Spinors in euclidean field theory, complex structures and discrete symmetries

TL;DR

This paper explores fermions in various dimensions and signatures, focusing on Euclidean space, defining generalized Majorana spinors, and analyzing complex structures and discrete symmetries to ensure consistent analytic continuation and real observables.

Contribution

It introduces a framework for defining generalized Majorana spinors across dimensions, clarifies complex structures for Euclidean and Minkowski signatures, and examines their impact on discrete symmetries and real actions.

Findings

Generalized Majorana spinors are defined for specific mod 8 dimensions.

A new complex structure involving Euclidean time reflection is necessary for Euclidean signature.

Real actions ensure hermitean observables are real, even in Euclidean space.

Abstract

We discuss fermions for arbitrary dimensions and signature of the metric, with special emphasis on euclidean space. Generalized Majorana spinors are defined for $d=2,3,4,8,9$ mod 8, independently of the signature. These objects permit a consistent analytic continuation of Majorana spinors in Min-kowski space to euclidean signature. Compatibility of charge conjugation with complex conjugation requires for euclidean signature a new complex structure which involves a reflection in euclidean time. The possible complex structures for Minkowski and euclidean signature can be understood in terms of a modulo two periodicity in the signature. The concepts of a real action and hermitean observables depend on the choice of the complex structure. For a real action the expectation values of all hermitean multi-fermion observables are real. This holds for arbitrary signature, including euclidean space. In particular, a chemical potential is compatible with a real action for the euclidean theory. We also discuss the discrete symmetries of parity, time reversal and charge conjugation for arbitrary dimension and signature.