An action principle for the quantization of parametric theories and nonlinear quantum cosmology

TL;DR

This paper introduces a novel quantization method for parametric and nonlinear quantum theories, with applications to quantum cosmology, enabling a unitary evolution description for constrained systems with nonlinear structures.

Contribution

It proposes a new action-based quantization approach that extends to nonlinear quantum mechanics and applies to cosmological models with gravitational degrees of freedom as geometric time.

Findings

Successfully quantized Friedmann, Bianchi I, and Bianchi IX models.

Demonstrated unitary evolution in nonlinear quantum cosmology.

Discussed potential for advancing quantum gravity theories.

Abstract

By parametrizing the action integral for the standard Schrodinger equation we present a derivation of the recently proposed method for quantizing a parametrized theory. The reformulation suggests a natural extension from conventional to nonlinear quantum mechanics. This generalization enables a unitary description of the quantum evolution for a broad class of constrained Hamiltonian systems with a nonlinear kinematic structure. In particular, the new theory is applicable to the quantization of cosmological models where a chosen gravitational degree of freedom acts as geometric time. This is demonstrated explicitly using three cosmological models: the Friedmann universe with a massless scalar field and Bianchi type I and IX models. Based on these investigations, the prospect of further developing the proposed quantization scheme in the context of quantum gravity is discussed.