Unimodular quantum cosmology in the connection representation: A minimal model

TL;DR

This paper develops a quantum cosmological model based on unimodular gravity in the connection representation, revealing how the cosmological constant influences the wave function behavior and discussing implications for quantum geometry fluctuations.

Contribution

It introduces a quantum unimodular gravity model in the connection representation for a simple cosmological setting, analyzing operator regularity and wave function behavior related to the cosmological constant.

Findings

Wave functions vanish at zero volume for positive cosmological constant.

Operator regularity conflicts with negative cosmological constant.

Semiclassical fluctuations are discussed in context of the cosmological constant problem.

Abstract

We present a quantization of unimodular gravity in the connection representation for a homogeneous, isotropic, and spatially flat cosmological model without matter. In this model, the wave function is governed by a Schr\"odinger-type equation derived from a reduced phase space approach. Our analysis suggests that, within this minimal setting, the regularity of the operators and the self-adjointness of the Hamiltonian operator are incompatible with a negative cosmological constant. For a positive cosmological constant, the wave functions vanish at zero spatial volume. This behavior emerges as a consequence of enforcing the unimodular condition at the quantum level. Semiclassical fluctuations of the geometry are evaluated and discussed in relation to the cosmological constant problem.