Generalized Uncertainty Principle Research Articles

On the generalized uncertainty principle and cosmology

In this work, we study the effects of the Generalized Uncertainty Principle (GUP) in cosmology. We start with the Friedmann–Robertson–Walker (FRW) model endowed with a scalar field. After introducing the GUP modification to the model, we solve for the quantum and classical cases. Finally, we find the GUP-modified Friedmann equations.

Duality in spontaneous broken time translation symmetry: Sisyphus dynamics and the Liénard equation

In this paper, we establish a connection between Sisyphus dynamics and the Liénard-II equation through branched Hamiltonians. Sisyphus dynamics stem from a higher-order Lagrangian. Surprisingly, when expressed in terms of velocity, the Sisyphus dynamical equations align closely with the Liénard-II equation. Sisyphus dynamics introduces velocity-dependent “mass functions”, a departure from conventional position-dependent mass, potentially linked to cosmological time crystals. Additionally, we demonstrate that spontaneously broken time translational symmetry results in a deformed symplectic structure, resembling the classical counterpart of the Generalized Uncertainty Principle (GUP).

Quantum gravity effects on the tachyon inflation from thermodynamic perspective

By considering the Friedmann equations emerging from the entropy-area law of black hole thermodynamics in the context of the generalized uncertainty principle, we study tachyon inflation in the early universe. The presence of a minimal length modifies the Friedmann equations and hence the slow-roll and perturbation parameters in the tachyon model. These modifications, though small, affect the viability of the tachyon inflation in confrontation with observational data. We compare the numerical results of the model with Planck2018 TT, TE, EE +lowE+lensing+BAO+BK14(18) data and Planck2018 TT, TE,EE +lowE+lensing+BK14(18) +BAO+LIGO & Virgo2016 data at 68% and 95% CL. We show that while the tachyon inflation with power-law, inverse power-law and inverse exponential potentials is not observationally viable in comparison with the 1σ and 2σ confidence levels of the new joint data, in the presence of the minimal length the model becomes observationally viable.

New models of d-dimensional black holes without inner horizon and with an integrable singularity

Theoretically, it has been proposed that objects traveling radially along regular black holes (RBHs) would not be destroyed because of finite tidal forces and the absence of a singularity. However, the matter source allows the creation of an inner horizon linked to an unstable de Sitter core due to mass inflation instability. This inner horizon also gives rise to the appearance of a remnant, inhibiting complete evaporation. We introduce here a d-dimensional black hole model with Localized Sources of Matter (LSM), characterized by the absence of an inner horizon and featuring a central integrable singularity instead of an unstable de Sitter core. In our model, any object tracing a radial and timelike world-line would not be crushed by the singularity. This is attributed to finite tidal forces, the extendability of radial geodesics, and the weak nature of the singularity. Our LSM model enables the potential complete evaporation down to r h = 0 without forming a remnant. In higher dimensions, complete evaporation occurs through a phase transition, which could occur at Planck scales and be speculatively driven by the Generalized Uncertainty Principle (GUP). Unlike RBHs, our model satisfies the energy conditions. We demonstrate a linear correction to the conventional area law of entropy, distinct from the RBH's correction. Additionally, we investigate the stability of the solutions through the speed of sound.

High-temperature modifications of charged Casimir wormholes

In this work, we extend the investigation of the consequences of thermal fluctuations on the Casimir effect within the context of a traversable wormhole, recently proposed by Garattini & Faizal, arXiv:2403.15174 [gr-qc], subject to charge contributions. Specifically, we focus on scenarios where the plates exhibit both constant and radial variations. In our analysis, we initially concentrate on the high temperature approximation, considering solely the influence of charge on the thermal Casimir wormholes. Additionally, upon incorporating Generalized Uncertainty Principle (GUP) corrections to the Casimir energy, we obtain a new class of wormhole solutions. Notably, we establish that the flare-out condition remains consistently satisfied. Intriguingly, our findings reveal that both the charge and GUP contributions serve to further enlarge the throat's size in the radial variation.

Bose and Fermi Gases in Metric-Affine Gravity and Linear Generalized Uncertainty Principle

Palatini-like theories of gravity have a remarkable connection to models incorporating linear generalized uncertainty principles. Considering this, we delve into the thermodynamics of systems comprising both Bose and Fermi gases. Our analysis encompasses the equations of state for various systems, including general Fermi gases, degenerate Fermi gases, Boltzmann gases, and Bose gases such as phonons and photons, as well as Bose–Einstein condensates and liquid helium.

Relativistic generalised uncertainty and the corrected vacuum energy

Abstract In this paper we make use of the recent covariant extension of the generalised uncertainty principle to study the corrections to the vacuum energy of the simplest scalar quantum field theory. We then calculate the modifications to the Casimir effect that such a correction to the vacuum energy would bring about. We emphasise that these corrections are indeed physical.

Resonant detectors of gravitational wave in the linear and quadratic generalized uncertainty principle framework

In this work, we consider a resonant bar detector of gravitational waves in the generalized uncertainty principle (GUP) framework with linear and quadratic momentum uncertainties. The phonon modes in these detectors vibrate due to the interaction with the incoming gravitational wave. In this uncertainty principle framework, we calculate the resonant frequencies and transition rates induced by the incoming gravitational waves on these detectors. We observe that the energy eigenstates and the eigenvalues get modified by the GUP parameters. We also observe non-vanishing transition probabilities between two adjacent energy levels due to the existence of the linear order momentum correction in the generalized uncertainty relation which was not present in the quadratic GUP analysis (Class Quantum Gravity 37:195006, 2020). We finally obtain bounds on the dimensionless GUP parameters using the form of the transition rates obtained during this analysis.

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Inspired by the generalization of scalar field gravitational models with a minimum length we study the equivalent theory in modified theories of gravity. The quadratic generalized uncertainty principle (GUP) gives rise to a deformed Heisenberg algebra in the application, resulting in the emergence of additional degrees of freedom described by higher-order derivatives. The new degrees of freedom can be attributed to the introduction of a new scalar field, transforming the resulting theory into a two-scalar field theory. Thus, in order to describe all the degrees of freedom we investigate special forms of the sixth-order modify fR,□R-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R,\\square R\\right) -$$\\end{document}theory of gravity, where the gravitational Lagrangian has similar properties to that of the GUP scalar field theory. Finally, the cosmological applications are discussed, and we show that the de Sitter universe can be recovered without introducing a cosmological constant.

Thermodynamics of harmonic oscillator with minimal length

The generalized uncertainty principle has been applied to the Schrödinger wave equation for a one-dimensional harmonic oscillator to generate a sixth-order generalized Schrödinger equation in the position representation. The energy eigenvalues and the eigenfunctions of the sixth-order equation have been obtained. The results show the approximate correction terms of the energies due to the modified uncertainty principle. The quantum partition functions derived from the energy eigenvalue have also been used to study the thermodynamic properties of the system. The results suggest a lower bound for the minimal length equivalent to the thermal wavelength of the oscillator at very high temperature.

Weak cosmic censorship conjecture and black hole shadow for black hole with generalized uncertainty principle

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.

A covariant tapestry of linear GUP, metric-affine gravity, their Poincaré algebra and entropy bound

Motivated by the potential connection between metric-affine gravity and linear generalized uncertainty principle (GUP) in the phase space, we develop a covariant form of linear GUP and an associated modified Poincaré algebra, which exhibits distinctive behavior, nearing nullity at the minimal length scale proposed by linear GUP. We use three-torus geometry to visually represent linear GUP within a covariant framework. The three-torus area provides an exact geometric representation of Bekenstein’s universal bound. We depart from Bousso’s approach, which adapts Bekenstein’s bound by substituting the Schwarzschild radius ( rs ) with the radius (R) of the smallest sphere enclosing the physical system, thereby basing the covariant entropy bound on the sphere’s area. Instead, our revised covariant entropy bound is described by the area of a three-torus, determined by both the inner radius rs and outer radius R where rs⩽R due to gravitational stability. This approach results in a more precise geometric representation of Bekenstein’s bound, notably for larger systems where Bousso’s bound is typically much larger than Bekensetin’s universal bound. Furthermore, we derive an equation that turns the standard uncertainty inequality into an equation when considering the contribution of the three-torus covariant entropy bound, suggesting a new avenue of quantum gravity.

Casimir wormhole with GUP correction in extended symmetric teleparallel gravity

Quantum mechanical concept such as the Casimir effect is explored to model traversable wormholes in an extended symmetric teleparallel gravity theory. The minimal length concept leading to the generalized uncertainty principle (GUP) is used to obtain the Casimir energy density. The effect of the GUP correction in the geometrical and physical properties of traversable Casimir wormholes are investigated. It is noted that the GUP correction has a substantial effect on the wormhole geometry and it modifies the energy condition. From a detailed calculation of the exotic matter content of the GUP corrected Casimir wormhole, it is shown that, a minimal amount of exotic matter is sufficient to support the stability of the wormhole.

On some quantum correction to the Coulomb potential in generalized uncertainty principle approach

Taking into account the importance of the unified theory of quantum mechanics and gravity, and the existence of a minimal length of the order of the Planck scale, we consider a modified Schrödinger equation resulting from a generalized uncertainty principle, which finds applications from the realm of quantum information to large-scale physics, with a quantum mechanically corrected gravitational interaction proposed very recently. As the resulting equation cannot be solved by common exact approaches, we propose a Bethe ansatz approach, which will be applied and whose results we will discuss, commenting on the analogy of the present study with some other interesting physical problems.

First-order quantum correction of thermodynamics in a charged accelerating AdS black hole with gauge potential

In this paper, we study the tunneling radiation from a charged-accelerating AdS black hole with gauge potential under the impact of quantum gravity. Using the semi-classical phenomenon known as the Hamilton–Jacobi ansatz, it is studied that tunneling radiation occurs via the horizon of a black hole and also employs the Lagrangian equation using the generalized uncertainty principle. Furthermore, we investigate the impact of charge, gauge potential, and first order correction parameters on the temperature as well as the stable and unstable states of the black hole. We also compute thermodynamic properties such as entropy, internal energy, Helmholtz free energy, enthalpy, specific heat, and Gibbs free energy under the impact of the correction parameter for the black hole. We calculate the logarithmic modification terms for entropy around the equilibrium state to analyze the impacts of logarithmic correction. In the presence of the correction terms, we also check the validity of the thermodynamics. It examines the graphical representation of the influence of logarithmic correction on the thermodynamic properties of black hole stability as well as charged, accelerating, and gauge potential parameters.

Non-singular gravitational collapse through modified Heisenberg algebra

We study the effects of cut-off physics, in the form of a modified algebra inspired by Polymer Quantum Mechanics and by the Generalized Uncertainty Principle representation, on the collapse of a spherical dust cloud. We analyze both the Newtonian formulation, originally developed by Hunter, and the general relativistic formulation, that is the Oppenheimer–Snyder model; in both frameworks we find that the collapse is stabilized to an asymptotically static state above the horizon, and the singularity is removed. In the Newtonian case, by requiring the Newtonian approximation to be valid, we find lower bounds of the order of unity (in Planck units) for the deformation parameter of the modified algebra. We then study the behaviour of small perturbations on the non-singular collapsing backgrounds, and find that for certain range of the parameters (the polytropic index for the Newtonian case and the sound velocity in the relativistic setting) the collapse is stable to perturbations of all scales, and the non-singular super-Schwarzschild configurations have physical meaning.

Conditional Effects, Observables and Instruments

We begin with a study of operations and the effects they measure. We define the probability that an effect $a$ occurs when the system is in a state $\rho$ by $P_{\rho}(a)=\textrm{Tr}(\rho a)$. If $P_{\rho}(a)\ne0$ and $\mathcal{I}$ is an operation that measures $a$, we define the conditional probability of an effect $b$ given $a$ relative to $\mathcal{I}$ by $P_{\rho}(b\mid a)=\textrm{Tr}[\mathcal{I}(\rho)b]/P_{\rho}(a)$. We characterize when Bayes' quantum second rule $P_{\rho}(b\mid a)=\frac{P_{\rho}(b)}{P_{\rho}(a)}\,P_{\rho}(a\mid b)$ holds. We then consider Lüders and Holevo operations. We next discuss instruments and the observables they measure. If $A$ and $B$ are observables and an instrument $\mathcal{I}$ measures $A$, we define the observable $B$ conditioned on $A$ relative to $\mathcal{I}$ and denote it by $(B\mid A)$. Using these concepts, we introduce Bayes' quantum first rule. We observe that this is the same as the classical Bayes' first rule, except it depends on the instrument used to measure $A$. We then extend this to Bayes' quantum first rule for expectations. We show that two observables $B$ and $C$ are jointly commuting if and only if there exists an atomic observable $A$ such that $B=(B\mid A)$ and $C=(C\mid A)$. We next obtain a general uncertainty principle for conditioned observables. Finally, we discuss observable conditioned quantum entropies. The theory is illustrated with many examples.Quanta 2024; 13: 1–10.

Uncertainty principle from the noise of gravitons

The effect of the noise induced by gravitons in the case of a freely falling particle from the viewpoint of an external observer has been recently calculated in Phys. Rev. D 107, 066024 (2023). There the authors have calculated the quantum gravity modified Newton’s law of free fall where the spacetime has been considered to be weakly curved. In our work, we extend this work by calculating the variance in the velocity and eventually the momentum of the freely falling massive particle. From this simple calculation, we observe that the product of the standard deviation in the position with that of the standard deviation in momentum picks up a higher order correction which is proportional to the square of the standard deviation in momentum. We also find out that in the Planck limit (both Planck length and Planck mass), this uncertainty product gives the well-known form of the generalized uncertainty principle. We then calculate a similar uncertainty product when the graviton is in a squeezed state, and eventually, we get back the same uncertainty product. Finally, we extend our analysis for the gravitons being in a thermal state and obtain a temperature-dependent uncertainty product. If one replaces this temperature with the Planck temperature and the mass of the particle by the Planck mass, the usual uncertainty product appears once again. We also obtain an upper bound of the uncertainty product thereby giving a range of the product of the variances in position and momentum.

Black hole evaporation and its remnants with the generalized uncertainty principle including a linear term

In recent years, researchers have investigated the evaporation of Schwarzschild black holes using various forms of the generalized uncertainty principle (GUP), metric quantum correction, and non-commutative geometry, respectively. However, there are differences between the GUP correction and the other two methods in terms of describing the later stages of black hole evaporation. Furthermore, some studies argue that the GUP with a negative parameter cannot effectively correct black hole evaporation, while others contend that the positivity or negativity of the GUP parameters should not affect the correction results. Taking the above into consideration, we reconsider black hole evaporation with the generalized uncertainty principle including a linear term (LGUP), and examine the case of negative parameters. The results indicate that the evaporation behavior of both Schwarzschild black holes and Reissner–Nordström black holes, under LGUP correction, is consistent with the results of metric quantum correction and non-commutative geometry. Additionally, the negative parameter LGUP can also effectively correct for black hole evaporation.

Neutral current neutrino-electron scattering and generalized uncertainty principle

In this article, we study the effect of quantum gravitational corrections on neutrino electron scattering in electroweak theory SU(2) L × U(1) Y . We consider a neutrino that scatters off an electron by an exchange of a massive neutral gauge boson. In the minimal length formalism, starting from the equation of motion for a massive gauge field we obtain the propagator. And currents are obtained from the action for a massless Dirac fermion. Scattering amplitudes and differential cross sections as function of inelasticity variable y are obtained using expressions of the propagator and leptonic current. All usual expressions are recovered when the deformation parameter α tends to zero. We find that quantum gravitational effects become important only for a highly energetic process with a momentum transfer being much greater than the neutral gauge boson mass.