The Superspinorial Field Theory in Riemannian Coordinates

TL;DR

This paper develops a Riemannian coordinate framework for the superspinorial dual-covariant field theory, revealing conformal invariance and enabling algebraic simplifications based on metric determinants.

Contribution

It introduces an orthogonal Riemannian coordinate frame in SSFT, demonstrating conformal invariance and reducing complex equations to determinant-based forms.

Findings

Constancy of the rotation matrix in RC simplifies the metric analysis.

Conformal invariance of the metric ratio is established in the orthogonal RC frame.

Reduction of SSFT equations to a determinant form of the metric is achieved.

Abstract

The Superspinorial Dual-covariant Field Theory (SSFT) developed in papers [1, 2] is treated in terms of Riemannian coordinates (RC) [7, 8] in space of the N dimensions unified manifold (UM). Metric tensor of UM (grand metric, GM) is built on the split metric matrices (SM) [1] which are a proportion of the Cartan's affinors (an extended analog of Dirac's matrices) of his Theory of Spinors [3] as explicated in [2]. Transition to RC based on consideration of geodesics is described. A principal property of an orthogonal RC frame (ORC) utilized in the present paper is constancy of the rotation matrix A of the Riemannian space of UM, while transformation matrix B of the dual superspinorial state vector field (DSV) varies together with Cartan's affinors according to the dynamical law of SSFT derived in [2]. The spinorial genesis of notion of the orthogonality as aspect of irreducible SSFT is pointed out in the present paper. The main outcome of resorting to an orthogonal RC frame (ORC) is explication of the conformal dynamic invariance of metric i.e. constancy of ratio between components of GM in ORC frame. This leads to possibility of reduction of algebraic equations of SSFT for the SM [2] to equation for GM determinant where GM signature plays a role of N discrete parameters non-specified in advance.