Hypergravity II

Abstract

This lecture completes the discussion of the basic formalism of the proposed generalization of General Relativity based on local automorphism invariance. It contains the proposed Lagrangian for this theory and the concomitant field equations. The fundamental Lagrangian is taken to be in the canonical gauge theory form (quadratic in the total curvature), and the coupling constant therefore is dimensionless. The dynamical variables of the theory are taken to be the spinor, the spin connection, the vierbeins, the Clifford connection, and the drehbeins. Internal consistency of the field equations, as well as congruency with the Bianchi identities, is demonstrated. The relationship of this theory to usual Einsteinian gravity is discussed. Since the proposed Lagrangian is quadratic in the total curvature, it does not contain the usual Einstein-Hilbert term (linear in the Ricci scalar curvature). However, the Clifford connection contains a scalar and a pseudoscalar field. When these fields are extracted from the Clifford connection, the Lagrangian is found to contain a Higgs type potential in these fields, as well as a coupling of these fields to the Ricci scalar. The minimum of the potential occurs for non-zero values of these fields and this induces a term in the Lagrangian linear in the Ricci curvature. In quantum language we would say that the automorphism gauge fields C _α obtain non-zero vacuum expectation values:

$$\left\langle {C_\alpha } \right\rangle = \sqrt {8\alpha _f } M_p \hat \gamma _\alpha ,$$

(26.1)

where $\hat \gamma _\alpha$ are a set of global (spacetime independent) matrices satisfying the Clifford algebra defining relation, M _p is the Planck mass, and $\alpha _f = \frac{{f^2 }} {{4\pi }}$ where f is the coupling constant for the automorphism gauge fields.