SciPost Submission Page

Euclidean Spinors and Twistor Unification

by Peter Woit

Submission summary

As Contributors: Peter Woit
Arxiv Link: (pdf)
Date submitted: 2021-04-26 22:43
Submitted by: Woit, Peter
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We argue that the necessity of picking an imaginary time direction for the analytic continuation to Minkowski signature gives a new point of view on the problem of Euclidean spinor fields, allowing an interpretation in Minkowski space-time of one of the chiral factors of Spin(4)=SU(2)xSU(2) as an internal symmetry. The imaginary time direction spontaneously breaks this SU(2), playing the role of the Higgs Field. Twistor geometry provides a compelling framework for formulating spinor fields in complexified four-dimensional space-time and implementing the above suggestion. Projective twistor space PT naturally includes an internal SU(3) symmetry as well as the above SU(2), and spinors on this space behave like a generation of leptons. Since only one chirality of the Euclidean Spin(4) is a space-time symmetry after analytic continuation and the Higgs field defines the imaginary time direction, the space-time geometry degrees of freedom are only a chiral SU(2) connection and a spatial frame. These may allow a consistent quantization of gravity in a chiral formulation, unified in the twistor framework with the degrees of freedom of the Standard Model. This unification proposal is incomplete, still requiring implementation as a theory with gauge symmetry on PT, perhaps related to known correspondences between super Yang-Mills theories and supersymmetric holomorphic Chern-Simons theories on PT.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

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Submission 2104.05099v1 on 26 April 2021

Reports on this Submission

Anonymous Report 1 on 2021-5-17 (Invited Report)


1- The background material covered in sections 2 and 3 is well-written and clearly presented.

2- The underlying proposal (that twistor theory could provide a framework to unify the degrees of freedom of the Standard Model and gravity) is close in spirit to Penrose's initial motivations for the subject, and provides a potentially interesting modern update in this regard. In particular, the role of Euclidean reality conditions is a key difference from the original Penrose formulation.


1- I found it difficult to pin down the precise nature of the central proposal.

2- The bulk of the manuscript is review material, with only Section 4 dealing with the actual unification proposal. It is also unclear how large parts of the review material (e.g., the Penrose transform or Ward correspondence) are actually being used in the proposal, leaving me to wonder why they were included.

3- There are a few places where references to the literature could be more complete (see Report for details).


This paper proposes using twistor theory, adapted to Euclidean reality conditions, as a framework to study the Standard Model and gravity. The paper includes a large amount of well-written review material on the distinction between Euclidean and Lortenzian QFT (with specific examples) and several aspects of twistor theory. While the underlying idea is intriguing (and reflects Penrose's initial motivation for developing twistor theory), I found the portions of the paper which were not review material to be somewhat vague.

In particular, various groups appearing in the Standard Model are identified in twistor space with Euclidean reality conditions, but there are very few details on how this could be used to concretely formulate the Standard Model in twistor space. How is an electron represented in twistor space? Where will dynamics (non-linear equations of motion, interactions, etc.) come from? Why should a theory of quantum gravity be UV finite in twistor space? For instance, the author alludes to twistor constructions of supersymmetric Yang-Mills theories (studied in the context of scattering amplitudes in recent years), but gives no ideas as to how massive particles (such as those appearing in the Standard Model) could be described in twistor space. Indeed, this is a long-standing difficulty for the twistor approach.

I was also unclear on what, if any, role gravity is actually playing in this story. At several points, the author comments that twistor theory could provide a chiral formulation of GR (indeed, it can-see below), but without saying how or if this twistor formulation would differ from its space-time counterpart at the quantum level. I do not see any reason to believe that this would be finite as a quantum theory.

Additionally, the author spends a large amounts of text unpacking twistor tools (e.g., the Penrose transform) which seem to have no concrete role to play in Section 4, where the central proposal is laid out. More generally, I felt that Section 4 requires substantial expansion and detail to truly constitute a unification proposal.

There were also a few places where I felt that the literature was not properly represented:
-page 2: The googly problem is not a purely gravitational issue; it also applies to Yang-Mills theories! What twistor actions and twistor string theory provide is a perturbative solution to the googly problem: good enough for computing perturbative observables (like scattering amplitudes) but not a truly non-linear, non-perturbative resolution.
-pages 27-28: The idea that Euclidean reality conditions are more natural for providing a chiral, twistor description of field theories is central to the twistor actions first proposed by Mason [hep-th/0507269] and subsequently developed by many others. Indeed, these twistor actions are only known to be equivalent to space-time field theories with Euclidean reality conditions.
-page 29: Identifying PT_0 with a fixed-time slice of space-time is related to Low's twistor causality/linking relations; see R. J. Low (1990) Class. Quantum Grav. 7, 177.
-page 30: The problem of giving a twistor description of "chiral" general relativity (i.e., Plebanski's formulation of GR) was recently solved at the level of twistor actions by Sharma [2104.07031].

In conclusion, I feel that the precise nature of the author's unification proposal needs to be fleshed out more for this paper to warrant publication in SciPost Physics. While the review portions of the paper are clear and well-written, the portions which deal with the actual proposal are not sufficiently well-developed to meet the standards of novelty and rigor for this journal. With substantial further development, it could be possible for this paper to truly open a new research direction, but only after significant additions to its key section (i.e., Section 4).

Requested changes

1- Section 4 expanded to provide significantly more detail on the nature of the unification proposal for Standard Model particle content.

2- Make clear the extent to which gravity is/is not playing a role in this proposal.

3- Minor additions to literature referencing (see Report).

  • validity: ok
  • significance: ok
  • originality: good
  • clarity: low
  • formatting: excellent
  • grammar: perfect

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