Classical and quantum Cosmology of the S\'aez-Ballester theory

TL;DR

This paper explores the classical and quantum aspects of the Se1ez-Ballester cosmological theory, demonstrating its limitations in solving dark matter issues and providing exact solutions in both regimes through Hamilton-Jacobi and Wheeler-DeWitt approaches.

Contribution

It clarifies the classical solutions and quantization of the Se1ez-Ballester theory, correcting misconceptions and simplifying the quantum analysis via the Einstein frame.

Findings

Classical solutions obtained via Hamilton-Jacobi method.

The theory cannot realistically address dark matter.

Exact quantum solutions found in the Einstein frame.

Abstract

We study the generalization of the S\'aez-Ballester theory applied to a flat FRW cosmological model. Classical exact solutions up to quadratures are easily obtained using the Hamilton-Jacobi approach. Contrary to claims in the specialized literature, it is shown that the S\'aez-Ballester theory cannot provide a realistic solution to the dark matter problem of Cosmology. Furthermore the quantization procedure of the theory can be simplified by reinterpreting the theory in the Einstein frame, where the scalar field can be interpreted as part of the matter content of the theory, in this approach, exact solutions are also found for the Wheeler-DeWitt equation in the quantum regime.