Kartik Prabhu
Peter, Being from a GR background, I agree that it is very convenient to work in the mostly plus signature and being in Chicago I’ll skip the East/West dichotomy! :) * I am not sure what significance one should attach to “Wick rotation” or analytic continuation procedures employed in textbook QFT. Those work only in analytic and static/stationary spacetimes, which again from a GR perspective is highly restrictive. * Yes, one can do 2-component spinor notation in either signature if one wants to carry around the “sigma matrices”. Penrose’s notation gets rid of the sigma matrices entirely ( See: Spinors and Spacetime Vol1 — Penrose & Rindler ) and one can write equations like g_ab = \epsilon_AB \epsilon_A′B′ which works only in the mostly minus signature. If using a mostly plus signature this would have a minus sign which is very inconvenient. So this notation by Penrose sacrifices the goodness of the mostly plus metric for the goodness of skipping sigma matrices entirely, which is also the convention Wald uses. I have also seen Bob Geroch use this notation in his lectures. In the http://www.niu.edu/spmartin/spinors/ link that you cite this corresponds to their Eq.2.48 which does have a sign change when changing the metric signature. * I still don’t understand the physical significance (if any) of pinors. As far as I have seen, physics theories only use spinors and the Spin group in which case this sign choice is irrelevant. * Again, I am pretty skeptical of the importance of analytic continuation to physics in general.
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Peter Woit
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Kartik, Thanks for the comments on the two-component formalism. About analytic continuation, I realize this doesn’t work in arbitrary backgrounds, but I do think it’s a deep issue in QFT. As far as I’ve ever been able to tell, there really is no such thing as a rigorous, non-perturbative formulation of QFT that does not use the analytic continuation. Even in the simplest considerations of the free field propagator, you have to add some extra piece of information (how do you integrate through the poles) that is very nicely expressed by the analytic continuation requirement. Anyway, analytic continuation issues are somewhat of a different topic, but the point here is just that they seem to not have been sorted out for spinors, with the general assumption being one of “only depends on complexification”, in the sense of assuming the analytic continuation can be done. Your comment led me to take a look at Penrose (at least to the extent of picking up “Road to Reality”), and I was intrigued to learn that he does use the West Coast metric, but that he also clearly thinks of the question in terms of complexified Minkowski space and analytic continuation. Quite possibly the right way to think about these questions is in terms of twistors.
John Baez
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Kartik wrote: “I still don’t understand the physical significance (if any) of pinors.” One place they can be important is when you’ve got a theory where spacetime is non-orientable. Then you need to know how your “spinors” transform under orientation-reversing symmetries – so what you really need is pinors, which are a representation of a double cover of O(p,q) rather than merely SO(p,q). Here I will not enter into the fact that O(p,q) has up to 8 different double covers! After choosing a sign convention for your Clifford algebras, the Clifford algebra will contain a group Pin(p,q) which is a specific double cover of O(p,q). We can hope that nature will be kind enough to use this one. I saw Cécile DeWitt-Morette give a talk about this stuff. Once she and her husband Bryce DeWitt both did computations for a free fermion field on a nonorientable spacetime. They got physically different answers! And it turned out the reason was that one was using a pinor representation of Pin(1,n), while the other was using a pinor representation of Pin(n,1). They wrote a paper about this.