  An approach to a Unified Field Theory (UFT) is developed as an attempt to
establish unification of the Theory of Quantum Fields (QFT) and General Theory
of Relativity (GTR) on the background of a covariant differential calculus. A
dual State Vector field (DSV)consisting of covariant and contravariant
N-component functions of variables of a N-dimensional unified manifod (UM)is
introduced to represents matter. DSV is supposed to transform in a way distinct
from that of the differentials of the UM variables. Consequently, the hybrid
tensors and a hybrid affine tensor (Dynamic Connection, DC) are introduced. The
hybrid curvature form (HCF) is introduced as a covariant derivative of DC. A
system of covariant Euler-Lagrange (EL) equations for DSV, DC, and a twin
couple of the triadic hybrid tensors (Split Metric, SM)is derived. A scalar
Lagrangian form is composed based on a set of principles suited for UFT,
including the homogeneity in the UM space, differential irreducibility and
scale invariance. The type of the manifold geometry is not specified in
advance, in neither local (signature) nor regional (topology) aspects.
Equations for DSV play role of the Schroedinger-Dirac equation in space of UM.
By the correspondent EL equations, DC and SM are connected to DSV and become
responsible for the non-linear features of the system i.e. interactions. In
this paper we mark breaking of a background paradigm of the modern QFT, the
superposition principle. The issue of the UM-MF dimensionality will be
addressed, and relations to the principles and methodology of QFT and GTR will
be discussed.
