Geon-type solutions of the nonlinear Heisenberg-Klein-Gordon equation

Abstract

As a model for a "unitary" field theory of extended particles we consider the nonlinear Klein-Gordon equation---associated with a "squared" Heisenberg-Pauli-Weyl nonlinear spinor equation---coupled to strong gravity. Using a stationary spherical ansatz for the complex scalar field as well as for the background metric generated via Einstein's field equation, we are able to study the effects of the scalar self-interaction as well as of the classical tensor forces. By numerical integration we obtain a continuous spectrum of localized, gravitational solitons resembling the geons previously constructed for the Einstein-Maxwell system by Wheeler. A self-generated curvature potential originating from the curved background partially confines the Schr\"odinger-type wave functions within the "scalar geon." For zero-angular-momentum states and normalized scalar charge the spectrum for the total gravitational energy of these solitons exhibits a branching with respect to the number of nodes appearing in the radial part of the scalar field. Preliminary studies for higher values of the corresponding "principal quantum number" reveal that a kind of fine splitting of the energy levels occurs, which may indicate a rich, particlelike structure of these "quantized geons."