I’ve finally finished writing up a new version of some ideas that I first wrote about here last summer. The latest draft is here, I may set up a web page with more info here.

Several people had very helpful comments on what I wrote last summer, especially in pointing out that I wasn’t providing sufficient justification for the most radical claim I was making, that the problems with analytic continuation of spinor fields indicated that one could interpret one of the Euclidean space rotation group SU(2)s as an internal symmetry. I then spent a lot of time mastering aspects of Euclidean QFT I had never properly understood. Section two of the current paper is the result. It’s in some sense quite elementary, people may find it of independent interest, even if you’re not interested in the ideas involving twistors. Section three, an exposition of relevant aspects of twistors, is pretty much unchanged. Section 4 is an outline of the ideas about how to get a unified theory out of twistors, much there is still sketchy. I understand a lot better than last year how what I’m proposing fits into some standard ideas about “chiral” formulations of gravity, also have learned a bit more about previous attempts to formulate chiral gravity and gauge theory on twistor space. Some highly speculative remarks that this might all be somewhat related to N=4 super Yang-Mills have been added.

Here’s a little bit more here about the hardest to believe claim being made (about analytically continuing spinors). The standard assumption (this is what I always thought) has been based on the analytic continuation behavior of correlation functions: Schwinger and Wightman functions are analytic continuations of each other, and one might think there’s nothing more to analytic continuation between Euclidean and Minkowski space theories. After learning more about the Euclidean QFT literature, I was struck by how different this is from the physical Minkowski space formalism: states and fields don’t just analytically continue, they’re quite different sorts of objects in the Euclidean case. Anyway, this is all explained in detail in the paper…

**Update**: No, this is not an April Fool’s joke. I’ve now created a twistor unification page where I’ll try and maintain updated information about this unification proposal

um, the date?

Can someone say what makes this April 1, aka what is the fake part?

I don’t know anything about twistors and QFT or relativity, but, from a cursory glance it gave a legit feeling. Perhaps it’s the LaTeX formatting?

One should understand that Schwinger’s comment about the doubling of spin 1/2 fields in order to transform to Euclidian space involved starting with HERMITIAN four-component spin 1/2 fields in space time.

All,

No, not an April Fool’s joke, perfectly serious.

Lowell Brown,

From Schwinger’s Euclidean Quantum Electrodynamics paper

“Thus the requirement of a Euclidean formulation excludes the simplest field in space-time, the four-component Hermitian spin-1/2 field (Majorana). In this context a a trivial observation may be worth repeating – a four-component Hermitian field is fully equivalent to a two-component non-Hermitian field.”

The issue of hermiticity of fields here is confusing, but my focus is on something else, the relation between the Spin(4)=SU(2)xSU(2) symmetry in Euclidean space and the Spin(3,1)=SL(2,C) symmetry in Minkowski space for a Weyl spinor theory. Looking just at the Wightman and Schwinger functions that are analytic continuations of each other, the analytic continuation relates the symmetries. But if you try and define states and operators, something different happens (the Euclidean theory state space is a different sort of thing than the Minkowski state space).

I was about to request an exegesis myself, as this either had either to be an inside joke of diabolical sophistication or, well, not a joke.

LMMI,

At some point it became clear I’d have the draft done around April 1 and I thought best to avoid that date. Later decided might as well post on April 1, maybe people would carefully read the thing looking (unsuccessfully) for the joke. As far as I can tell, that didn’t work out…

In this subject these days it is sometimes difficult to distinguish between an April Fool’s joke and an idea that is meant to be taken seriously. Perhaps that is not such a bad thing – there is many a true word spoken in jest.

Do you plan on publishing a 2nd edition of Quantum Theory, Groups and Representations: An Introduction, with added material on twistors?

Hi Peter. I was wondering if you could comment on Eric Weinstein’s article on Geometric Unity.

Justin,

I would at some point like to write a revised and expanded version of that textbook, now have a long list of notes to myself about ways it could be improved based on teaching the class again this year. Unclear when I’ll have the time and energy for this project, and whether I’d be able to explain about twistors in a useful way that wouldn’t take writing hundreds of pages. So, not any time soon, but maybe some day.

Jamie.

No, especially certainly not here, where I think there’s a much more interesting set of ideas about unification to discuss…