Thanks for pointing that out and for the link to the DeWitts’ paper. I was completely ignoring orientability issues.
In this case, wouldn’t one want the universal cover of O(p,q) rather than merely a double cover? If I am remembering correctly for d > 3 the Spin(p,q) group is also the universal cover of the oriented part of SO(p,q). For instance in d=3, I thought the universal cover gave rise to representations like anyons.Kartik Prabhu
* I also think that twistors are in some sense capturing the some part of this issue. I would highly recommend both volumes of Spinors and Spacetime — Penrose & Rindler. They present the formalism in a very clear style.
* Another related annoyance wrt complex vs real forms of groups is that we physicists do not distinguish these in Yang-Mills theory. I have had the hardest time finding a good reference on the issues of group structure, complex vs real forms, Cartan subalgebras… as it relates to Yang-Mills theories.Kartik Prabhu
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.Kartik Prabhu
* The link to Figueroa-O’Farrill’s notes is broken. You need a www. instead of ww. in the URL.
* I think the choice of mostly minus or mostly plus metric when not dealing with spinors, is largely an issue of convenience, nothing to do with positivity of energy or action terms.
* For spinors, the relevant issue is the relative sign choice in the metric and the definition of the Clifford algebra. Different relative signs would give inequivalent spinor representations as already pointed out. I don’t know of a good physical argument for choosing one over the other, unless physics some how only cares about the representation of the complexified group in which case either convention is fine.
* As for Wald’s switch in metric signature. That is done to use the 2-spinor notation where one wants to identify an abstract index “a” on the tangent space to a pair of spinor indices “AA′”. In particular, one wants the metric \eta_ab -> \epsilon_AB \epsilon_A′B′ instead of with a minus sign where \epsilon is a real symplectic 2x2 matrix. Though I am sure this again has to do with a choice of sign for the defining equation of the Clifford algebra.
* But overall as pointed out in the notes by Figueroa-O’Farrill “There is no difference between 3+1 and 1+3—there never is at the level of Spin group” so maybe this point is moot!?