Classical Universe Arising from Quantum Cosmology

TL;DR

This paper applies deformation quantization to quantum cosmology, demonstrating how classical universe features emerge from quantum states and exploring the role of canonical transformations in defining time within the model.

Contribution

It introduces a novel application of the Weyl--Wigner--Groenewold--Moyal formalism to quantum cosmology, providing exact solutions and insights into the classical limit and time emergence.

Findings

Classical universe emerges from quantum states with large quantum numbers.

Wigner functions peak along classical trajectories in phase space.

Canonical transformation can disentangle time from dynamical variables.

Abstract

In this paper, we study the classical limit and unitary evolution of quantum cosmology by applying the Weyl--Wigner--Groenewold--Moyal formalism of deformation quantization to quantum cosmology of a homogeneous and isotropic universe with positive spatial curvature and conformally coupled scalar field. The corresponding quantum cosmology (similar to the Schr\"{o}dinger interpretation in canonical quantization scheme of quantum cosmology) is described by the Moyal--Wheeler--DeWitt equation which has an exact solution in Moyal phase space, resulting in Wigner quasiprobability distribution function, peaking over the classical solutions. We show that for a large value of the quantum number $n$, the emerged classical universe is filled with radiation with quantum mechanical origin. Also, we introduce a canonical transformation on the scalar field sector of the model such that the conjugate momenta of the new canonical variable appear linearly in the transformed total Hamiltonian. Using this canonical transformation, we show that, it may lead to disentangle the time from the true dynamical variables. We obtain the time-dependent Wigner function for a coherent as well as for squeezed states. We show that the peak of these Wigner functions follows the classical trajectory in the phase space.