M. F. Gusson | Searcharxiv

arXiv2021

Quantum Cosmology with Dynamical Vacuum in a Minimal-Length Scenario

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 $[\hat{X},\hat{P}] := \frac{i\hbar}{ 1 - β\hat{P}^2 }$ . We determine the wave function of the Universe $ ψ_{qp}(ξ,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 $ψ_{qp}(ξ,t)$ is a physically acceptable state it is not a realizable state of the Universe because $ ψ_{qp}(ξ,t)$ has infinite norm, as in the ordinary case with no minimal length.

gr-qc↗

arXiv2018

Dirac $δ$-function potential in quasiposition representation of a minimal-length scenario

A minimal-length scenario can be considered as an effective description of quantum gravity effects. In quantum mechanics the introduction of a minimal length can be accomplished through a generalization of Heisenberg's uncertainty principle. In this scenario, state eigenvectors of the position operator are no longer physical states and the representation in momentum space or a representation in a quasiposition space must be used. In this work, we solve the Schroedinger equation with Dirac $δ$-function potential in quasiposition space. We calculate the bound state energy and the coefficients of reflection and transmission for scattering states. We show that leading corrections are of order of the minimal length $({\sl O}(\sqrtβ))$ and the coefficients of reflection and transmission are no longer the same for the Dirac delta well and barrier as in ordinary quantum mechanics. Furthermore, assuming that the equivalence of the 1s state energy of the hydrogen atom and the bound state energy of the Dirac $δ$-function potential in 1-dim is kept in a minimal-length scenario, we also find that the leading correction term for the ground state energy of the hydrogen atom is of order of the minimal length and $Δx_{min} \le 10^{-25}$ m.

hep-th↗