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{ihbar }{ 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.
Highlights
We determine the wave function of the Universe ψqp(ξ, t), which is solution of the modified
The most common modified or deformed commutation relation (MCR) associated with a Generalized Uncertainty Principle (GUP) (KMM GUP [37]) is quadratic in the momentum operator, effects of a MCR with a linear term in the momentum operator, which leads to a minimal length and a maximum momentum [38], have been studied in contexts of cosmological models
Since GUP corresponds to a modification of the commutation relation between the operator and its conjugate operator, the WDW equation in a minimal-length scenario can be obtained imposing that some or all of those operators and its conjugate momentum operators satisfy modified commutation relations
Summary
We can obtain a minimal-length scenario by modifying the Heisenberg uncertainty principle (HUP) between the position and momentum operators. In a minimal-length scenario that scalar field and its conjugate momentum have to obey the modified commutation relations [43,44,45,46] As it is well known, the ordinary approach of the third quantization of the Wheeler–DeWitt (WDW) equation leads to the vanishing of the cosmological constant. Since GUP corresponds to a modification of the commutation relation between the operator and its conjugate operator, the WDW equation in a minimal-length scenario can be obtained imposing that some or all of those operators (which have come from the quantization of remaining degrees of freedom in the mini-superspace approach) and its conjugate momentum operators satisfy modified commutation relations. We determine the modified WDW equation up to O(β2) considering a quantum cosmology model of dynamical vacuum in a minimal-length scenario induced by the commutation relation proposed by Pedram [66,67],.
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