Ostrogradski approach for the Regge-Teitelboim type cosmology

TL;DR

This paper introduces a geometric derivation of quantum cosmology for a brane universe using Ostrogradski formalism, emphasizing the second-order derivative nature of the Regge-Teitelboim model and its quantization.

Contribution

It provides a novel geometric approach to quantize the Regge-Teitelboim cosmological model via Ostrogradski formalism, clarifying its second-order derivative structure.

Findings

The Wheeler-DeWitt equation matches previous results.

The approach manages first- and second-class constraints effectively.

The geometric derivation aligns with existing Hamiltonian methods.

Abstract

We present an alternative geometric inspired derivation of the quantum cosmology arising from a brane universe in the context of {\it geodetic gravity}. We set up the Regge-Teitelboim model to describe our universe, and we recover its original dynamics by thinking of such field theory as a second-order derivative theory. We refer to an Ostrogradski Hamiltonian formalism to prepare the system to its quantization. Our analysis highlights the second-order derivative nature of the RT model and the inherited geometrical aspect of the theory. A canonical transformation brings us to the internal physical geometry of the theory and induces its quantization straightforwardly. By using the Dirac canonical quantization method our approach comprises the management of both first- and second-class constraints where the counting of degrees of freedom follows accordingly. At the quantum level our Wheeler-De Witt Wheeler equation agrees with previous results recently found. On these lines, we also comment upon the compatibility of our approach with the Hamiltonian approach proposed by Davidson and coworkers.