Euclidean Twistor Unification


This is work in progress towards creating a theory unifying the standard model and gravitation, based on the idea of defining the theory on the Euclidean signature version of twistor space. Unlike other unification proposals, this does not introduce new, very different degrees of freedom, but just involves considering the Standard Model and a chiral formulation of gravity on the Euclidean twistor space. The new idea here is that the Euclidean theory has features not previously considered: one of the chiral SU(2) factors of the four-dimensional spin group can be taken to behave like an internal symmetry, and the breaking of this symmetry introduced by picking an imaginary time direction (necessary to define analytic continuation to physical Minkowski space-time) corresponds to the observed phenomenon of electroweak symmetry breaking.

This picture has several 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.
  • Electroweak symmetry breaking has a novel origin in the breaking of Euclidean SO(4) symmetry necessary to define a physical theory.
  • 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.
  • Points in space time are described by the $p=\infty$ analog of the Fargues-Fontaine description of the “points” p of number theory.

For a very old embryonic version of this idea, see my 1988 Supersymmetric quantum mechanics, spinors and the Standard Model. A new paper about this was posted on the arXiv 4/11/21, much improved version 10/14/21. Blog posts about this are here. Slides from a September 2020 talk at OIST about this are here, from a September 2021 talk at Brown are here (video here), from a seminar talk on 2/14/22 here, and from a colloquium talk at UT Dallas are here.

A related paper with notes on the twistor $P^1$ is available here.


I’ve written a new and shorter paper, concentrating on one piece of this set of ideas which has to do with how spinor geometry works in four dimensions, exploiting the very different ways things work in Minkowski and Euclidean signature. The larger argument is that to describe spacetime degrees of freedom, one only needs right-handed spinor geometry. Besides the paper, there are also blog posts here and here.