Conformal structures and twistors in the paravector model of spacetime

TL;DR

This paper explores the use of Clifford algebras and the paravector model to construct conformal maps and twistors in spacetime, providing new algebraic insights and simplifying the mathematical framework.

Contribution

It introduces a novel algebraic approach to construct twistors using lower-dimensional Clifford algebras within the paravector model.

Findings

Established isomorphisms between key Clifford algebras for conformal mappings.

Described conformal maps as twisted adjoint actions in Minkowski spacetime.

Presented a new algebraic construction of twistors using Clifford algebras.

Abstract

Some properties of the Clifford algebras Cl(3,0), Cl(1,3), Cl(1,3)(C), Cl(4,1) and Cl(2,4) are presented, and three isomorphisms between the Dirac-Clifford algebra C x Cl(1,3) and Cl(4,1) are exhibited, in order to construct conformal maps and twistors, using the paravector model of spacetime. The isomorphism between the twistor space inner product isometry group SU(2,2) and the group Spin+(2,4) is also investigated, in the light of a suitable isomorphism between C x Cl(1,3) and Cl(4,1). After reviewing the conformal spacetime structure, conformal maps are described in Minkowski spacetime as the twisted adjoint representation of Spin+(2,4), acting on paravectors. Twistors are then presented via the paravector model of Clifford algebras and related to conformal maps in the Clifford algebra over the Lorentzian R(4,1) spacetime. We construct twistors in Minkowski spacetime as algebraic spinors associated with the Dirac-Clifford algebra C x Cl(1,3) using one lower spacetime dimension than standard Clifford algebra formulations, since for this purpose the Clifford algebra over R(4,1) is also used to describe conformal maps, instead of R(2,4). Our formalism sheds some new light on the use of the paravector model and generalizations.