Wick Rotating Spinors and Twistors

Key Takeaways

• Wick rotation fails for Weyl spinors across Minkowski and Euclidean signatures.
• Proposed Lorentz group reformulation uses right‑handed SL(2,C) only.
• Quantum evolution holomorphic in lower half‑plane defines rotation.
• Projective twistor space splits like complex time half‑planes.
• AI‑generated figures showcase novel visual tools for abstract concepts.

Pulse Analysis

Wick rotation, the analytic continuation between Minkowski and Euclidean spacetimes, has long been a cornerstone of quantum field theory, yet it encounters a fundamental obstacle when applied to Weyl spinors. In four dimensions, complex spacetime admits a clean description via two copies of SL(2,ℂ), allowing spinors and vectors to be treated as simple tensor products. The difficulty arises because real Minkowski space requires only one chiral spinor type, whereas Euclidean space demands both, breaking the continuity needed for a straightforward rotation. This mismatch hampers calculations in gauge theories and limits the elegance of unification attempts that rely on seamless signature changes.

A recent proposal reframes the problem by treating the Minkowski Lorentz group solely as the right‑handed SL(2,ℂ), discarding the traditional embedding of conjugate left‑handed elements. Coupled with the insight that quantum dynamics, expressed as U(z)=e^{-izH}, is holomorphic in the lower half of the complex time plane, the approach suggests that Wick rotation is essentially a choice of contour within a holomorphic domain. Twistor theory reinforces this view: projective twistor space naturally decomposes into PT+, PT–, and PT0, mirroring the upper half‑plane, lower half‑plane, and real axis of complex time. By defining physics on PT+ and extracting Minkowski physics from its boundary PT0, the need for a distinguished Euclidean counterpart disappears, offering a more flexible framework for theoretical extensions.

Beyond the mathematical elegance, the talk highlighted practical experimentation with AI‑generated visualizations of spinor‑twistor geometry. While still rudimentary, these figures demonstrate how machine‑learning tools can accelerate the communication of abstract concepts, a trend that could reshape research dissemination in high‑energy physics. If the revised Wick rotation scheme proves robust, it may streamline perturbative calculations, influence quantum computing algorithms that simulate field theories, and provide fresh pathways toward a unified description of fundamental forces.