On Euclidean spinors and Wick rotations

TL;DR

The paper introduces a continuous Wick rotation for fermions that unifies the treatment of spinors in Minkowski and Euclidean spaces, enabling the construction of supersymmetric Euclidean theories from Minkowski counterparts.

Contribution

It presents a novel Wick rotation method for spinors that treats fermions and bosons uniformly, linking traditional approaches and maintaining hermiticity without external metrics.

Findings

Provides a unified Wick rotation framework for Dirac, Majorana, and Weyl spinors.

Establishes a connection to complex Lorentz boosts in five dimensions.

Enables supersymmetric Euclidean theories derived from Minkowski theories.

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.