Unimodular quantum cosmology in the connection representation: A minimal model

TL;DR

This paper studies unimodular gravity in the connection representation for a minimal, homogeneous, isotropic, spatially flat universe without matter, yielding a Schrödinger-type evolution with a reduced Hamiltonian whose eigenvalues correspond to the unimodular cosmological constant $\Lambda$. Exact solutions in the $p$-representation show that negative $\Lambda$ is incompatible with both regularity of the basic operators and self-adjointness, while positive $\Lambda$ produces oscillatory Airy-type eigenstates that vanish at $p=0$ due to the unimodular constraint. A WKB analysis demonstrates how unimodularity ties semiclassicality to a superposition over $\Lambda$, leading to a coherence condition where relative fluctuations scale as $r\sim \hbar k/(ΛV_4)$; using the present de Sitter-like expansion yields an extremely small $r\sim10^{-120}$, illustrating a distinct connection between the cosmological constant and quantum coherence in this framework. The findings highlight the radical re-interpretation of $\Lambda$ as an eigenvalue rather than a fixed input and expose model-dependent limitations, motivating exploration of more general unimodular cosmologies with matter or curvature. Nevertheless, the work clarifies how unimodularity modifies the canonical quantum cosmology structure and provides a concrete link between cosmological-constant issues and quantum coherence in the connection formalism.

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 Schrödinger-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.