I gave a “Spacetime is Right-handed” talk yesterday, part of a series entitled Octonions, standard model and unification. The slides are here, video should appear here.
Much of the talk was devoted to explaining the usual relation between spinors and vectors and how analytic continuation in complexified spacetime works then, from both the spinor and twistor point of view. This is contrasted to a new proposal for the relation between vectors and spinors in which the space-time degrees of freedom see only one of the two SL(2,C) factors of the usual complexified Lorentz group.
Nothing in the talk about using this for unification, where the idea is to exploit the other factor, which now appears as an internal symmetry. Starting from the point of view of Euclidean spacetime, the spacetime vectors and spinors that are related by Wick rotation to Minkowski spacetime degrees of freedom behave differently than usual, with a distinguished imaginary time direction. The general idea is that in standard Euclidean spacetime, where the geometry is governed by the rotation group SU(2) x SU(2), so splits into self-dual and anti-self-dual parts, one of these parts Wick rotates to spacetime symmetry, the other to an internal symmetry.
It was an excellent talk. Looking forward for the new manuscript. I hope it is self-contained and builds everything from scratch so that also who only have a mathematical background (no physical background) can understand the deep motivations of this work.
If I may, I’d like to expand a little on the comments I made after the talk, concerning the first section (slides 1-12). The classification of representations into real, complex and quaternionic on slide 8 only applies to complex irreducible representations, not to arbitrary representations. When you consider the Dirac spinors, on slide 10, the representation is not irreducible, and it has not only the “real” interpretation, that is as the complexification of a real Majorana spinor, but also a “quaternionic” interpretation, that is as a restriction of a quaternionic 2-spinor to a complex 4-spinor. To see this, it is enough to negate \sigma on one of the Weyl spinors, in order to change from \sigma^2=1 to \sigma^2=-1.
In fact, the difficulties with the standard formalism that you encounter on slide 12 occur mainly as a result of ignoring the quaternionic case, and working exclusively with the real case. Several people have pointed out that the quaternionic case introduces a *choice* into the theory, that is not there in the real case: the copy of C inside H, that is used to define the standard Dirac spinor, is not canonical. This choice has been used, for example, to extend from one generation (choice of i) to three (i, j, k).
In terms of Clifford algebras, what is going on here is that Cl(3,1) and Cl(2,2) are 4×4 real matrix algebras, while Cl(1,3), Cl(4,0) and Cl(0,4) are 2×2 quaternion matrix algebras. Hence it is possible to unify an external Minkowski spacetime with an internal Euclidean spacetime in a single Clifford algebra, if and *only if* the Minkowski signature is (1,3). This is not what is done in the SM, of course, in which complexification is used to obliterate the signature completely. But it could be done, and it might resolve the problems with the SM that you point out on slide 12.
Danielle Corradetti,
While the symmetry group structure largely determines the Standard Model QFTs, it’s a very long and complex story to get from facts about the basic groups and their representations to the actual quantum field theory.
This spring I’m teaching an advanced graduate course with a goal of explaining to mathematicians as much as possible what the SM QFT actually is, using a great deal of geometry and representation theory. I’ll be writing notes, the end result may answer your question, you might not find the answer so easy to follow…
Robert A. Wilson,
Many thanks for the comment, which is very helpful. I have to think more about the role of Clifford algebras in what I’m doing. They work in general dimension/signature, while I’m focused on the very special properties of the 4d case.
Naive question: How non-trivial would it be to apply these ideas to a QFT defined on the lattice in order to avoid the doubler problem and/or define a lattice chiral theory?
lun,
I hope ultimately these ideas will give new possibilities for how to formulate spinor fields on the lattice, but haven’t yet thought very much about this. I’m concentrating now on trying to understand how to exploit the Euclidean left-SU(2) symmetry as an internal symmetry.
A major problem with spinors on the lattice as always been that differential forms have a natural lattice formulation, but spinors are “square roots” of differential forms (the relation between vectors=1-forms and spinors discussed in the talk is part of this story).
You naturally end up with lots of copies of spinors (e.g. Kogut-Susskind fermions). I hope that with this new interpretation where some space-time symmetries become internal, you might have a new lattice formulation not for a single chiral fermion (where you expect anomaly problems), but for a multiplet of them. But, still a ways from knowing how to even exactly formulate this kind of proposal.
Have you thought about / commented at all on the potential relationship between your Euclidean twistor theory and Connes’ spectral approach and NCG? As a lay person, there seem to be some striking parallels:
– Both are Euclidean to start
– Twistor coordinates define a noncommutative space
– Derivation of the SM from the eigenvalues of the spectral action gives Pati–Salam, with the same SU(2) – left / right splitting you use to encode spacetime chirality
– Particle representations are given by the inner-automorphisms / internal symmetries of the full metric, rather than spatial symmetries that have to be compactified
– The spectral view (observables are incoming energy differences) seems philosophically complementary to the projective space of twistor theory
– Both do away with the classical notion of a point (NCG is “pointless” in that one cannot localize arbitrarily; the fixed points in twistor theory are quaternionic lines)
– Both connect with the homotopy of spheres and k-theory
Searching the archives, you’ve linked to Connes before and mentioned his work on the spectra of the RZF, but I don’t think you’ve ever explicitly discussed the relationship between his programme and your own. A friendly request!
Sam,
There are some points of contact such as you mention between what I’ve been working on and NCG a la Connes. But from the little time I’ve spent so far trying to better understand NCG, I haven’t found a deep enough connection to help with the problems I’ve been thinking about.