Minimal and maximal lengths of quantum gravity from non-Hermitian
position-dependent noncommutativity
- URL: http://arxiv.org/abs/2106.03586v2
- Date: Wed, 22 Dec 2021 15:18:24 GMT
- Title: Minimal and maximal lengths of quantum gravity from non-Hermitian
position-dependent noncommutativity
- Authors: Lat\'evi M.Lawson
- Abstract summary: A minimum length scale of the order of Planck length is a feature of many models of quantum gravity.
We show that a simultaneous measurement of both lengths form a family of discrete spaces.
- Score: 0.0
- License: http://creativecommons.org/licenses/by-nc-nd/4.0/
- Abstract: A minimum length scale of the order of Planck length is a feature of many
models of quantum gravity that seek to unify quantum mechanics and gravitation.
Recently, Perivolaropoulos in his seminal work [Phys. Rev.D 95, 103523 (2017)]
predicted the simultaneous existence of minimal and maximal length measurements
of quantum gravity. More recently, we have shown that both measurable lengths
can be obtained from position-dependent noncommutativity [J. Phys. A:
Math.Theor. 53, 115303 (2020)]. In this paper, we present an alternative
derivation of these lengths from non-Hermitian position-dependent
noncommutativity. We show that a simultaneous measurement of both lengths form
a family of discrete spaces. In one hand, we show the similarities between the
maximal uncertainty measurement and the classical properties of gravity. On the
other hand, the connection between the minimal uncertainties and the
non-Hermicity quantum mechanic scenarios. The existence of minimal
uncertainties are the consequences of non-Hermicities of some operators that
are generators of this noncommutativity. With an appropriate Dyson map, we
demonstrate by a similarity transformation that the physically meaningfulness
of dynamical quantum systems is generated by a hidden Hermitian
position-dependent noncommutativity. This transformation preserves the
properties of quantum gravity but removes the fuzziness induced by minimal
uncertainty measurements at this scale. Finally, we study the eigenvalue
problem of a free particle in a square-well potential in these new Hermitian
variables.
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