Abstract
We study the correction of the energy spectrum of a gravitational quantum well due to the combined effect of the braneworld model with infinite extra dimensions and generalized uncertainty principle. The correction terms arise from a natural deformation of a semiclassical theory of quantum gravity governed by the Schrödinger–Newton equation based on a minimal length framework. The two fold correction in the energy yields new values of the spectrum, which are closer to the values obtained in the GRANIT experiment. This raises the possibility that the combined theory of the semiclassical quantum gravity and the generalized uncertainty principle may provide an intermediate theory between the semiclassical and the full theory of quantum gravity. We also prepare a schematic experimental set-up which may guide to the understanding of the phenomena in the laboratory.
Highlights
The fact that the minimal observable length can be useful to impose an effective cut-off in the ultraviolet domain in order to make the theory of quantum fields renormalizable was suggested very early by Heisenberg
The most interesting fact is that the new theory resulting from the combination of generalized uncertainty principle (GUP) and semi-classical quantum gravity (SCQG) may provide an intermediate theory between a full quantum theory of gravity and the SCQG and, this is precisely the issue that we address in the present manuscript
We have studied a two-fold correction in the energy spectrum of a gravitational quantum well (GQW)
Summary
It was Snyder, who formalized the idea for the first time in the form of an article, and showed that the noncommutative structure of space-time characterizes the minimal measurable length in a very natural way [1]. Some interesting consequences may follow from the combination of the above two frameworks, namely, the GUP and SCQG, and it is allowed since both of the effects occur at small scales This can be achieved by imposing the minimal length structure into the semi-classical scheme of gravity by means of deforming the Schrodinger-Newton equation in accordance with the laws of GUP. It becomes important to understand the effects of the combined theory, especially when there is a strong viability that the theory may be tested by using a similar type of experimental set-up in near future
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