World Spinors Revisited

TL;DR

This paper reviews the mathematical structure and physical interpretation of world spinors, focusing on their infinite-dimensional representations and their role in various symmetry groups relevant to gravitational and field theories.

Contribution

It introduces new results on the infinite dimensionality of spinorial representations and constructs explicit $ar{SL}(4,R)$ representations within finite-dimensional $SL(2,C)$ bases.

Findings

Explicit construction of $ar{SL}(4,R)$ representations

Regrouping of tensorial and spinorial fields in arbitrary spin theories

Generalization to $SL(5,R)$ for infinite-component fields

Abstract

World spinors are objects that transform w.r.t. double covering group $\bar{Diff}(4,R)$ of the Group of General Coordinate Transformations. The basic mathematical results and the corresponding physical interpretation concerning these, infinite-dimensional, spinorial representations are reviewed. The role of groups $Diff(4,R)$, $GA(4,R)$, $GL(4,R)$, $SL(4,R)$, $SO(3,1)$ and the corresponding covering groups is pointed out. New results on the infinite dimensionality of spinorial representations, explicit construction of the $\bar{SL}(4,R)$ representations in the basis of finite-dimensional non-unitary $SL(2,C)$ representations, $SL(4,R)$ representation regrouping of tensorial and spinorial fields of an arbitrary spin lagrangian field theory, as well as its $SL(5,R)$ generalization in the case of infinite-component world spinor and tensor field theories are presented.