A nonlinear quantum model of the Friedmann universe

TL;DR

This paper introduces a nonlinear quantum model for the Friedmann universe, where time is treated as a classical variable interacting with quantum states, resulting in a Schrödinger-type equation that is generally nonlinear but consistent with quantum probability.

Contribution

It presents a novel nonlinear quantisation scheme for cosmological models that maintains a probabilistic interpretation and is applied to a scalar field-driven Friedmann universe.

Findings

Numerical analysis of the model shows consistent dynamical behavior.

The scheme preserves a Hilbert space structure with a Euclidean norm.

The nonlinear Schrödinger-type equation effectively describes quantum cosmology.

Abstract

A discussion is given of the quantisation of a physical system with finite degrees of freedom subject to a Hamiltonian constraint by treating time as a constrained classical variable interacting with an unconstrained quantum state. This leads to a quantisation scheme that yields a Schrodinger-type equation which is in general nonlinear in evolution. Nevertheless it is compatible with a probabilistic interpretation of quantum mechanics and in particular the construction of a Hilbert space with a Euclidean norm is possible. The new scheme is applied to the quantisation of a Friedmann Universe with a massive scalar field whose dynamical behaviour is investigated numerically.