A space-time calculus based on symmetric 2-spinors

TL;DR

This paper introduces a novel space-time calculus for symmetric spinors that simplifies tensor operations by avoiding index manipulations, and provides a computer algebra implementation for practical use.

Contribution

It develops a formalism for symmetric spinors that eliminates the need for index manipulations and can translate any covariant tensor expression in 4D Lorentzian spacetime.

Findings

Formalism simplifies spinor calculations without index manipulations

Implementation SymSpin integrated into xAct for Mathematica

Applicable to all covariant tensor expressions in 4D spacetime

Abstract

In this paper we present a space-time calculus for symmetric spinors, including a product with a number of index contractions followed by symmetrization. As all operations stay within the class of symmetric spinors, no involved index manipulations are needed. In fact spinor indices are not needed in the formalism. It is also general because any covariant tensor expression in a 4-dimensional Lorentzian spacetime can be translated to this formalism. The computer algebra implementation SymSpin as part of xAct for Mathematica is also presented.