It’s pretty common these days for people to refer to successfully quantizing general relativity as “the Holy Grail of Physics”, but it seems to me that there is a different problem that better deserves this name:
“Why does the vacuum state break electroweak gauge symmetry?”
If we could answer this question, we’d probably understand where masses of particles come from, as well as just about all of the undetermined parameters of the standard model (except for a couple ratios of the strengths of the gauge interactions). The exciting thing about this problem is that we have good reason to expect experiments to give us some new clues about it in 2008 when data from the LHC begins to come in.
The standard unification paradigm these days explains this in terms of the potential for a Higgs field, with various grand and super-unification schemes allowing an appropriate Higgs field but somehow never being able to predict anything at all about it. Even worse, such schemes not only don’t explain anything about this field, but also require the addition of extra Higgs fields beyond the single one required by the standard model.
An idea I’ve always found appealing is that this spontaneous gauge symmetry breaking is somehow related to the other mysterious aspect of electroweak gauge symmetry: its chiral nature. SU(2) gauge fields couple only to left-handed spinors, not right-handed ones. In the standard view of the symmetries of nature, this is very weird. The SU(2) gauge symmetry is supposed to be a purely internal symmetry, having nothing to do with space-time symmetries, but left and right-handed spinors are distinguished purely by their behavior under a space-time symmetry, Lorentz symmetry. So SU(2) gauge symmetry is not only spontaneously broken, but also somehow knows about the subtle spin geometry of space-time. Surely there’s a connection here…
This idea has motivated various people, including Roman Jackiw, who has several papers about chiral gauge theories that are very much worth reading. The problem you quickly get into is that the gauge symmetry of chiral gauge theories is generally anomalous. People mostly believe that theories with an anomalous gauge symmetry make no sense, but it is perhaps more accurate to say that no one has yet found a unitary, Lorentz-invariant, renormalizable way of quantizing them. In the standard model, the contributions to the anomaly from different particles cancel, so you can at least make sense of the standard perturbation expansion. Outside of perturbation theory, chiral gauge theories remain quite mysterious, even when the overall anomaly cancels.
So, this is my candidate for the Holy Grail of Physics, together with a guess as to which direction to go looking for it. There is even a possible connection to the other Holy Grail, I’ll probably get around to writing about that some other time.
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