Abstract

This paper follows one of our earlier publications detailing an extended theory of General Relativity (GR) with two types of curvatures.
In this framework, we first extend the Riemannian spinor theory in the formalism of A.Lichnerowicz.
We then suggest that the non vanishing covariant derivative of the metric tensor linked to the extra curvature of our extended GR theory, be related to the trace of a specific Hermitean matrix.
In the Riemannian theory, this matrix reduces to the classical gamma matrix that is precisely designed to distinguish the spin one half fields from its anti Fermionic counterpart.
Our extended General Relativity theory therefore appears as unifying the Fermion-anti Fermion field states, and would legitimate the hidden medium concept introduced by Louis de Broglie in his attempt to explain the matter-antimatter symmetry.

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