Weyl Spinor Fields and Right-Handed Spacetime

Key Takeaways

• Single Weyl spinor resists standard Wick rotation.
• Proposal uses only right‑handed spinors for vectors.
• Euclidean spacetime retains only SU(2)_R action on vectors.
• Left‑handed sector becomes internal gauge symmetry.
• Approach may streamline QFT constructions and twistor links.

Summary

The article explains why a single Weyl spinor field cannot be Wick‑rotated using the conventional Euclidean continuation, highlighting a fundamental mismatch between Minkowski and Euclidean spinor representations. It proposes a new framework that employs only right‑handed Weyl spinors to encode spacetime vectors, effectively treating spacetime as intrinsically right‑handed. In this scheme the Euclidean rotation group acts solely through the SU(2)₍R₎ factor, while the left‑handed sector is relegated to an internal symmetry. The author suggests this perspective could reshape how chiral fields are handled in quantum field theory and twistor constructions.

Pulse Analysis

Weyl spinors form the backbone of massless fermion descriptions in the Standard Model, yet their chiral nature creates a technical obstacle when transitioning from Minkowski to Euclidean space. Traditional Wick rotation assumes a one‑to‑one mapping between four‑vectors and 2×2 matrices, but this breaks down for a solitary right‑handed spinor because the left‑handed counterpart, essential in Euclidean constructions, disappears. The resulting inconsistency forces theorists to double the field content, complicating the path‑integral formulation and obscuring the underlying geometry.

The new proposal sidesteps this hurdle by insisting that spacetime vectors be built exclusively from right‑handed spinors, even after Wick rotation. In four dimensions the rotation group Spin(4) splits into SU(2)₍L₎×SU(2)₍R₎; the author retains only the SU(2)₍R₎ factor for vector transformations, relegating SU(2)₍L₎ to an internal gauge role. This re‑interpretation preserves the original degrees of freedom while delivering a self‑adjoint Euclidean Schwinger function without the usual field‑doubling, offering a cleaner mathematical foundation for chiral gauge theories.

Beyond formal elegance, the right‑handed‑only viewpoint resonates with twistor theory, where points in spacetime correspond directly to right‑handed spinor spaces. By aligning the Euclidean and Minkowski descriptions, the approach could simplify the construction of supersymmetric models, facilitate lattice simulations of chiral fermions, and inspire novel internal symmetry structures. Researchers are now poised to explore these implications, potentially redefining how chirality is encoded in quantum field theory and advancing the quest for a unified geometric language in high‑energy physics.