Minimum Measurable Length Research Articles

Time Evolution in Quantum Mechanics with a Minimal Time Scale

The existence of a minimum measurable length scale was suggested by various theories of quantum gravity, string theory and black hole physics. Motivated by this, we examine a quantum theory exhibiting a minimum measurable time scale. We use the Page–Wootters formalism to describe time evolution of a quantum system with the modified commutation relations between the time and frequency operator. Such modification leads to a minimal uncertainty in the measurement of time. This causes breaking of the time-translation symmetry and results in a modified version of the Schrödinger equation. A minimal time scale also allows us to introduce a discrete Schrödinger equation describing time evolution on a lattice. We show that both descriptions of time evolution are equivalent. We demonstrate the developed theory on a couple simple quantum systems.

Minimum measurable length inspired thermodynamic characteristics of the Schwarzschild black hole surrounded by the cloud of strings and quintessence

We explore the thermodynamic properties of the Schwarzschild black hole surrounded by quintessence matter and a cloud of strings within the framework of the generalized Heisenberg algebra. We provide precise computations of temperature, heat capacity, and entropy associated with this black hole and investigate the impacts of various deformations of the Heisenberg algebra on these thermodynamic parameters. The findings are compared with those for the usual Schwarzschild black hole. Additionally, the behavior of the isotherm related to this black hole is studied. It is anticipated that remnants will hold the key to solving the information loss problem. It is observed that different deformations of the Heisenberg algebra, which introduce minimum measurable length, maximum measurable momenta, or both simultaneously, could lead to the existence of a black hole remnant. Furthermore, it is noted that both exotic fields, cloud of strings and quintessence contribute to the quantum gravity-corrected entropy of the composite system.

Relativistic generalized uncertainty principle for a test particle in four-dimensional spacetime

The generalized uncertainty principle provides a promising phenomenological approach to reconciling the fundamentals of general relativity with quantum mechanics. As a result, noncommutativity, measurement uncertainty, and the fundamental theory of quantum mechanics are subject to finite gravitational fields. The generalized uncertainty principle (RGUP) on curved spacetime allows for the imposition of quantum-induced elements on general relativity (GR) in four dimensions. In the relativistic regimes, the determination of a test particle’s spacetime coordinates, [Formula: see text], becomes uncertain. There exists a specific range of coordinates where the accessibility to [Formula: see text] is notably limited. Consequently, the spacetime coordinates lack both smoothness and continuity. The quantum-mechanical calculations of the spacetime coordinates are directly linked to their measurement through the expectation value [Formula: see text]. This expectation value is dependent on [Formula: see text], which is itself limited by a minimum measurable length [Formula: see text]. A crucial finding presented in this script is the existence of a lower bound for the Hamiltonian, which implies the stability of the quantum nature of spacetime. The direct correlation between [Formula: see text] and [Formula: see text] is a significant discovery, suggesting a proportional relationship. The conjecture is made that the primary metric g carries all essential information regarding spacetime curvature and serves a role akin to the Jacobian determinant in general relativity. Moreover, the linear relationship between [Formula: see text] and the Planck length [Formula: see text] is established with a proportionality factor of [Formula: see text], where [Formula: see text] denotes the RGUP parameter. The discretization of spacetime coordinates results in the discontinuity of the test particle’s wavefunction [Formula: see text], leading to an unrealistic [Formula: see text]. It is also noted that the lower limit of [Formula: see text] is directly proportional to the fundamental tensor.

On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor

The minimal length conjecture is merged with a generalized quantum uncertainty formula, where we identify the minimal uncertainty in a particle’s position as the minimal measurable length scale. Thus, we obtain a quantum-induced deformation parameter that directly depends on the chosen minimal length scale. This quantum-induced deformation is conjectured to require the generalization of Riemannian spacetime geometry underlying the classical theory of general relativity to an eight-dimensional spacetime fiber bundle, which dictates the deformation of the line element, metric tensor, Levi-Civita connection, Riemann curvature tensor, etc. We calculate the deformation thus produced in the Levi-Civita connection and find it to explicitly and exclusively depend on the product of the minimum measurable length and the particle’s spacelike four-acceleration vector, L2x¨2. We find that the deformed Levi-Civita connection preserves all properties of its undeformed counterpart, such as torsion freedom and metric compatibility. Accordingly, we have constructed a deformed version of the Riemann curvature tensor whose expression can be factorized in all its terms with different functions of L2x¨2. We also show that the classical four-manifold status of being Riemannian is preserved when the quantum-induced deformation is negligible. We study the dependence of a parallel-transported tangent vector on the spacelike four-acceleration. We illustrate the impact of the minimal-length-induced quantum deformation on the classical geometrical objects of the general theory of relativity using the unit radius two-sphere example. We conclude that the minimal length deformation implies a correction to the spacetime curvature and its contractions, which is manifest in the additional curvature terms of the corrected Riemann tensor. Accordingly, quantum-induced effects endow an additional spacetime curvature and geometrical structure.

Radial oscillations and dynamical instability analysis for linear-quadratic GUP-modified white dwarfs

A modification to the Heisenberg uncertainty principle is called the generalized uncertainty principle (GUP), which emerged due to the introduction of a minimum measurable length, common among phenomenological approaches to quantum gravity. One approach to GUP is called linear-quadratic GUP (LQGUP) which satisfies both the minimum measurable length and the maximum measurable momentum, resulting to an infinitesimal phase space volume proportional to the first-order momentum (1−αp)−4d3xd3p, where α is the still-unestablished GUP parameter. In this study, we explore the mass–radius relations of white dwarfs whose equation of state has been modified by LQGUP, and provide them with radial perturbations to investigate the dynamical instability arising from the oscillations. We find from the mass–radius relations that the main effect of LQGUP is to worsen the gravitational collapse by decreasing the mass of the relatively massive white dwarfs (including their limiting mass, while increasing their limiting radius). This effect gets more prominent with larger values of α. We then compare the results with available observational data. To further investigate the impact of the GUP parameter, a dynamical instability analysis of the white dwarf was conducted, and we find that instability sets in for all values of α. With increasing α, we also find that the central density at which instability occurs decreases, resulting to a lower maximum mass. This is in contrast to quadratic GUP, where instability only sets in below a critical value of the quadratic GUP parameter.

Accretion in an asymptotically safe environment inspired by minimum measurable length

We investigate the impact of the incorporation of minimum measurable length on accretion onto Schwarzschild black hole within the asymptotically safe ambiance. The Schwarzschild black hole is modified by the minimum measurable length associated with the generalized uncertainty principle (GUP), enabling it to accept the quantum gravity correction. We look at potential critical point locations and calculate the critical speed of the matter accreting. We determine the temperature and total integrated flux correction at the event horizon for the polytropic matter using the least measurable length conjecture offered by the GUP. We note that the incorporation of minimum measurable length has a significant impact on the accretion process. Furthermore, the quantum gravity regime associated with modification maintains an asymptotically safe ambiance.

Thermodynamical character and the radiation process of the thin accretion disk of minimum measurable length inspired regular black hole

Thermodynamical character and the radiation process of the thin accretion disk of minimum measurable length inspired regular black hole

Quantum Gravity effect on accretion disk of Schwarzschild black hole

This paper is concerned with the effect of quantum gravity, that is represented by generalized uncertainty principle (GUP), which admits a presence of a minimum measurable length, on the accretion disk about Schwarzschild black hole. The modification in the metric due to GUP was determined. The problem of spherically symmetric, steady state, adiabatic accretion onto a modified Schwarzschild black hole was considered. By assuming a polytropic accreting gas, analytical formulation of the problem is achieved in general relativity framework. Despite the decrease in the effective mass of the black hole due GUP affect, the rate of mass accretion is accelerated and there is an increase in the critical radius and in critical speed of sound with a decrease in critical gas velocity. Also, the gas pressure and temperature profile of the gas below the critical point and at event horizon were considered and is found they increased due to GUP effect.

The effects of the covariant generalized uncertainty principle on quantum mechanics

One can use the generalized uncertainty principle (GUP) to incorporate the minimum measurable length in quantum gravity. It may be interesting to have a minimal time interval as well as the minimal length in the relativistic version of quantum mechanics in the presence of the gravitational effects. In this paper, we considered a covariant version of the GUP to investigate the effects of both the minimal time and the minimal length. Using the covariant GUP, the energy–momentum dispersion relation was modified. Starting with the modified dispersion relation, the corrections to the wave function were obtained and the problems of the particle in a box and the hydrogen atom were revisited. Considering the minimal time may be an interesting new topic, which is not widely studied before. It may lead to new results that can open a new window into quantum gravity.

On quantum‐induced revisiting Einstein tensor in the relativistic regime

AbstractFor a rigorous approach to reconcile principles of general relativity (GR) and quantum mechanics (QM), we suggest applying the generalized relativistic noncommutative Heisenberg algebra on spacetime coordinates and momenta of a test particle on curved Riemann geometry manifold, which in turn is extended to an eight‐dimensional manifold. The natural generalization of the four‐dimensional Riemann geometry is the Finsler geometry, in which the quadratic restriction on the length measure is relaxed. With the minimum measurable length derived from the relativistic generalized four‐dimensional uncertainty principle, the quantum‐induced corrections to the fundamental tensor could be determined in the relativistic regime. Accordingly, the affine connections could be revisited. With the quantum‐induced revisiting Riemann curvature tensor and its contractions, the Ricci curvature tensor, and then the Ricci scalar, we have been able to construct a quantum‐induced revision of the Einstein tensor, in which besides quantum‐induced corrections, additional geometric structures emerge. On the surface of the 2‐sphere, we compared the quantum‐induced revisiting and non‐revisiting Einstein tensor and concluded that the difference between both versions strongly depends on the minimum measurable length and the local geodesics of the test particle through the additional curvature.

On possible quantization of the fundamental tensor in the relativistic regime

AbstractInspired by the quantum gravity prescriptions, for instance, string theory, doubly‐special relativity, and quantum loop gravity, we apply the generalized relativistic noncommutative Heisenberg algebra on the eight‐dimensional Finsler manifold, the natural generalization of the four‐dimensional Riemann manifold. The Finsler manifold is conjectured to relax the quadratic restriction on the length measure, especially in the relativistic regime. The additional curvature coupled with the Finsler manifold is assumed to mimic the quantization on the Riemann manifold. The second ingredient we suggest is the existence of a minimum measurable length, whose value is determined from the relativistic generalized uncertainty principle, and then applying in to the coordinates of the Finsler manifold. The quantum‐induced corrections to the fundamental tensor are determined under extreme conditions of a relativistic energy. Whether this proposal is adequate for a full quantization, the various limitations of the proposed theory are elaborated and discussed, especially the limitations of the curved spacetime model assumed for the line element measure. We have conservatively formulated our conclusion that the quantization of the fundamental tensor might be conditioned to an adequate quantization of the line element measure, at least the one on the eight‐dimensional Finsler manifold.

Maximal momentum GUP leads to quadratic gravity

Quantum theories of gravity predict interesting phenomenological features such as a minimum measurable length and maximum momentum. We use the Generalized Uncertainty Principle (GUP), which is an extension of the standard Heisenberg Uncertainty Principle motivated by Quantum Gravity, to model the above features. In particular, we use a GUP with modelling maximum momentum to establish a correspondence between the GUP-modified dynamics of a massless spin-2 field and quadratic (referred to as Stelle) gravity. In other words, Stelle gravity can be regarded as the classical manifestation of a maximum momentum and the related GUP. We explore the applications of Stelle gravity to cosmology and specifically show that Stelle gravity applied to a homogeneous and isotropic background leads to inflation with an exit. Using the above, we obtain strong bounds on the GUP parameter from CMB observations. Unlike previous works, which fixed only upper bounds for GUP parameters, we obtain both lower and upper bounds on the GUP parameter.

Minimal length, maximal momentum and stochastic gravitational waves spectrum generated from cosmological QCD phase transition

We investigate thoroughly the temporal evolution of the universe temperature as a function of the Hubble parameter associated with the Stochastic Gravitational Wave (SGW), that formed at the cosmological QCD phase transition epoch to the current epoch, within the Generalized Uncertainty Principle (GUP) framework. Here we use GUP version which provide constraints on the minimum measurable length and the maximum observable momentum, that characterized by a free parameter α. We study the effect of this parameter on the SGW background. We show that the effect can slightly enhance the SGW frequency in the lower frequency regime which might be important in the detection of SGW in the future GW detection facilities.

Schwarzschild Black Hole Thermodynamics and Generalized Uncertainty Principle

In this paper, a linear generalized uncertainty principle (GUP), which suggests a minimum measurable length and a maximal measurable momentum, is used in analyzing COW experiment and Einstein-Bohr’s a photon box experiment. Based in these experiments a modified Schwarzschild metric was obtained in order to calculate a quantum corrections in Schwarzschild black hole thermodynamics and tunneling probability. It was found that GUP placed restrictions on the minimum mass, size and temperature of the black hole, indicating that there was a black hole remnant after the completion of the evaporation process. The resulting values of minimum mass and size of the black hole are identical to that values predicted by quadratic GUP, which points out that the two models are equivalence in Planck scale measurements.

Generalized Uncertainty Principle in Quantum Cosmology

The effects of gravity which manifest themselves when performing the simultaneous measurement of two non-commuting observables in the quantum theory are discussed. Matter and gravity are considered as quantum fields. The Schr¨odinger-type time equation is given for the case of a finite number of degrees of freedom: one for the matter field and one for geometry. For a spatially closed system filled with dust and radiation being in definite quantum states, the solutions to the quantum equations are found, and the existence of the minimum measurable length and the minimum momentum is shown. It appears that the simultaneous measurement of fluctuations of the intrinsic and extrinsic curvatures of the spacelike hypersurface in spacetime cannot be performed with an accuracy exceeding the Planck constant. Unruh’s and Bronstein’s uncertainty relations are discussed.

A new type of GUP with commuting coordinates

In this paper, using Kempf and Mangano’s generalized uncertainty principle (GUP) [A. Kempf and G. Mangano, Phys. Rev. D 55, 7909 (1997)], we introduce a GUP which is called GUP*. The minimum measurable length predicted by the GUP* leads to the existence of a maximum momentum, in agreement with various previous studies. Then, we investigate some deformed calculus including both exponential and differential cases. Finally, as some applications of this GUP*, we discuss some quantum mechanical problems and obtain the results.

Potential tests of the generalized uncertainty principle in the advanced LIGO experiment

The generalized uncertainty principle and a minimum measurable length arise in various theories of gravity and predict Planck-scale modifications of the canonical position-momentum commutation relation. Postulating a similar modified commutator between the canonical variables of the electromagnetic field in quantum optics, we compute Planck-scale corrections to the radiation pressure noise and shot noise of Michelson–Morley interferometers, with particular attention to gravity wave detectors such as LIGO. We show that advanced LIGO is potentially sensitive enough to observe Planck-scale effects and thereby indirectly a minimal length. We also propose estimates for the bounds on quantum gravity parameters from current and future advanced LIGO experiments.

Topological defects in a deformed gauge theory

In this paper, we will analyse the topological defects in a deformation of a non-abelian gauge theory using the Polyakov variables. The gauge theory will be deformed by the existence of a minimum measurable length scale in the background spacetime. We will construct the Polyakov loops for this deformed non-abelian gauge theory, and use these deformed loop space variables for obtaining a deformed loop space curvature. It will be demonstrated that this curvature will vanish if the deformed Bianchi identities are satisfied. However, it is possible that the original Bianchi identities are satisfied, but the deformed Bianchi identities are violated at the leading order in the deformation parameter, due to some topological defects. Thus, topological defects could be produced purely from a deformation of the background geometry.

Inflationary universe in the presence of a minimal measurable length

In this paper, we will study the effect of having a minimum measurable length on inflationary cosmology. We will analyze the inflationary cosmology in the Jacobson approach. In this approach, gravity is viewed as an emergent thermodynamical phenomena. We will demonstrate that the existence of a minimum measurable length will modify the Friedmann equations in the Jacobson approach. We will use this modified Friedmann equation to analyze the effect of minimum measurable length scale on inflationary cosmology. This analysis will be performed using the Hamiltonian-Jacobi approach. We compare our results to recent data, and find that our model may agree with the recent data.

Euclidean path integral formalism in deformed space with minimum measurable length

We study time-evolution at the quantum level by developing the Euclidean path-integral approach for the general case where there exists a minimum measurable length. We derive an expression for the momentum-space propagator which turns out to be consistent with recently developed β-canonical transformation. We also construct the propagator for maximal localization which corresponds to the amplitude that a state which is maximally localized at location ξ′ propagates to a state which is maximally localized at location ξ″ in a given time. Our expression for the momentum-space propagator and the propagator for maximal localization is valid for any form of time-independent Hamiltonian. The nonrelativistic free particle, particle in a linear potential, and the harmonic oscillator are discussed as examples.