Quantum Cosmology with Dynamical Vacuum in a Minimal-Length Scenario

TL;DR

This paper explores quantum cosmology incorporating a dynamical vacuum and a minimal length scale via GUP, deriving the universe's wave function in a modified Wheeler-DeWitt framework, highlighting issues with state normalizability.

Contribution

It introduces a modified Wheeler-DeWitt equation accounting for a minimal length and derives the universe's wave function in this context, addressing the implications of GUP on quantum cosmology.

Findings

Wave function solutions in quasi-position space

Identification of non-normalizable states

Insights into minimal length effects on quantum cosmology

Abstract

In this work, we consider effects of the dynamical vacuum in quantum cosmology in presence of a minimum length introduced by the GUP (generalized uncertainty principle) related to the modified commutation relation $[\hat{X},\hat{P}] := \frac{i\hbar}{ 1 - \beta\hat{P}^2 }$ . We determine the wave function of the Universe $ \psi_{qp}(\xi,t)$, which is solution of the modified Wheeler-DeWitt equation in the representation of the quasi-position space, in the limit where the scale factor of the Universe is small. Although $\psi_{qp}(\xi,t)$ is a physically acceptable state it is not a realizable state of the Universe because $ \psi_{qp}(\xi,t)$ has infinite norm, as in the ordinary case with no minimal length.