Quantum evolution of the Universe from $\tau=0$ in the constrained quasi-Heisenberg picture

TL;DR

This paper develops a quantum cosmology model using a quasi-Heisenberg approach, demonstrating that the Universe's evolution persists despite the Hamiltonian constraint, and analyzes observable dispersions during inflation.

Contribution

It introduces a quantization scheme where all observables, including the scale factor, are treated as quasi-Heisenberg operators, providing analytical solutions and numerical analysis of inflationary dispersions.

Findings

Operators evolve over time despite the H=0 constraint.

Singular classical observables remain singular quantum mechanically.

Dispersions of observables do not vanish at inflation's end.

Abstract

The Heisenberg picture of the minisuperspace model is considered. The suggested quantization scheme interprets all the observables including the Universe scale factor as the (quasi)Heisenberg operators. The operators arise as a result of the re-quantization of the Heisenberg operators that is required to obtain the hermitian theory. It is shown that the DeWitt constraint H=0 on the physical states of the Universe does not prevent a time-evolution of the (quasi)Heisenberg operators and their mean values. Mean value of an observable, which is singular in a classical theory, is also singular in a quantum case. The (quasi)Heisenberg operator equations are solved in an analytical form in a first order on the interaction constant for the quadratic inflationary potential. Operator solutions are used to evaluate the observables mean values and dispersions. A late stage of the inflation is considered numerically in the framework of the Wigner-Weyl phase-space formalism. It is found that the dispersions of the observables do not vanish at the inflation end.