On Euclidean spinors and Wick rotations

TL;DR

This work addresses how to consistently rotate fermionic fields from Minkowski to Euclidean space in a unified, non-metric-dependent way to construct Euclidean supersymmetric theories from Minkowski ones. It achieves this via a continuous spinor index rotation generated by S(theta)=exp(gamma^4 gamma^5 theta/2), interpreted as a complex Lorentz boost in a five-dimensional setting, yielding a coherent mapping for Dirac, Majorana and Weyl spinors. For Dirac spinors the approach produces a Hermitian, SO(4)-invariant Euclidean action L_E = psi_E^dagger gamma^5_E (gamma_E^mu partial_mu + m) psi_E, linking to Schwinger and Zumino; for Majorana and Weyl spinors, reality conditions must be relaxed, yielding OS-like non-Hermitian Euclidean actions. Overall, the method provides a uniform, field-theoretic Wick rotation that preserves SUSY structure and offers avenues for canonical formulations, curved-space extensions, and higher-dimensional generalizations, while clarifying how axial terms acquire factors of i in regulators and observables.

Abstract

We propose a continuous Wick rotation for Dirac, Majorana and Weyl spinors from Minkowski spacetime to Euclidean space which treats fermions on the same footing as bosons. The result is a recipe to construct a supersymmetric Euclidean theory from any supersymmetric Minkowski theory. This Wick rotation is identified as a complex Lorentz boost in a five-dimensional space and acts uniformly on bosons and fermions. For Majorana and Weyl spinors our approach is reminiscent of the traditional Osterwalder Schrader approach in which spinors are ``doubled'' but the action is not hermitean. However, for Dirac spinors our work provides a link to the work of Schwinger and Zumino in which hermiticity is maintained but spinors are not doubled. Our work differs from recent work by Mehta since we introduce no external metric and transform only the basic fields.