[Paper Review] Euclidean Spinors and Twistor Unification

Peter Woit|arXiv (Cornell University)|Apr 11, 2021

Particle physics theoretical and experimental studies9 references4 citations

TL;DR

The paper proposes a twistor-based unification of gravity and the Standard Model by interpreting one chiral SU(2) factor of Euclidean Spin(4) as a space-time symmetry after analytic continuation, with the Higgs field emerging from the imaginary time direction. This framework embeds leptonic generations via SU(3) and chiral SU(2) gauge structures in projective twistor space, enabling a chiral, geometric unification of gravity and matter degrees of freedom.

ABSTRACT

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.

Motivation & Objective

• To resolve the challenge of defining Euclidean spinor fields in a way compatible with Minkowski space-time geometry.
• To reinterpret the chiral components of Spin(4) = SU(2)×SU(2) as physical symmetries, with one SU(2) acting as a space-time symmetry after analytic continuation.
• To show how the choice of imaginary time direction spontaneously breaks one SU(2) factor, realizing it as a Higgs field.
• To embed the Standard Model’s leptonic generations within projective twistor space (PT), which naturally includes SU(3) and chiral SU(2) symmetries.
• To propose a chiral formulation of gravity unified with matter in a twistor framework, potentially realizable via known correspondences with holomorphic Chern-Simons theories.

Proposed method

• Use analytic continuation from Euclidean to Minkowski signature to interpret the imaginary time direction as a spontaneous symmetry-breaking mechanism.
• Apply twistor geometry to complexified four-dimensional space-time, leveraging projective twistor space (PT) as the natural arena for spinor fields.
• Identify one chiral SU(2) factor of Spin(4) as a space-time symmetry in Minkowski space, while the other is interpreted as an internal symmetry.
• Utilize the natural SU(3) symmetry of PT to model the three generations of leptons via spinor fields on PT.
• Construct a chiral formulation of gravity based on a single chiral SU(2) connection and spatial frame, reducing the gravitational degrees of freedom.
• Explore connections to super Yang-Mills and holomorphic Chern-Simons theories on PT as a path toward a full gauge-theoretic realization of the unification.

Experimental results

Research questions

• RQ1 How can Euclidean spinor fields be consistently interpreted in Minkowski space-time through the choice of an imaginary time direction?
• RQ2 What is the physical role of the imaginary time direction in breaking one SU(2) factor of Spin(4) and realizing it as a Higgs field?
• RQ3 How do spinor fields on projective twistor space PT naturally realize the structure of a lepton generation?
• RQ4 Can a chiral formulation of gravity be consistently derived from the remaining chiral SU(2) connection and spatial frame degrees of freedom?
• RQ5 What is the potential correspondence between the proposed unification and known dualities involving super Yang-Mills and holomorphic Chern-Simons theories on PT?

Key findings

• The imaginary time direction in analytic continuation spontaneously breaks one SU(2) factor of Spin(4), realizing it as a Higgs field that defines the Minkowski signature.
• One chiral SU(2) factor of Euclidean Spin(4) survives as a space-time symmetry after analytic continuation, while the other becomes an internal symmetry.
• Projective twistor space PT naturally incorporates both SU(3) and chiral SU(2) symmetries, allowing spinor fields on PT to model a generation of leptons.
• The gravitational degrees of freedom reduce to a chiral SU(2) connection and a spatial frame, suggesting a chiral quantization approach.
• The unification framework is incomplete but potentially realizable via known dualities with holomorphic Chern-Simons theories on PT.
• The proposal suggests a geometric unification of gravity and the Standard Model’s matter content within a single twistor-theoretic framework.