There’s a well-known list of high-profile problems in fundamental theoretical physics that have gotten most of the attention of the field during the past few decades (examples would be the problems of quantizing gravity, solving QCD, explaining dark energy, finding a model of dark matter, breaking supersymmetry and connecting it to experiment, etc.). Progress on these problems has been rather minimal, and in reaction one recent trend has organizations such as FQXI promoting research into questions that are much more “philosophical” (for instance, they are now asking for grant proposals to study “The Nature of Time”). In this posting I’d like to discuss a different class of problems, ones which I believe haven’t gotten anywhere near the attention they deserve, for an interesting reason.
The three problems share the characteristic of being apparently of a purely technical nature. The argument against paying much attention to them is that, in each case, even if one were to find a satisfactory solution, it might not be very interesting. It’s possible that all one would discover is that the conventional wisdom about these problems, that they’re just “technical” and thus not of much significance, is correct. The argument for paying more attention is that the technical problem may be an indication that we’re doing something wrong, that there is something of significance about the Standard Model that we haven’t yet understood. Achieving this understanding may lead us to the insight needed to successfully get beyond the Standard Model. At the moment all eyes are on the LHC, with the hope that experiment will lead to new insight. Whether this will work out is still to be seen, but in any case it looks like it’s going to take a few years. Perhaps theorists with nothing better to do but wait will want to consider thinking about these problems.
Non-Perturbative BRST
The BRST method used to deal with the gauge symmetry of perturbative Yang-Mills theory does not appear to generalize to the full non-perturbative theory, for a rather fundamental reason. This was first pointed out by Neuberger back in 1986 (Phys. Lett. B, 183 (1987), p337-40.), who argued that, non-perturbatively, the phenomenon of Gribov copies implies that expectation values of gauge-invariant observables will vanish. I’ve written elsewhere about a different approach to BRST that I’m working on (see here), which is still at a stage where I only fully understand what is going on in some toy quantum-mechanical models. My own point of view is that there’s still a lot of very non-trivial things to be understood about gauge symmetry in QFT and that the BRST sort of homological techniques for dealing with it are of deep significance. Others will disagree, arguing that gauge symmetry is just an un-physical redundancy in our description of nature, and how one treats it is a technical problem that is not of a physically significant nature.
One reaction to this question is to just give up on BRST outside of perturbation theory as something unnecessary. In lattice gauge theory computations, one doesn’t fix a gauge or need to invoke BRST. However, one can only get away with this in vector-like theories, not chiral gauge theories like the Standard Model. Non-perturbative chiral gauge theories have their own problems…
Non-perturbative Chiral Gauge Theory
Since the early days of lattice gauge theory, it became apparent that chiral symmetry was problematic on the lattice. One way of seeing this is that naively there should be no chiral anomaly on the lattice. The problem was made more precise by a well-known argument of Nielsen-Ninomiya. More recently, it has become clear that one can consistently introduce chiral symmetry on the lattice, at the cost of using fermion fields that take values in an infinite dimensional space. One such construction is known as “overlap fermions”, which have the crucial property of satisfying relations first written down by Ginsparg and Wilson. This kind of construction solves the problem of dealing with the global chiral symmetry in theories like QCD, but it still leaves unsolved the problem of how to deal with a gauged chiral symmetry, such as the gauge symmetry of the Standard Model.
Poppitz and Shang have recently written a nice review of the problem, entitled Chiral Lattice Gauge Theories Via Mirror-Fermion Decoupling: A Mission (im)Possible? They comment about the significance of the problem as follows:
Apart from interest in physics of the Standard Model — which, at low energies, is a weakly-coupled spontaneously broken chiral gauge theory that does not obviously call for a lattice study — interest in strong chiral gauge dynamics has both intensified and abated during the past few decades. From the overview in the next Section, it should be clear that while there exist potential applications of strong chiral gauge dynamics to particle physics, at the moment it appears difficult to identify “the” chiral theory most relevant to particle physics model-building (apart from the weakly-coupled Standard Model, of course). Thus, the problem of a lattice formulation of chiral gauge theories is currently largely of theoretical interest. This may or may not change after the LHC data is understood. Regardless, we find the problem sufficiently intriguing to devote some effort to its study.
In a footnote they compare two points of view on this: Creutz who argues that the question is important since otherwise we don’t know if the Standard Model makes sense, and Kaplan who points out that if there is some complicated and un-enlightening solution to the problem, it won’t be worth the effort to implement.
You can read more about the problem in the references given in the Poppitz-Shang article.
Euclideanized Fermions
Another peculiarity of chiral theories arises when one tries to understand how they behave under Wick rotation. Non-perturbative QFT calculations are well-defined not in Minkowski space, but in Euclidean space, with physical observables recovered by analytic continuation. But the behavior of spinors in Minkowski and Euclidean space is quite different, leading to a very confusing situation. Despite several attempts over the years to sort this out for myself, I remain confused, and can’t help suspecting that there is more to this than a purely technical problem. One natural mathematical setting for trying to think about this is the twistor formalism, where complexified, compactified Minkowski space is the Grassmanian of complex 2-planes in complex 4-space. The problem though is that thinking this way requires taking as basic variables holomorphic quantities, and how this fits into the standard QFT formalism is unclear. Perhaps the current vogue for twistor methods to study gauge-theory amplitudes will shed some light on this.
On the general problem of Wick rotation, about the deepest thinking that I’ve seen has been that of Graeme Segal, who deals with the issue in the 2d context in his famous manuscript “The Definition of Conformal Field Theory”. I saw recently that he’s given some talks in Europe on “Wick Rotation in Quantum Field Theory”, which makes me quite curious about what he had to say on the topic.
For some indication of why this confusion over Minkowski versus Euclidean spinors remains and doesn’t get cleared up, you can take a look at what happened recently when Jacques Distler raised it in related form on his blog here (he was asking about it in the context of the pure spinor formulation of the superstring). I’m not convinced by his claim that the thing to do is to go to Euclidean space-time variables, while keeping Minkowski spinors. Neither is Lubos, and he and Jacques manage to have an argument about this that sheds more heat than light. It ends up with Lubos accusing Jacques of behaving like Peter Woit, which lead to him being banned from commenting on the blog. While this all is, as Jacques describes it “teh funny”, it would be interesting to see a serious discussion of the issue. Since it in some sense is all about how one treats time, perhaps one could get FQXI funding to study this subject.
Update: Lubos Motl has immediately come up with a long posting explaining why these are all non-problems, of concern only to those like myself who are “hopeless students”, “confused by many rudimentary technicalities that prevented him from thinking about serious, genuinely physical topics.” If I would just understand AdS/CFT and Matrix theory I would realize that gauge symmetry is an irrelevance. Few in the theoretical physics community are as far gone as Lubos, but unfortunately he’s not the only one that thinks that concern with these “technicalities” is evidence that someone just doesn’t understand the basics of the subject.
