Spinorial Reduction of the Superdimensional Dual-covariant Field Theory

TL;DR

This paper advances a superdimensional dual-covariant field theory by specifying its geometric and algebraic properties, emphasizing invariance principles, and interpreting key objects as spin-affinors and dual spin-fields within a unified geometric framework.

Contribution

It introduces a geometric and algebraic specification of the superdimensional dual-covariant field theory, linking it to spinor geometry and invariance principles for a potential unified field theory.

Findings

Establishment of rotational invariance of the split metric and grand metric tensor.

Interpretation of the split metric and dual state vector as spin-affinors and dual spin-fields.

Derivation of Euler-Lagrange equations consistent with the invariance requirements.

Abstract

In this paper we produce further specification of the geometric and algebraic properties of the earlier introduced superdimensional dual-covariant field theory (SFT) in a N-dimensional manifold [1] as an approach to a unified field theory (UFT). Considerations in the present paper are directed by a requirement of transformational invariance of connections of derivatives of dual state vector (DSV) and unified gauge field (UGF matrices) to these objects themselves established by mean of N split metric matrices of a rank {\mu} (SM, an extended analog of Dirac matrices) in frame of the related Euler-Lagrange equations for DSV, UGF and SM derived in [1]. This requirement is posed on SFT as one of the aspects of the general demand of irreducibility claimed to UFT; it leads to rotational invariance of SM and grand metric tensor (GM) as being structured on SM. Study in this work has led to explication of geometrical nature of SM and DSV as spin-affinors (variable in space of the unified manifold) and dual spin-field, respectively, in accordance with the E. Cartan's theory of spinors [2, 3, 4].