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 (Perivolaropoulos in 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 (Lawson in 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.
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 (Perivolaropoulos in 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 (Lawson in 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.
More recently, we have proposed a set of noncommutative space that describes the quantum gravity at the Planck scale (J. Phys. A: Math. Theor53, 115303 (2020)). The interesting significant result, we found is that, the generalized uncertainty principle induced a maximal measurable length of quantum gravity. This measurement revealed strong quantum gravitational effects at this scale and predicted the measurement of gravity particles with low energies. In the present paper, to make an evidence this prediction, we study in this space, the dynamics of a particle with position-dependent mass (PDM) trapped in an infinite square well. We show that, by increasing the quantum gravitational effect, the PDM of the particle increases and induces contractions of the quantum energy levels. These contractions are more pronounced as one increases the quantum levels, allowing the particle to jump from one state to another with low energies and with high probability densities. Moreover, with the spectrum of this system at hand, we develop generalized coherent states based on the Gazeau-Klauder formalism. These states are shown to satisfy the Klauder's minimum conditions. Some statistical properties are calculated and analyzed. We find that these states are sub-Poissonian in nature.
Candidate theories of quantum gravity predict the presence of a minimal measurable length at high energies. Such feature is in contrast with the Heisenberg Uncertainty Principle. Therefore, phenomenological approaches to quantum gravity introduced models spelled as modifications of quantum mechanics including a minimal length. The effects of such modification are expected to be relevant at large energies/small lengths. One first consequence is that position eigenstates are not included in such models due to the presence of a minimal uncertainty in position. Furthermore, depending on the particular modification of the position–momentum commutator, when such models are considered from momentum space, the position operator is changed, and a measure factor appears to let the position operator be self-adjoint. As a consequence, the (quasi-)position representation acquires numerous issues. For example, the position operator is no longer a multiplicative operator, and the momentum of a free particle does not correspond directly to its wave number. Here, we will review such issues, clarifying aspects of minimal length models, with particular reference to the representation of the position operator. Furthermore, we will show how such a (quasi-)position description of quantum mechanical models with a minimal length affects results concerning simple systems.
Several phenomenological approaches to quantum gravity predict the existence of a minimal measurable length and/or a maximum measurable momentum near the Planck scale. When embedded into the framework of quantum mechanics, such constraints induce a modification of the canonical commutation relations and thus a generalization of the Heisenberg uncertainty relations, commonly referred to as generalized uncertainty principle (GUP). Different models of quantum gravity imply different forms of the GUP. For instance, in the framework of string theory the GUP is quadratic in the momentum operator, while in the context of doubly special relativity it includes an additional linear dependence. Among the possible physical consequences, it was recently shown that the quadratic GUP induces a universal decoherence mechanism, provided one assumes a foamy structure of quantum spacetime close to the Planck length. Along this line, in the present work we investigate the gravitational decoherence associated to the linear-quadratic GUP and we compare it with the one associated to the quadratic GUP. We find that, despite their similarities, the two generalizations of the Heisenberg uncertainty principle yield decoherence times that are completely uncorrelated and significantly distinct. Motivated by this result, we introduce a theoretical and experimental scheme based on cavity optomechanics to measure the different time evolution of nonlocal quantum correlations corresponding to the two aforementioned decoherence mechanisms. We find that the deviation between the two predictions occurs on time scales that are macroscopic and thus potentially amenable to experimental verification. This scenario provides a possible setting to discriminate between different forms of the GUP and therefore different models of quantum gravity.
Several quantum gravity theories predict a minimal length at the order of magnitude of the Planck length, under which the concepts of space and time lose their physical meaning. In quantum mechanics, the insurgence of such a minimal length can be described by introducing a modified position-momentum commutator, which in turn yields a generalized uncertainty principle, where the uncertainty on position measurements has a lower bound. The value of the minimal length is not predicted by theories and must be estimated experimentally. In this paper, we address the quantum bound to the estimability of the minimal uncertainty length by performing measurements on a harmonic oscillator, which is analytically solvable in the deformed algebra induced by the deformed commutation relations.
Quantum gravity theories predict a minimal length at the order of magnitude of the Planck length, under which the concepts of space and time lose every physical meaning. In quantum mechanics, the insurgence of such minimal length can be described by introducing a modified position-momentum commutator, which in turn yields a generalized uncertainty principle, where the uncertainty on the position measurement has a lower bound. The value of the minimal length is not predicted by theories and must be evaluated experimentally. In this paper, we address the quantum bound to estimability of the minimal uncertainty length by performing measurements on a harmonic oscillator, which is analytically solvable in the deformed algebra of the Hilbert subspace.
Several approaches to quantum gravity imply the presence of a minimal measurable length at high energies. This is in tension with the Heisenberg Uncertainty Principle. Such a contrast is then considered in phenomenological approaches to quantum gravity by introducing a minimal length in quantum mechanics via the Generalized Uncertainty Principle. Several features of the standard theory are affected by such a modification. For example, position eigenstates are no longer included in models of quantum mechanics with a minimal length. Furthermore, while the momentum-space description can still be realized in a relatively straightforward way, the (quasi-)position representation acquires numerous issues. Here, we will review such issues, clarifying aspects regarding models with a minimal length. Finally, we will consider the effects of such models on simple quantum mechanical systems.