad.
But really, how bad is it? When Einstein put together special relativity on the basis of Lorentz invariance, he was arguing that there was no absolute space nor absolute time in the sense of Sir Isaac Newton. If two physicists traveling freely through empty space passed by each other at a high relative velocity, we couldn't tell in any universal sense which one was stationary and which was moving -- it's all relative, if you like. If we violated Lorentz invariance by having a vector field get a nonzero value in the vacuum, we could tell who was stationary and who was moving -- the vector would define a preferred rest frame. But that's not quite the same as going all the way back to Newtonian spacetime. The underlying theory is still Lorentz invariant -- if we can't easily detect this vector field (and we obviously haven't thus far), Lorentz invariance could be spontaneously violated while remaining in complete accord with all experimental tests. I was in on the ground floor for this idea -- it was the first project I worked on in graduate school (with George Field and Roman Jackiw), and was sufficiently non-mainstream that I worried for my career prospects. Alas, those were more freewheeling times, and you could get a good postdoc without necessarily jumping on a major bandwagon. Subsequently, I was surprised to see Lorentz violation actually become it's own (relatively tiny) bandwagon! A group of researchers, led by Alan Kostelecky at Indiana, have really pushed the idea of writing down ways to spontaneously violate Lorentz invariance, and have spawned an active experimental program to test these ideas using precision data from astophysics, particle physics, and atomic physics. (Alan has a FAQ on the whole idea of violating Lorentz symmetries.) So I occasionally return to the idea, as in work with my former graduate student Eugene Lim on the gravitational effects of Lorentz-violating vectors. And now I've returned to it again, this time with current student Jing Shu, as we try to understand a fundamental question in physics: why is there more matter than antimatter? This issue goes under the name of "baryogenesis," as it is baryons (protons, neutrons, and other heavy particles made of three quarks) that have an actual verifiable excess over antibaryons in our observable universe. We could also contemplate an asymmetry in leptons -- electrons, muons, tau particles, and the various neutrinos -- but it is hard to measure, since we have no direct handle on the total number of neutrinos vs. anti-neutrinos. But there are certainly a lot more baryons than antibaryons, as Mark (who is one of the world's experts) will tell you. I am not one of the world's experts, but Jing and I have contemplated an interesting idea: that baryogenesis becomes easier in the presence of Lorentz violation. Ordinarily, successful baryogenesis requires three ingredients, as first elucidated by Andrei Sakharov: violation of baryon number (i.e., processes which produce different numbers of baryons than antibaryons), violation of charge and charge-parity symmetries (i.e., processes which behave differently for particles and antiparticles), and a departure from thermal equilibrium (i.e., things don't have a chance to settle down in to a quiescent state, in which baryons and antibaryons would presumably be equally abundant). Sakharov's argument, sensibly enough, assumes that everything is nice and Lorentz invariant. If you violate that assumption, an interesting thing happens -- you can get different numbers of baryons and anti-baryons even in thermal equilibrium! This is an old idea, actually -- suggested by Cohen, Kaplan and Nelson under the name "spontaneous baryogenesis," and explored more recently in the context of evolving dark-energy (quintessence) fields by Mark and his students Antonio De Felice and Salah Nasri, as well as in the context of simple Lorentz-violating vector fields by Bertolami et al. The loophole is easy enough to state (although more difficult to appreciate). In quantum field theory there is something called the CPT theorem, which (among other things) guarantees that particles and antiparticles have equal masses. But Lorentz invariance is an assumption of the CPT theorem, and a vector field with a nonzero value in the vacuum can violate it. If the vector interacts with baryons in a certain way (not so hard to arrange, really), it can make antibaryons be just a bit heavier than baryons. That means that the baryons can be more abundant, even in equilibrium, and this slight asymmetry can persist to this day -- and provide the particles out of which we are all made. (You know, the tiny yellow slice of the cosmic pie chart.) What Jing and I did in our recent paper was basically to investigate this idea from various angles, suggesting what kinds of ways it could be successfully implemented. We found a few ideas, some of which were nicer than others. Amusingly enough, it's the new experimental constraints on violating Lorentz invariance that get in our way -- for a model of the type we consider to really work, the violation has to be strong in the early universe, and fade away by the time we are doing are experiments today. That's plausible enough, but puts an extra limit on the imagination of we theorists. I don't know if the models we looked at will ultimately be judged to be very promising or not -- like I said, I'm not the world's expert. But it's fun to imagine that we owe our very existence to a tiny violation of a cherished symmetry of our natural world.
