Nonperturbative effects of the minimal length uncertainty on the relativistic quantum mechanics

We study the nonperturbative effects of the minimal length on the energy spectrum of a relativistic particle in the context of the generalized uncertainty principle (GUP). This form of GUP is consistent with various candidates of quantum gravity such as string theory, loop quantum gravity, and black-hole physics and predicts a minimum measurable length proportional to the Planck length. Using a recently proposed formally self-adjoint representation, we solve the generalized Dirac and Klein–Gordon equations in various situations and find the corresponding exact energy eigenvalues and eigenfunctions. We show that for the Dirac particle in a box, the number of the solutions renders to be finite as a manifestation of both the minimal length and the theory of relativity. For the case of the Dirac oscillator and the wave equations with scalar and vector linear potentials, we indicate that the solutions can be obtained in a more simpler manner through the self-adjoint representation. It is also shown that, in the ultrahigh frequency regime, the partition function and the thermodynamical variables of the Dirac oscillator can be expressed in a closed analytical form. The Lorentz violating nature of the GUP-corrected relativistic wave equations is discussed finally.     

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.     

Some aspects of gravitational quantum mechanics

String theory, quantum geometry, loop quantum gravity and black hole physics all indicate the existence of a minimal observable length on the order of Planck length. This feature leads to a modification of Heisenberg uncertainty principle. Such a modified Heisenberg uncertainty principle is referred as gravitational uncertainty principle(GUP) in literatures. This proposal has some novel implications on various domains of theoretical physics. Here, we study some consequences of GUP in the spirit of Quantum mechanics. We consider two problem: a particle in an one-dimensional box and free particle wave function. In each case we will solve corresponding perturbational equations and compare the results with ordinary solutions.     

Generalized Uncertainty Principle in a Simple Varying Speed of Light Model

In this paperwewill derive a generalized uncertainty principle (GUP) in a simple varying speed of light (VSL) model. First we will show that VSL is an immediate consequence of GUP. Then, within the framework of a simple VSL model, we will show that GUP can be expressed as a function of cosmological scale factor. This expression gives two main results: uncertainties in position and momentum are actually cosmological models dependent and these uncertainties depend on mass and momentum of the particle under consideration. The relationship between matter content of the Universe and the values of uncertainties in early stages of the evolution of the Universe will be discussed in a mini-superspace approach.     

Why have we Never Observed the Massless Charged Particle?

In this paper we try to explore the possible contact between quantum gravity and the least mass of a charged particle in de Sitter spacetime. The effect of Generalized Uncertainty Principle (GUP) on the thermodynamics of de Sitter spacetime is discussed in a heuristic manner. We find a maximal entropy/probability that corresponds to the absence of charge of a massless particle. Furthermore, the holographic principle provides a possible lower limit to the mass of a charged particle. PACS Numbers: 04.70.Dy, 04.70.-s, 98.80.Es.     

Minimal Length, Maximal Momentum and the Entropic Force Law

Different candidates of quantum gravity proposal such as string theory, noncommutative geometry, loop quantum gravity and doubly special relativity, all predict the existence of a minimum observable length and/or a maximal momentum which modify the standard Heisenberg uncertainty principle. In this paper, we study the effects of minimal length and maximal momentum on the entropic force law formulated recently by E. Verlinde.     

Generalized Uncertainty Principle and Thermostatistics: A Semiclassical Approach

We present an exact treatment of the thermodynamics of physical systems in the framework of the generalized uncertainty principle (GUP). Our purpose is to study and compare the consequences of two GUPs that one implies a minimal length while the other predicts a minimal length and a maximal momentum. Using a semiclassical method, we exactly calculate the modified internal energies and heat capacities in the presence of generalized commutation relations. We show that the total shift in these quantities only depends on the deformed algebra not on the system under study. Finally, the modified internal energy for an specific physical system such as ideal gas is obtained in the framework of two different GUPs.     

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.     

Quantum theory of the generalised uncertainty principle

We extend significantly previous works on the Hilbert space representations of the generalized uncertainty principle (GUP) in 3 + 1 dimensions of the form \([X_i,P_j] = i F_{ij}\) where \(F_{ij} = f({\mathbf {P}}^2) \delta _{ij} + g({\mathbf {P}}^2) P_i P_j\) for any functions f. However, we restrict our study to the case of commuting X’s. We focus in particular on the symmetries of the theory, and the minimal length that emerge in some cases. We first show that, at the algebraic level, there exists an unambiguous mapping between the GUP with a deformed quantum algebra and a quadratic Hamiltonian into a standard, Heisenberg algebra of operators and an aquadratic Hamiltonian, provided the boost sector of the symmetries is modified accordingly. The theory can also be mapped to a completely standard Quantum Mechanics with standard symmetries, but with momentum dependent position operators. Next, we investigate the Hilbert space representations of these algebraically equivalent models, and focus specifically on whether they exhibit a minimal length. We carry the functional analysis of the various operators involved, and show that the appearance of a minimal length critically depends on the relationship between the generators of translations and the physical momenta. In particular, because this relationship is preserved by the algebraic mapping presented in this paper, when a minimal length is present in the standard GUP, it is also present in the corresponding Aquadratic Hamiltonian formulation, despite the perfectly standard algebra of this model. In general, a minimal length requires bounded generators of translations, i.e. a specific kind of quantization of space, and this depends on the precise shape of the function f defined previously. This result provides an elegant and unambiguous classification of which universal quantum gravity corrections lead to the emergence of a minimal length.     

Klein-Gordon-Wheeler-DeWitt-Schrödinger equation

We start from the Einstein-Hilbert action for the gravitational field in the presence of a “point particle” source, and cast the action into the corresponding phase space form. The dynamical variables of such a system satisfy the point particle mass shell constraint, the Hamilton and the momentum constraints of the canonical gravity. In the quantized theory, those constraints become operators that annihilate a state. A state can be represented by a wave functional Ψ that simultaneously satisfies the Klein-Gordon and the Wheeler-DeWitt-Schrödinger equation. The latter equation, besides the term due to gravity, also contains the Schrödinger like term, namely the derivative of Ψ with respect to time, that occurs because of the presence of the point particle. The particle?s time coordinate, X⁰, serves the role of time. Next, we generalize the system to p-branes, and find out that for a quantized spacetime filling brane there occurs an effective cosmological constant, proportional to the expectation value of the brane?s momentum, a degree of freedom that has two discrete values only, a positive and a negative one. This mechanism could be an explanation for the small cosmological constant that drives the accelerated expansion of the universe.     

Essay: Supersymmetry, the Cosmological Constant, and a Theory of Quantum Gravity in Our Universe

There are many theories of quantum gravity, depending on asymptotic boundary conditions, and the amount of supersymmetry. The cosmological constant is one of the fundamental parameters that characterizes different theories. If it is positive, supersymmetry must be broken. A heuristic calculation shows that a cosmological constant of the observed size predicts superpartners in the TeV range. This mechanism for SUSY breaking also puts important constraints on low energy particle physics models.     

Gravitational induced uncertainty and dynamics of harmonic oscillator

As a consequence of gravitational induced uncertainty, equation of motion for harmonic oscillator differs considerably from usual quantum mechanical situation. This paper considers the dynamics of a simple harmonic oscillator in the context of Generalized (Gravitational) Uncertainty Principle (GUP). Using Heisenberg Picture of quantum mechanics, we find time evolution of position and momentum operators and we will show that expectation values have an unusual complicated mass dependence. Also we will show that since the notion of locality breaks down, Ehrenfest theorem is not satisfied for harmonic oscillator in GUP.     

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.     

A mesoscopic quantum gravity effect

We explore the symmetry reduced form of a non-perturbative solution to the constraints of quantum gravity corresponding to quantum de Sitter space. The system has a remarkably precise analogy with the non-relativistic formulation of a particle falling in a constant gravitational field that we exploit in our analysis. We find that the solution reduces to de Sitter space in the semi-classical limit, but the uniquely quantum features of the solution have peculiar property. Namely, the unambiguous quantum structures are neither of Planck scale nor of cosmological scale. Instead, we find a periodicity in the volume of the universe whose period, using the observed value of the cosmological constant, is on the order of the volume of the proton.     

Black hole entropy and the modified uncertainty principle: A heuristic analysis

Recently Ali et al. (2009) proposed a Generalized Uncertainty Principle (or GUP) with a linear term in momentum (accompanied by Plank length). Inspired by this idea here we calculate the quantum corrected value of a Schwarzschild black hole entropy and a Reissner-Nordström black hole with double horizon by utilizing the proposed generalized uncertainty principle. We find that the leading order correction goes with the square root of the horizon area contributing positively. We also find that the prefactor of the logarithmic contribution is negative and the value exactly matches with some earlier existing calculations. With the Reissner-Nordström black hole we see that this model-independent procedure is not only valid for single horizon spacetime but also valid for spacetimes with inner and outer horizons.     

Black hole remnants due to GUP or quantum gravity?

Based on the micro-black hole gedanken experiment as well as on general considerations of quantum mechanics and gravity the generalized uncertainty principle (GUP) is analyzed by using the running Newton constant. The result is used to decide between the GUP and quantum gravitational effects as a possible mechanism leading to the black hole remnants of about Planck mass.     

Effects of quantum gravity on the inflationary parameters and thermodynamics of the early universe

The effects of generalized uncertainty principle (GUP) on the inflationary dynamics and the thermodynamics of the early universe are studied. Using the GUP approach, the tensorial and scalar density fluctuations in the inflation era are evaluated and compared with the standard case. We find a good agreement with the Wilkinson Microwave Anisotropy Probe data. Assuming that a quantum gas of scalar particles is confined within a thin layer near the apparent horizon of the Friedmann-Lemaitre-Robertson-Walker universe which satisfies the boundary condition, the number and entropy densities and the free energy arising form the quantum states are calculated using the GUP approach. A qualitative estimation for effects of the quantum gravity on all these thermodynamic quantities is introduced.     

Excitation spectrum and correlation functions of theZ3 chiral Potts quantum spin chain

We study the excitation spectrum and the correlation functions of theZ₃ chiral Potts model in the massive high-temperature phase using perturbation expansions and numerical diagonalization. We are mainly interested in results for general chiral angles but we consider also the superintegrable case. For the parameter values considered, we find that the band structure of the low-lying part of the excitation spectrum has the form expected from a quasiparticle picture with two fundamental particles.Studying the chain-size dependence of the spectrum, we find a remarkable stability of the second fundamental particle in a limited range of the momentum, even when its energy becomes so high that it lies very high up among the multiparticle scattering states. This is not a phenomenon restricted to the superintegrable line.Calculating a non-translationally invariant correlation function, we give evidence that it is oscillating. Within our numerical accuracy we find a relation between the oscillation length and the dip position of the momentum dispersion of the lightest particle which seems to be quite independent of the chiral angles.