Wigner-Ville distribution associated with the quaternion linear canonical transform and their generalized uncertainty principles

The generalized uncertainty principle (GUP) is a common feature among several approaches related to quantum gravity. An approach to GUP was recently developed that contains both linear and quadratic terms of momenta, from which an infinitesimal phase space volume was derived up to the linear term of momenta. We studied the effects of this linear GUP approach on the structure equations and mass–radius relation of zero-temperature white dwarfs. We formulated a linear GUP-modified Chandrasekhar equation of state (EoS) by deriving exact forms of the thermodynamic properties of ideal Fermi gases. This was then used to obtain the analytical form of the modified Newtonian structure equations for the white dwarfs. By imposing a constraint on the momenta of the particles in the white dwarf due to linear GUP, the structure equations were solved and the modified mass–radius relation of the white dwarfs were obtained. This was then extended in the context of general relativity (GR), which, like linear GUP, affects white dwarfs significantly in the high-mass regime. We found that linear GUP displays a similar overall effect as in GR — linear GUP supports gravitational collapse of the white dwarf, by decreasing its limiting (maximum) mass and increasing its corresponding limiting (minimum radius). We also found that GUP effects become evident only at large values of the GUP parameter, but these values are still within the estimated bounds. This effect gets more prominent as we increase the as-of-yet unestablished value of the parameter.

Based on string theory, loop quantum gravity, black hole physics, and other theories of quantum gravity, physicists have proposed generalized uncertainty principle (GUP) modifications. In this work, within the framework of GUP gravity theory, we successfully derive an exact solution to Einstein’s field equation, and discuss the possibility of using EHT to test GUP and how GUP changes the weak cosmic censorship conjecture for black holes. We analyze two different ways of constructing GUP rotating black holes (model I and model II). Model I takes into account the modification of mass by GUP, i.e., the change in mass by quantization of space, and the resulting GUP rotating black hole metric (18) is similar in form to the Kerr black hole metric. Model II takes into account the modification of the rotating black hole when GUP is an external field, where GUP acts like an electric charge, and the resulting GUP rotating black hole metric (19) is similar in form to the Kerr–Newman black hole metric. The difference between (18) and (19) in the spacetime linear structure provides a basis for us to examine the physical nature of GUP rotating black holes from observation. By analyzing the shadow shape of the GUP rotating black hole, we discover intriguing characteristics regarding the impact of first-order and second-order momentum correction coefficients on the black hole’s shadow shape. These findings will be instrumental in future GUP testing using EHT. Additionally, by incident test particle and scalar field with a rotating GUP black hole, the weak cosmic censorship conjecture is not violated in either extreme black holes or near-extreme black holes.

We explore the implications of the generalized uncertainty principle (GUP) on the nuclear equation of state (EoS) and on the structure of neutron stars. Two approaches of GUP are used: the quadratic GUP approach, satisfying minimal length and the linear GUP approach, satisfying both minimal length and maximal momentum. The resulting invariant phase space volumes from these GUP approaches are applied to the [Formula: see text]-[Formula: see text] or Walecka model, serving as a starting point for neutron matter in the relativistic mean field theory. We find that linear GUP increases the range of energy densities corresponding to instabilities in the [Formula: see text]-[Formula: see text] EoS, while quadratic GUP decreases it. A stable EoS was constructed from the GUP-modified EoS via Maxwell construction, and this was fed into the Tolman–Oppenheimer–Volkoff equations and the mass–radius relation of neutron stars was obtained. We observe linear GUP to decrease both the maximum mass and limiting radius of the neutron star, while shifting the whole mass–radius relation to the low-radius regime. Meanwhile, quadratic GUP increases the maximum mass and limiting radius, and the mass-radius relation is shifted to the high-radius regime. The effects that are observed for both GUP modifications in the EoS and mass–radius relations get more prominent as we increase the values of the still unknown GUP parameters.

We investigate shadows, deflection angle, quasinormal modes (QNMs), and sparsity of Hawking radiation of the Schwarzschild string cloud black hole’s solution after applying quantum corrections required by the Generalised Uncertainty Principle (GUP). First, we explore the shadow’s behaviour in the presence of a string cloud using three alternative GUP frameworks: linear quadratic GUP (LQGUP), quadratic GUP (QGUP), and linear GUP. We then used the weak field limit approach to determine the effect of the string cloud and GUP parameters on the light deflection angle, with computation based on the Gauss–Bonnet theorem. Next, to compute the quasinormal modes of Schwarzschild string clouds incorporating quantum correction with GUP, we determine the effective potentials generated by perturbing scalar, electromagnetic and fermionic fields, using the sixth-order WKB approach in conjunction with the appropriate numerical analysis. Our investigation indicates that string and linear GUP parameters have distinct and different effects on QNMs. We find that the greybody factor increases due to the presence of string cloud while the linear GUP parameter shows the opposite. We then examine the radiation spectrum and sparsity in the GUP corrected black hole with the cloud of string framework, which provides additional information about the thermal radiation released by black holes. Finally, our inquiries reveal that the influence of the string parameter and the quadratic GUP parameter on various astrophysical observables is comparable, however the impact of the linear GUP parameter is opposite.

Significant evidence is available to support the quantum effects of gravity that lead to the generalized uncertainty principle (GUP) and the minimum observable length. Usually in quantum mechanics, statistical physics does not take gravity into account. Thermodynamic properties of ideal Bose gases in different external power-law potentials are studied under the GUP with a statistical physical method. Critical temperature, internal energy, heat capacity, entropy, particle number of ground state and excited state are calculated analytically to ideal Bose gases in the external potentials under the GUP. Below the critical temperature, taking the rubidium and sodium atoms, ideal Bose gases whose particle densities are under standard and experimental conditions, respectively, as examples, the relations of internal energy, heat capacity and entropy with temperature are analyzed numerically. Theoretical and numerical calculations show that: (1) the GUP leads to an increase in the critical temperature. (2) When the temperature is lower than the critical temperature and slightly higher than 0 K, the GUP’s amendments to internal energy, heat capacity and entropy etc are positive. As the temperature increases to a certain value, these amendments become negative. (3) The external potentials can increase or decrease the influence of the GUP on thermodynamic properties. When ε = 1 J, ε is the quantity that reflects the external potential intensity, and atomic density n = 2.687 × 1025 m−3, the GUP’s amendments to the internal energy, heat capacity and entropy of the rubidium atoms ideal Bose gas first decrease and then increase with the increase of X (where X ≡ Σi1/ti is sum of the reciprocal of the exponents of the power function). In three-dimensional harmonic potential, the relative correction term of the GUP is 26 orders of magnitude larger than that of a free-particle system in a fixed container. (4) When ε ≈10−31 J and n ≈ 1020 m−3 (which are the experimental data when BEC was first verified by sodium atomic gas), the influence of the GUP can be completely ignored. (5) Under certain conditions, GUP may become the dominant factor governing the thermodynamic properties of the system.

Various frameworks of quantum gravity predict a modification in the Heisenberg uncertainty principle to a so-called generalized uncertainty principle (GUP). Introducing quantum gravity effect makes a considerable change in the density of states inside the volume of the phase space which changes the statistical and thermodynamical properties of any physical system. In this paper we investigate the modification in thermodynamic properties of ideal gases and photon gas. The partition function is calculated and using it we calculated a considerable growth in the thermodynamical functions for these considered systems. The growth may happen due to an additional repulsive force between constitutes of gases which may be due to the existence of GUP, hence predicting a considerable increase in the entropy of the system. Besides, by applying GUP on an ideal gas in a trapped potential, it is found that GUP assumes a minimum measurable value of thermal wavelength of particles which agrees with discrete nature of the space that has been derived in previous studies from the GUP.

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.

We review on further new developments of Generalized Uncertainty Principle (GUP) and implications for the cosmological vacuum energy. First, we introduce basic aspects of GUP as well as several possible different and viable formulation of it. Second, we move on discussing two recent new types of higher D-dimensional nonperturbative GUP models; which we dub D-Type-I and D-Type-II GUPs. The D-Type-I and D-Type-II GUPs are both related to the existence of a critical conspiracy between a minimal uncertainty length and a maximal observable momentum. Finally, we show direct implications of D-Type-I and D-Type-II on the cosmological vacuum energy obtained in quantum mechanical systems such as the typical quantum harmonic oscillator. Such a computation goes through investigations of the density of states for D-dimensional coordinate systems in the momentum space. We will also comment on several possible connections with fundamental issues of quantum gravity such as black hole physics and gravitational radiative aspects.

Inspired by string theory, Heisenberg's uncertainty principle can be generalized to include the photon-electron gravitational interaction, which leads to the Generalized Uncertainty Principle (GUP). Although GUP considers gravitational uncertainty at the minimum fundamental length scale in physics, it does not consider the effects of spacetime curvature on quantum mechanical uncertainty relations. The Extended Uncertainty Principle (EUP) is a generalization of Heisenberg's Uncertainty Principle that, unlike the GUP, applies to large length scales. GEUP is also a linear combination of EUP and GUP that creates minimal uncertainty on large length scales. The Einstein-Gauss-Bonnet theory (EGB) can be considered as one of the most promising candidates for modified gravity. In this paper, by using GUP, EUP, and GEUP, we intend to obtain the Hawking temperature of a four-dimensional EGB black hole in the asymptotically flat and (Anti)-de Sitter spacetime. We show that coupling constant, cosmological constant, mass, and radius significantly affect Hawking temperature and decrease or increase Hawking temperature depending on the chosen horizons.

When gravitation is combined with quantum theory, the Heisenberg uncertainty principle could be extended to the generalized uncertainty principle accompanying a minimal length. To see how the generalized uncertainty principle works in the context of black hole complementarity, we calculate the required energy to duplicate information for the Schwarzschild black hole. It shows that the duplication of information is not allowed and black hole complementarity is still valid even assuming the generalized uncertainty principle. On the other hand, the generalized uncertainty principle with the minimal length could lead to a modification of the conventional dispersion relation in light of Gravity's Rainbow, where the minimal length is also invariant as well as the speed of light. Revisiting the gedanken experiment, we show that the no-cloning theorem for black hole complementarity can be made valid in the regime of Gravity's Rainbow on a certain combination of parameters.

Diverse theories of quantum gravity expect modifications of the Heisenberg's uncertainty principle near the Planck scale to a so-called Generalized uncertainty principle (GUP). It was shown by some authors that the GUP gives rise to corrections to the Schrodinger , Klein-Gordon, and Dirac equations. By solving the GUP corrected equations, the authors arrived at quantization not only of energy but also of box length, area, and volume. In this paper, we extend the above results to the case of curved spacetime (Schwarzschild metric). We showed that we arrived at the quantization of space by solving Dirac equation with GUP in this metric.

In quantum gravity theories, when the scattering energy is comparable to the Planck energy, the usual Heisenberg uncertainty principle breaks down and is replaced by generalized uncertainty principle (GUP). In this paper, the Dirac equation is studied for a single particle with spin and pseudospin symmetry in the presence of GUP, in [Formula: see text] dimensions. For arbitrary wave [Formula: see text], the Dirac equation with multiparameter exponential-type potential is solved by applying the approximation of the centrifugal term and the Nikiforov–Uvarov method. The corresponding energy spectra and eigenvalue function are obtained in the closed form and depend on the GUP parameter. In addition, several interesting cases have been discussed.

We first briefly revisit the original Hamilton-Jacobi method and show that the Hamilton-Jacobi equation for the action I of tunneling of a fermionic particle from a charged black hole can be written in the same form of that for a scalar particle. On the other hand, various theories of quantum gravity suggest the existence of a minimal length scale, incorporating of which into quantum mechanics implies a modification of the uncertainty principle. In the scenario incorporating the generalized uncertainty principle (GUP) into a quantum field theory (QFT) in a covariant way, we derive the deformed model-independent KG/Dirac and Hamilton-Jacobi equations using the methods of effective field theory. For this Lorentz invariant GUP modified QFT, we find that the effect of GUP on the Hamilton-Jacobi equations is simply to “renormalize” the mass of the emitted particles, from m to meff. Therefore, in this scenario, the Hawking temperature of a black hole does not receive any corrections from the GUP effect.

The idea of a fundamental scale can be formalized modifying the canonical uncertainty principle as (we adopt units such that ~ = c = 16πG = 1) ∆q∆p ≥ 1 2 (1 + β(∆p) ), where β > 0. This is the so-called generalized uncertainty principle (GUP) appeared in string theory. From the string theory point of view, the relation above is a consequence of the fact that strings can not probe distances below the string scale. The GUP implies a minimal uncertainty in the position ∆q0 = √ β. Here we describe the dynamics of the Bianchi cosmological models in the GUP framework reviewing the results of. The GUP approach has been previously implemented to the FRW model filled with a massless scalar field as well as to the Taub Universe. The Bianchi Universes are spatially homogeneous cosmological models and their dynamics, taking into account the ADM reduction of the dynamics, is given by the constraint

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