I’ve completely re-organized and largely rewritten my paper from earlier this year on Euclidean Spinors and Twistor Unification. Soon I’ll upload this as a revision to the arXiv, for now it’s available here. This new version starts from a very basic point of view about 4d geometry, leaving the technicalities about Euclidean QFT for spinors and the expository material about twistors to appendices.
Most ideas I’ve worked on over the years that seemed initially promising ultimately became more and more problematic the more I looked at them. This set of ideas keeps looking more and more solid. There are several (to me at least…) attractive aspects:
- Spinors are tautological objects (a point in space-time is a space of Weyl spinors), rather than complicated objects that must be separately introduced in the usual geometrical formalism.
- Analytic continuation between Minkowski and Euclidean space-time can be naturally performed, since twistor geometry provides their joint complexification.
- Exactly the internal symmetries of the Standard Model occur.
- The intricate transformation properties of a generation of Standard Model fermions correspond to a simple construction.
- One gets a new chiral formulation of gravity, unified with the Standard Model.
- Conformal symmetry is built into the picture in a fundamental way.
There’s more in this version about how quantum gravity fits into this, when formulated in terms of chiral variables (i.e. Ashtekar variables). This gives a new context for old questions about quantizing in these variables (this is in Eucldean signature, the other chirality is not space-time geometry but internal Yang-Mills geometry, and the imaginary time component of the vierbein is distinguished and given the dynamics of a Higgs field). I haven’t spent much time on this yet, but suspect this new context may help overcome problems that people trying to pursue quantum gravity in this chiral connection framework have run into in the past.
One common reaction I’ve gotten to these ideas is the one I myself had in the past: analytic continuation relates expectation values of field operators in Euclidean and Minkowski signature, so my left-handed SU(2) after analytic continuation gives part of Lorentz symmetry, not an internal symmetry. What took me a long time to realize is just how different Euclidean and Minkowski signature QFT is. Yes, Schwinger functions and Wightman functions can be related by analytic continuation (in a rather subtle way, the Wightman functions aren’t functions, but boundary values of holomorphic functions). But at the level of states and operators things are very different. It’s just not true that there is some holomorphic formulation of QFT states and operators, with Euclidean and Minkowski space restrictions related by analytic continuation. There’s a lot of explanation about this in the paper.
One objection I’ve run into is that by distinguishing a direction in Euclidean space I’m breaking Lorentz symmetry. What’s true is quite the opposite: having such a distinguished direction is needed to get Lorentz symmetry after analytic continuation. If you want to start in Euclidean space and get Lorentz symmetry, you have to do something like distinguish a direction and get an Osterwalder-Schrader reflection in that direction, which you need to get from SO(4) to SL(2,C). From the other direction, if you start in Minkowski space-time and analytically continue, you have a choice of lots of possible Euclidean slices to analytically continue to. You need to pick one, and that will distinguish an imaginary time direction. This is most easily seen in the twistor formalism, where the Minkowski space-time geometry is determined by a quadratic form that picks out a 5-dimensional hypersurface in PT. This will project down to an imaginary time = 0 subspace of Euclidean space-time, which picks out the imaginary time direction.