On correspondence between tensors and bispinors

TL;DR

This paper presents an algorithm to reconstruct bispinors from given tensors in four-dimensional Riemannian space, revealing gauge degrees of freedom and extending to higher dimensions.

Contribution

It introduces a method to invert the tensor-bispinor correspondence using Hermitean matrices and eigenvalue spectra, including gauge transformation insights.

Findings

The inverse mapping exists when the Hermitean matrix is nonnegatively definite.

A matrix Z satisfying M=ZZ+ can be constructed, leading to bispinors.

The approach extends to higher-dimensional Riemannian spaces.

Abstract

It is known that in the four-dimensional Riemannian space the complex bispinor generates a number of tensors: scalar, pseudo-scalar, vector, pseudo-vector, antisymmetric tensor. This paper solves the inverse problem: the above tensors are arbitrarily given, it is necessary to find a bispinor (bispinors) reproducing the tensors. The algorithm for this mapping constitutes construction of Hermitean matrix $M$ from the tensors and finding its eigenvalue spectrum. A solution to the inverse problem exists only when $M$ is nonnegatively definite. Under this condition a matrix $Z$ satisfying equation $M=ZZ^{+}$ can be found. One and the same system of tensor values can be used to construct the matrix $Z$ accurate to an arbitrary factor on the left-hand side, viz. unitary matrix $U$ in polar expansion $Z=HU$. The matrix $Z$ is shown to be expandable to a set of bispinors, for which the unitary matrix $U$ is responsible for the internal (gauge) degrees of freedom. Thus, a group of gauge transformations depends only on the Riemannian space dimension, signature, and the number field used. The constructed algorithm for mapping tensors to bispinors admits extension to Riemannian spaces of a higher dimension.