Spinor Representations of Surfaces in 4-Dimensional Pseudo-Riemannian Manifolds

TL;DR

This paper develops a framework for representing surfaces in 4D pseudo-Riemannian manifolds using spinor fields, classifies these spinors and Dirac operators, and applies the theory to Lorentzian and Minkowski spacetimes.

Contribution

It introduces a novel spinor representation approach for surfaces in 4D pseudo-Riemannian manifolds, including classifications and applications to specific spacetime geometries.

Findings

Defined spinor representations using Clifford algebra decompositions

Classified spinor fields and Dirac operators on immersed surfaces

Applied Dirac-Hestenes spinors to Lorentzian and Minkowski geometries

Abstract

Spinor representations of surfaces immersed into 4-dimensional pseudo-riemannian manifolds are defined in terms of minimal left ideals and tensor decompositions of Clifford algebras. The classification of spinor fields and Dirac operators on the immersed surfaces is given. The Dirac-Hestenes spinor field on surfaces immersed into Lorentzian manifolds and on surfaces conformally immersed into Minkowski spacetime is defined.