Graded Majorana spinors

TL;DR

This paper introduces a new type of Majorana condition for spinors called graded Majorana, which uses pseudo-conjugation and can be applied in spacetimes where standard Majorana spinors are not possible, with applications to supersymmetry.

Contribution

The paper defines graded Majorana spinors using pseudo-conjugation and demonstrates their application in supersymmetric theories in spacetimes lacking standard Majorana spinors.

Findings

Introduces graded Majorana spinors using pseudo-conjugation.

Shows graded Majorana spinors can be imposed on single Dirac spinors.

Constructs a supersymmetric field theory in 3D Euclidean space.

Abstract

In many mathematical and physical contexts spinors are treated as Grassmann odd valued fields. We show that it is possible to extend the classification of reality conditions on such spinors by a new type of Majorana condition. In order to define this graded Majorana condition we make use of pseudo-conjugation, a rather unfamiliar extension of complex conjugation to supernumbers. Like the symplectic Majorana condition, the graded Majorana condition may be imposed, for example, in spacetimes in which the standard Majorana condition is inconsistent. However, in contrast to the symplectic condition, which requires duplicating the number of spinor fields, the graded condition can be imposed on a single Dirac spinor. We illustrate how graded Majorana spinors can be applied to supersymmetry by constructing a globally supersymmetric field theory in three-dimensional Euclidean space, an example of a spacetime where standard Majorana spinors do not exist.