Generalized Uncertainty Principle Research Articles

Gravitational effects on the Heisenberg Uncertainty Principle: A geometric approach

The Heisenberg Uncertainty Principle (HUP) limits the accuracy in the simultaneous measurements of the position and momentum variables of any quantum system. This is known to be true in the context of non-relativistic quantum mechanics. Based on a semiclassical geometric approach, here we conjecture an effective generalization of this principle, which is well-suited to be extended to general relativity scenarios as well. We apply our formalism to Schwarzschild and de Sitter spacetime, showing that the ensuing uncertainty relations can be mapped into well-known deformations of the HUP. We also infer the form of the perturbed metric that mimics the emergence of a discrete spacetime structure at Planck scale, consistently with the predictions of the Generalized Uncertainty Principle. Finally, we discuss our results in connection with other approaches recently appeared in the literature.

Thermodynamics of the black holes under the extended generalized uncertainty principle with linear terms

In this paper, we employ the extended generalized uncertainty principle with linear terms (LEGUP) to investigate the thermodynamics properties of the Schwarzschild and Reissner–Nordström (RN) black holes. Firstly, by constructing the theoretical framework of LEGUP, the minimal temperature of the Schwarzschild black hole and the modified mass–temperature function for the black hole are calculated. Furthermore, the heat capacity function for the Schwarzschild black hole is obtained. After that, we compare LEGUP black hole thermodynamics with EGUP black hole and with the usual forms. Besides, the modification of black hole entropy is discussed, which involves a heuristic analysis of particles absorbed by the black hole. Finally, we derive the LEGUP-corrected temperature, heat capacity and entropy functions of the RN black hole.

Comparison of the semiclassical and quantum dynamics of the Bianchi I cosmology in the polymer and GUP extended paradigms

We compare in some detail Polymer Quantum Mechanics and the Generalized Uncertainty Principle approach to clarify to what extent we can treat them on the same footing. We show that, while on a semiclassical level they may be formulated as similar modifications of the Poisson algebra, on a quantum level they intrinsically differ because PQM implies no absolute minimal uncertainty on position. Then we implement these schemes to Bianchi I cosmology on a semiclassical level deforming only the algebra of the Universe volume, searching for alternative formulations able to account for the modified Friedmann equations emerging in Brane Cosmology and Loop Quantum Cosmology. On a pure quantum level, we implement the two approaches through their original setups and reduce the two resulting Wheeler–DeWitt equations to the same morphological structure, showing how the polymer formalism is associated with a bouncing dynamics while in the Generalized Uncertainty Principle case the singularity is still present. The implications of the wavepacket spreading are also discussed in both approaches, outlining that, when the singularity survives, the Planckian era must necessarily be approached by a fully quantum (non-peaked) state of the Universe.

The influence of approximation in generalized uncertainty principle on black hole evaporation

The generalized uncertainty principle is often used to modify various thermodynamics systems by regarding the greater-than-equal relation as an approximate relation. We give a method to improve this approximation and compare the differences between the original and improved methods during the evaporation of black hole from two aspects of positive and negative parameters. Finally, we prove the rationality of the improved method and give some guiding opinions.

Generalized uncertainty principle effects in the Hořava-Lifshitz quantum theory of gravity

The Wheeler-DeWitt equation for a Kantowski-Sachs metric in Hořava-Lifshitz gravity with a set of coordinates in minisuperspace that obey a generalized uncertainty principle is studied. We first study the equation coming from a set of coordinates that obey the usual uncertainty principle and find analytic solutions in the infrared as well as a particular ultraviolet limit that allows us to find the solution found in Hořava-Lifshitz gravity with projectability and with detailed balance but now as an approximation of the theory without detailed balance. We then consider the coordinates that obey the generalized uncertainty principle by modifying the previous equation using the relations between both sets of coordinates. We describe two possible ways of obtaining the Wheeler-DeWitt equation. One of them is useful to present the general equation but it is found to be very difficult to solve. Then we use the other proposal to study the limiting cases considered before, that is, the infrared limit that can be compared to the equation obtained by using general relativity and the particular ultraviolet limit. For the second limit we use a ultraviolet approximation and then solve analytically the resulting equation. We find that and oscillatory behaviour is possible in this context but it is not a general feature for any values of the parameters involved.

Generalized uncertainty principles associated with the quaternionic offset linear canonical transform

The quaternionic offset linear canonical transform (QOLCT) can be defined as a generalization of the quaternionic linear canonical transform (QLCT). In this paper, we define the QOLCT, we derive the relationship between the QOLCT and the quaternion Fourier transform (QFT). Based on this fact, we prove the Rayleigh formula and some properties related to the QOLCT. Then, we generalize some different uncertainty principles (UPs), including Heisenberg-Weyl's UP, Hardy's UP, Beurling's UP, and logarithmic UP to the QOLCT domain.

Theory and Phenomenology of a Four-Dimensional String–Corrected Black Hole

We construct an effective four-dimensional string-corrected black hole (4D SCBH) by rescaling the string coupling parameter in a D-dimensional Callan–Myers–Perry black hole. From the theoretical point of view, the most interesting findings are that the string corrections coincide with the so-called generalized uncertainty principle (GUP) corrections to black hole solutions, Bekenstein–Hawking entropy acquires logarithmic corrections, and that there exists a critical value of the coupling parameter for which the black hole temperature vanishes. We also find that, due to the string corrections, the nature of the central singularity may be altered from space-like to time-like singularity. In addition, we study the possibility of testing such a black hole with astrophysical observations. Since the dilaton field does not decouple from the metric, it is not a priori clear that the resulting 4D SCBH offers only small corrections to the Schwarzschild black hole. We used motion of the S2 star around the black hole at the center of our galaxy to constrain the parameters (the string coupling parameter and ADM mass) of the 4D SCBH. To test the weak gravity regime, we calculate the deflection angle in this geometry and apply it to gravitational lensing. To test the strong field regime, we calculate the black hole shadow radius. While we find that the observables change as we change the string coupling parameter, the magnitude of the change is too small to distinguish it from the Schwarzschild black hole. With the current precision, to the leading order terms, the 4D SCBH cannot be distinguished from the Schwarzschild black hole.

A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty Principle

In this work, we point out an overlooked and subtle feature of the generalized uncertainty principle (GUP) approach to quantizing gravity: namely that different pairs of modified operators with the same modified commutator, [X^,P^]=iħ(1+βp2), may have different physical consequences such as having no minimal length at all. These differences depend on how the position and/or momentum operators are modified rather than only on the resulting modified commutator. This provides guidance when constructing GUP models since it distinguishes those GUPs that have a minimal length scale, as suggested by some broad arguments about quantum gravity, versus GUPs without a minimal length scale.

Quantum entanglement with generalized uncertainty principle

We explore how the quantum entanglement is modified in the generalized uncertainty principle (GUP)-corrected quantum mechanics by introducing the coupled harmonic oscillator system. Constructing the ground state ρ0 and its reduced substate ρA=TrBρ0, we compute two entanglement measures of ρ0, i.e. EEoF(ρ0)=Svon(ρA) and Eγ(ρ0)=Sγ(ρA), where Svon and Sγ are the von Neumann and Rényi entropies, up to the first order of the GUP parameter α. It is shown that Eγ(ρ0) increases with increasing α when γ=2,3,⋯. The remarkable fact is that EEoF(ρ0) does not have first-order of α. Based on there results we conjecture that Eγ(ρ0) increases or decreases with increasing α when γ>1 or γ<1 respectively for nonnegative real γ.

Hawking radiation of scalar particles and fermions from squashed Kaluza–Klein black holes based on a generalized uncertainty principle

We study the Hawking radiation from the five-dimensional charged static squashed Kaluza–Klein black hole by the tunneling of charged scalar particles and charged fermions. In contrast to the previous studies of Hawking radiation from squashed Kaluza–Klein black holes, we consider the phenomenological quantum gravity effects predicted by the generalized uncertainty principle with the minimal measurable length. We derive corrections of the Hawking temperature to general relativity, which are related to the energy of the emitted particle, the size of the compact extra dimension, the charge of the black hole and the existence of the minimal length in the squashed Kaluza–Klein geometry. We obtain some known Hawking temperatures in five and four-dimensional black hole spacetimes by taking limits in the modified temperature. We show that the generalized uncertainty principle may slow down the increase of the Hawking temperature due to the radiation, which may lead to the thermodynamic stable remnant of the order of the Planck mass after the evaporation of the squashed Kaluza–Klein black hole. We also find that the sparsity of the Hawking radiation modified by the generalized uncertainty principle may become infinite when the mass of the squashed Kaluza–Klein black hole approaches its remnant mass.

Probing the generalized uncertainty principle through quantum noises in optomechanical systems

In this work we have considered a simple mechanical oscillator interacting with a single mode optical field inside a cavity in the generalized uncertainty principle (GUP) framework. Our aim is to calculate the modified noise spectrum and observe the effects of the GUP. The commutation relation that we have considered has an extra linear order momentum term along with a quadratic order term. Confronting our theoretical results with the observational results, we observe that we get a much tighter bound on the GUP parameters from the noise spectrum using the values of the system parameters from different experiments.

Position-dependent mass in strong quantum gravitational background fields

More recently, we have proposed a set of noncommutative space that describes the quantum gravity at the Planck scale (2020 J. Phys. A: Math. Theor. 53 115303). The interesting significant result we found is that, the generalized uncertainty principle induces a maximal measurable length of quantum gravity. This measurement revealed strong quantum gravitational effects at this scale and predicted a detection of gravity particles with low energies. In the present paper, to make 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 deformations of the quantum energy levels. These deformations 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.

Thermodynamics of the Reissner-Nordström black hole with quintessence matter on the EGUP framework

In this paper, we investigate the quantum correction on thermodynamics of the Reissner-Nordström black hole in the presence of the quintessence matter associated with dark energy. To this end, the modified Hawking temperature, heat capacity, and entropy functions of the black hole are derived. Investigation reveals that the modified uncertainty principles and normalization factor α restrict the lower bound value of horizon radius to affect the Hawking temperature and the parameter η0 has a non-negligible effect on the heat capacity threshold. Furthermore, the correction of entropy does not depend on the quintessence matter. It is also examined the equation of state associated with the pressure and the volume in this framework. In addition, we employ graphical methods to compare and discuss the findings within the generalized uncertainty principle (GUP) and extended uncertainty principle (EUP).

Constraining the generalized uncertainty principle through black hole shadow, S2 star orbit, and quasiperiodic oscillations

In this paper, we study the effect of the Generalized Uncertainty Principle (GUP) on the shadow of GUP-modified Kerr black hole and the correspondence between the shadow radius and the real part of the quasinormal modes (QNMs). We find that the shadow curvature radius of the GUP-modified Kerr black hole is bigger compared to the Kerr vacuum solution and increases linearly monotonically with the increase of the GUP parameter. We then investigate the characteristic points of intrinsic curvature of the shadow from a topological point of view to calculate the angular size for these curvature radii of the shadow. To this end, we have used the EHT data for the M87* black hole to constrain the upper limits of the GUP parameter and our result is [Formula: see text]. Finally, we have explored the connection between the shadow radius and the scalar/electromagnetic/gravitational QNMs. Using the orbit of S2 star we have obtained a bound for the GUP parameter [Formula: see text]. The GUP-modified Kerr black hole is also used to provide perfect curve fitting of the particle oscillation upper and lower frequencies to the observed frequencies for three microquasars and to restrict the values of the correction parameter in the metric of the modified black hole to very reasonable bound [Formula: see text].

Spectral geometry and black holes degrees of freedom

In this work we show that the introduction of an appropriate cutoff in the spectra of the Laplacian of an spherically symmetric and static black hole reveals an equivalence between shape and holographic degrees of freedom. Even more, the aforementioned cutoff introduces a correction to the Bekenstein-Hawking entropy which resembles the corresponding holographic loop quantum gravity, generalized uncertainty principle and entanglement entropy corrections. The difficulty of extending our results to nonspherically symmetric black holes is pointed out.

Weakly interacting Bose gases with generalized uncertainty principle: Effects of quantum gravity

We investigate quantum gravity corrections due to the generalized uncertainty principle on three-dimensional weakly interacting Bose gases at both zero and finite temperatures using the time-dependent Hatree–Fock–Bogoliubov theory. We derive useful formulas for the depletion, the anomalous density, and some thermodynamic quantities such as the chemical potential, the ground-state energy, the free energy, and the superfluid density. It is found that the presence of a minimal length leads to modify the fluctuations of the condensate and its thermodynamic properties in the weak and strong quantum gravitational regimes. Unexpectedly, the interplay of quantum gravity effects and quantum fluctuations stemming from interactions may lift both the condensate and the superfluid fractions. We show that quantum gravity minimizes the interaction force between bosons leading to the formation of ultradilute Bose condensates. Our results which can be readily probed in current experiments may offer a new attractive possibility to understand gravity in the framework of quantum mechanics.

Quantum corrected polymer black hole thermodynamics: mass relations and logarithmic entropy correction

In this paper, we continue the analysis of the effective model of quantum Schwarz­schild black holes recently proposed by some of the authors in [1,2]. In the resulting quantum-corrected spacetime the central singularity is resolved by a black-to-white hole bounce, quantum effects become relevant at a unique mass-independent curvature scale, while they become negligible in the low curvature region near the horizon and classical Schwarzschild geometry is approached asymptotically. This is the case independently of the relation between the black and white hole masses, which are thus freely specifiable independent observables. A natural question then arises about the phenomenological implications of the resulting non-singular effective spacetime and whether some specific relation between the masses can be singled out from a phenomenological perspective. Here we focus on the thermodynamic properties of the effective polymer black hole and analyze the corresponding quantum corrections as functions of black and white hole masses. The study of the relevant thermodynamic quantities such as temperature, specific heat, and horizon entropy reveals that the effective spacetime generically admits an extremal minimal-sized configuration of quantum-gravitational nature characterized by vanishing temperature and entropy. For large masses, the classically expected results are recovered at leading order and quantum corrections are negligible, thus providing us with a further consistency check of the model. The explicit form of the corrections depends on the specific relationship among the masses. In particular, a first-order logarithmic correction to the black hole entropy is obtained for a quadratic mass relation. The latter corresponds to the case of proper finite-length effects which turn out to be compatible with a minimal length generalized uncertainty principle associated with an extremal Planck-sized black hole.

Quantum gravity and the square of Bell operators

The Bell’s inequality is a strong criterion to distinguish classical and quantum mechanical aspects of reality. Its violation is the net effect of the existence of non-locality in systems, an advantage for quantum mechanics over classical physics. The quantum mechanical world is under the control of the Heisenberg uncertainty principle that is generalized by quantum gravity scenarios, called generalized uncertainty principle (GUP). Here, the effects of GUP on the square of Bell operators of qubits and qutrits are studied. The achievements claim that the violation quality of the square of Bell inequalities may be a tool to get a better understanding of the quantum features of gravity. In this regard, it is obtained that the current accuracy of the Stern–Gerlach experiments implies upper bounds on the values of the GUP parameters.

Constraining the Generalized Uncertainty Principle with the light twisted by rotating black holes and M87*

We test the validity of the Generalized Heisenberg's Uncertainty principle in the presence of strong gravitational fields nearby rotating black holes; Heisenberg's principle is supposed to require additional correction terms when gravity is taken into account, leading to a more general formulation also known as the Generalized Uncertainty Principle. Using as probes electromagnetic waves acquiring orbital angular momentum when lensed by a rotating black hole, we find from numerical simulations a relationship between the spectrum of the orbital angular momentum of light and the corrections needed to formulate the Generalized Uncertainty Principle, here characterized by the rescaled parameter β0, a function of the Planck's mass and the bare mass of the black hole. Then, from the analysis of the observed twisted light due to the gravitational field of the compact object observed in M87, we find new limits for the parameter β0. With this method, complementary to black hole shadow circularity analyses, we obtain more precise limits from the experimental data of M87*, confirming the validity of scenarios compatible with General Relativity, within the uncertainties due to the experimental errors present in EHT data and those due to the numerical simulations and analysis.

Nonlinear electrodynamics effects on the black hole shadow, deflection angle, quasinormal modes and greybody factors

In this paper, we discuss the effects of nonlinear electrodynamics (NED) on non-rotating black holes, parametrized by the field coupling parameter β and magnetic charge parameter P in detail. Particularly, we survey a large range of observables and physical properties of the magnetically charged black hole, including the thermodynamic properties, observational appearance, quasinormal modes and absorption cross sections. Initially, we show that the NED black hole is always surrounded by an event horizon and any magnetic charge is permissible. We then show that the black hole gets colder with increasing charge. Investigating the heat capacity, we see that the black hole is thermally stable between points of phase transition. Introducing a generalized uncertainty principle (GUP) with a quantum gravity parameter λ extends the range of the stable region, but the effect on temperature is negligible. Then we compute the deflection angle at the weak field limit, by the Gauss-Bonnet theorem and the geodesic equation, and find that even at the first order, the magnetic charge has a contribution due to the “field mass” term. Small changes of the charge contributes greatly to the paths of null geodesics due to the P 2 dependence of the horizon radius. Using a ray-tracing code, we simulate the observational appearance of a NED black hole under different emission profiles, thin disk and spherical accretion. We find that the parameter P has a very strong effect on the observed shadow radius, in agreement with the deflection angle calculations. We finally consider quasinormal modes under massless scalar perturbations of the black hole and the greybody factor. We find that the charge introduces a slight difference in the fundamental frequency of the emitted waveform. We find that the greybody factor of the NED black hole is strongly steepened by the introduction of increasing charge. To present observational constrains, we show that the magnetic charge of the M87* black hole is between 0 ≤ P ≤ 0.024 in units of M, in agreement with the idea that real astrophysical black holes are mostly neutral. We also find that LIGO/VIRGO and LISA could detect NED black hole perturbations from BHs with masses between 5 M ☉ and 8.0 · 108 M ☉. We finally show that for black holes with masses detected with LIGO so far, charged NED black holes would deviate from Schwarzschild by 5∼10 Hz in their fundamental frequencies.