A higher order GUP with minimal length uncertainty and maximal momentum II: Applications

In a recent paper, we presented a nonperturbative higher order Generalized Uncertainty Principle (GUP) that is consistent with various proposals of quantum gravity such as string theory, loop quantum gravity, doubly special relativity, and predicts both a minimal length uncertainty and a maximal observable momentum. In this Letter, we find exact maximally localized states and present a formally self-adjoint and naturally perturbative representation of this modified algebra. Then we extend this GUP to D dimensions that will be shown it is noncommutative and find invariant density of states. We show that the presence of the maximal momentum results in upper bounds on the energy spectrum of the free particle and the particle in box. Moreover, this form of GUP modifies blackbody radiation spectrum at high frequencies and predicts a finite cosmological constant. Although it does not solve the cosmological constant problem, it gives a better estimation with respect to the presence of just the minimal length.     

Quantum black hole and the modified uncertainty principle

Recently Ali et al. (2009) [13] proposed a Generalized Uncertainty Principle (or GUP) with a linear term in momentum (accompanied by Planck length). Inspired by this idea we examine the Wheeler-DeWitt equation for a Schwarzschild black hole with a modified Heisenberg algebra which has a linear term in momentum. We found that the leading contribution to mass comes from the square root of the quantum number n which coincides with Bekenstein?s proposal. We also found that the mass of the black hole is directly proportional to the quantum number n when quantum gravity effects are taken into consideration via the modified uncertainty relation but it reduces the value of mass for a particular value of the quantum number.     

Discreteness of space from the generalized uncertainty principle

Various approaches to Quantum Gravity (such as String Theory and Doubly Special Relativity), as well as black hole physics predict a minimum measurable length, or a maximum observable momentum, and related modifications of the Heisenberg Uncertainty Principle to a so-called Generalized Uncertainty Principle (GUP). We propose a GUP consistent with String Theory, Doubly Special Relativity and black hole physics, and show that this modifies all quantum mechanical Hamiltonians. When applied to an elementary particle, it implies that the space which confines it must be quantized. This suggests that space itself is discrete, and that all measurable lengths are quantized in units of a fundamental length (which can be the Planck length). On the one hand, this signals the breakdown of the spacetime continuum picture near that scale, and on the other hand, it can predict an upper bound on the quantum gravity parameter in the GUP, from current observations. Furthermore, such fundamental discreteness of space may have observable consequences at length scales much larger than the Planck scale.     

Feynman’s Proof of Maxwell Equations: in?the?Context?of Quantum Gravity

Many of us are familiar with Feynman’s “proof” of 1948, as revealed by Dyson, which demonstrates that Maxwell equations of electromagnetism are a consequence of Newton’s laws of motion of classical mechanics and the commutation relations of coordinate and momentum of quantum mechanics. It was Feynman’s purpose to explore the universality of dynamics of particles while making the fewest assumptions. We re-examine this formulation in the context of quantum gravity and show how Feynman’s derivation can be extended to include quantum gravity.     

Charged Fermions Tunnel from the Kerr-Newman Black Hole Influenced by Quantum Gravity Effects

Taking into account quantum gravity effects, we investigate the tunnelling radiation of charged fermions in the Kerr-Newman black hole. The result shows that the corrected Hawking temperature is determined not only by the parameters of the black hole, but also by the energy, angular momentum and mass of the emitted fermion. Meanwhile, an interesting found is that the temperature is affected by the angle ??. The quantum gravity correction slows down the evaporation.     

Minimal length in quantum mechanics and non-Hermitian Hamiltonian systems

Deformations of the canonical commutation relations lead to non-Hermitian momentum and position operators and therefore almost inevitably to non-Hermitian Hamiltonians. We demonstrate that such type of deformed quantum mechanical systems may be treated in a similar framework as quasi/pseudo and/or -symmetric systems, which have recently attracted much attention. For a newly proposed deformation of exponential type we compute the minimal uncertainty and minimal length, which are essential in almost all approaches to quantum gravity.     

Quantization of Space in the Presence of a Minimal Length

In this article,we apply the Generalized Uncertainty Principle(GUP),which is consistent with quantum gravity theories to an elementary particle in a finite potential well,and study the quantum behavior in this system.The generalized Hamiltonian contains two additional terms,which are proportional to αp~3(the result of the maximum momentum assumption) and α~2p~4(the result of the minimum length assumption),where α ~ 1/M_(PIC) is the GUP parameter.On the basis of the work by Ali et al.,we solve the generalized Schrodinger equation which is extended to include the α~2 correction term,and find that the length L of the finite potential well must be quantized.Then a generalization to the double-square-well potential is discussed.The result shows that all the measurable lengths especially the distance between the two potential wells are quantized in units of α_0l_(PI) in GUP scenario.     

Maximally incompatible quantum observables

The existence of maximally incompatible quantum observables in the sense of a minimal joint measurability region is investigated. Employing the universal quantum cloning device it is argued that only infinite dimensional quantum systems can accommodate maximal incompatibility. It is then shown that two of the most common pairs of complementary observables (position and momentum; number and phase) are maximally incompatible.     

Overspinning a nearly extreme charged black hole via a quantum tunneling process

We examine a nearly extreme macroscopic Reissner-N?rdstrom black hole in the context of semiclassical gravity. The absorption rate associated with the quantum tunneling process of scalar particles whereby this black hole can acquire enough angular momentum to violate the weak cosmic-censorship conjecture is shown to be nonzero.     

Corrections to the Hawking Tunneling Radiation from MDR

We investigate some aspects of black hole (BH) thermodynamics in the framework of a modified dispersion relation. We calculate a minimal length and a maximal momentum to find a relation between spacetime dimensions and the presence of logarithmic prefactor in the black hole entropy relation. We show that the logarithmic prefactor appears not only in an even number of dimensions but also in an odd number of dimensions. In addition, the sign of the logarithmic factor is different for positive values of α in all dimensions. Using the corrected entropy, the black hole radiation probability is calculated in the tunneling formalism, which is corrected up to the same order of the Planck length and shows a more probable quantum tunneling.     

Lorentz invariance violation and generalized uncertainty principle

There are several theoretical indications that the quantum gravity approaches may have predictions for a minimal measurable length, and a maximal observable momentum and throughout a generalization for Heisenberg uncertainty principle. The generalized uncertainty principle (GUP) is based on a momentum-dependent modification in the standard dispersion relation which is conjectured to violate the principle of Lorentz invariance. From the resulting Hamiltonian, the velocity and time of flight of relativistic distant particles at Planck energy can be derived. A first comparison is made with recent observations for Hubble parameter in redshift-dependence in early-type galaxies. We find that LIV has two types of contributions to the time of flight delay Δt comparable with that observations. Although the wrong OPERA measurement on faster-than-light muon neutrino anomaly, Δt, and the relative change in the speed of muon neutrino Δv in dependence on redshift z turn to be wrong, we utilize its main features to estimate Δv. Accordingly, the results could not be interpreted as LIV. A third comparison is made with the ultra high-energy cosmic rays (UHECR). It is found that an essential ingredient of the approach combining string theory, loop quantum gravity, black hole physics and doubly spacial relativity and the one assuming a perturbative departure from exact Lorentz invariance. Fixing the sensitivity factor and its energy dependence are essential inputs for a reliable confronting of our calculations to UHECR. The sensitivity factor is related to the special time of flight delay and the time structure of the signal. Furthermore, the upper and lower bounds to the parameter, a that characterizes the generalized uncertainly principle, have to be fixed in related physical systems such as the gamma rays bursts.     

Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras

Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one cyclic representation invariant under spatial diffeomorphisms.While this result is particularly important for loop quantum gravity, we are rather general: The precise definition of the abstract *-algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group. The variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection. The uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory.     

Quantum Gravity Corrections to the Tunneling Radiation of Scalar Particles

The original derivation of Hawking radiation shows the complete evaporation of black holes. However, theories of quantum gravity predict the existence of the minimal observable length. In this paper, we investigate the tunneling radiation of the scalar particles by introducing the quantum gravity effects influenced by the generalized uncertainty principle. The Hawking temperatures are not only determined by the properties of the black holes, but also affected by the quantum numbers of the emitted particles. The quantum gravity corrections slow down the increase of the temperatures. The remnants are found during the evaporation.     

The Vilkovisky-DeWitt effective action for quantum gravity

The Vilkovisky-DeWitt effective action for gauge theories is reviewed and then discussed in the context of N-dimensional quantum gravity and quantum Kaluza-Klein theory. The formalism gives an effective action which is gauge-independent and gauge and field-parametrization invariant. These features are illustrated for the vacuum energy of N-dimensional gravity. The Bunch-Parker local momentum space approach is used to calculate also the induced Ricci scalar term in the expansion of the effective action in powers of the curvature. The effective field equations are applied to the self-consistent dimensional reduction of five-dimensional Kaluza-Klein theory. A solution exists, but is found to be physically unacceptable.     

Quantum regime for dielectric Cherenkov radiation and graphene surface plasmons

In this article, we propose a quantum regime for Cherenkov free-electron laser (CFEL) and surface plasmon polaritons (SPPs) excited in dielectric and multilayer graphene waveguides, respectively. This quantum regime is realized when the momentum spread induced in the interaction is smaller than the photon recoil. The discrete momentum exchange characterizing this interaction yields a significantly narrow single emission line. To determine the condition of the quantum regime, we derive an expression for the gain in the Cherenkov effect using a quantum mechanical treatment. It is assumed that the effective spread in momentum is due to the finite interaction length L (or the propagation length in the case of SPPs). For both cases, CFEL and SPPs, the effects of electron beam and waveguide parameters on the possibility of the quantum regime are studied. We conclude that the quantum regime can be basically verified at low electron beam energy (<40 keV) and at emission wavelengths in the near infrared range (<5 μm) when L is in the order of millimeters. In the case of SPPs, we also show that the feasibility to realize quantum SPPs is enhanced by increasing the chemical potential and number of graphene layers.     

Polarized vector bosons on the de Sitter expanding universe

The quantum theory of the vector field minimally coupled to the gravity of the de Sitter spacetime is built in a canonical manner starting with a new complete set of quantum modes of given momentum and helicity derived in the moving chart of conformal time. It is shown that the canonical quantization leads to new vector propagators which satisfy similar equations as the propagators derived by Tsamis and Woodard (J Math Phys 48:052306, 2007) but having a different structure. The one-particle operators are also written down pointing out that their properties are similar with those found already in the quantum theory of the scalar, Dirac and Maxwell free fields.     

Internal symmetry groups of quantum geons

In canonical quantum gravity asymptotically trivial diffeomorphisms not deformable to the identity can act nontrivially on the quantum state space. We show that for many 3-manifolds, the inequivalent diffeomorphisms comprise coverings in SU(2) of crystallographic groups. When the diffeomorphism R_2π associated with 2π-rotation is nontrivial, state vectors can have half-integral angular momentum; we list all 3-manifolds with R_2π trivial.     

Quantum groups and quantum field theory with nonzero minimal uncertainties in positions and momenta

Various regularisation techniques are presently emerging from the field of noncommutative geometry. We focus on the possibility of introducing nonzero minimal uncertainties in positions and momenta into the quantum field theoretical framework. Thereby we use techniques that have been developed in the field of quantum groups. The regularisation of a quadratic ultraviolet divergency inφ ⁴ theory is given explicitely. We also discuss the underlying idea that such uncertainties in position or momentum measurements originate in gravity.     

Quantum gravity effect on the Hawking radiation of charged rotating BTZ black hole

In this study, the quantum gravity effect on the tunnelling radiation of charged massive spin-0 scalar particle from \(2+1\) dimensional charged rotating Banados–Teitelboim–Zanelli (BTZ) black hole is looked into by using the Hamilton–Jacobi approach. For this, we calculate the modified Hawking temperature of the black hole by using the modified Klein–Gordon equation based on the generalized uncertainty principle, and we noticed that the modified Hawking temperature of the black hole depends not only on the black hole properties, but also on the angular momentum, energy, charge and mass of the tunnelling scalar particle. Using the modified Hawking temperature, we discussed the stability of the black hole in the context of the modified heat capacity, and observed that it might undergo both first and 1 phase transitions in the presence of the quantum gravity effect, but just a first-type transition in the absence of the quantum gravity effect. Furthermore, we investigated the modified Hawking temperature of the black hole by using the tunnelling processes of the charged massive Dirac and vector boson particles. We observed that scalar, Dirac and vector particles are tunnelled from the black hole completely differently from each other in the presence of the quantum gravity effect.     

Half-integral spin from quantum gravity

For a certain class of three-manifolds, the angular momentum of an asymptotically flat quantum gravitational field can have half-integral values. In the absence of a full theory of quantum gravity, this result relies on a set of apparently natural assumptions governing the kinematics of such a theory. A key feature is that state vectors are in general invariant only under asymptotically trivial diffeomorphisms that can be continuously deformed to the identity. Angular momentum is associated with diffeomorphisms that look asymptotically like rotations; and the question of whether half-integral values occur depends on whether the diffeomorphism associated with a 2 rotation is itself deformable to the identity.This essay received the fourth award from the Gravity Research Foundation for the year 1981-Ed.