Dirac-Bergmann algorithm and canonical quantization of $k$-essence cosmology

TL;DR

This paper develops a canonical quantization scheme for $k$-essence cosmology using the Dirac-Bergmann algorithm, leading to a Wheeler-DeWitt equation and exploring quantum effects like phantom crossing and singularity avoidance.

Contribution

It introduces a novel canonical quantization approach for $k$-essence cosmology, simplifying the Hamiltonian and analyzing quantum phenomena such as tunneling and boundary condition effects.

Findings

Hamiltonian reduces to a quadratic form with no potential.

Quantum tunneling can induce phantom crossing.

Boundary conditions influence singularity avoidance.

Abstract

We develop a general canonical quantization scheme for $k$-essence cosmology in scalar-tensor theory. Utilizing the Dirac-Bergmann algorithm, we construct the Hamiltonian associated with the cosmological field equations and identify the first- and second-class constraints. The introduction of appropriate canonically conjugate variables with respect to Dirac brackets, allows for the canonical quantization of the model. In these new variables, the Hamiltonian constraint reduces to a quadratic function with no potential term. Its quantum realization leads to a Wheeler-DeWitt equation reminiscent of the massless Klein-Gordon case. As an illustrative example, we consider the action of a tachyonic field and investigate the conditions under which a phantom crossing can occur as a quantum tunneling effect. For the simplified constant potential case, we investigate the consequences of different boundary conditions on the singularity avoidance and to the mean expansion rate.