Generalized uncertainty principle and self-adjoint operators

In this paper, we study two generalized uncertainty principles (GUPs) including [Formula: see text] and [Formula: see text] which imply minimal measurable lengths. Using two momentum representations, for the former GUP, we find eigenvalues and eigenfunctions of the free particle and the harmonic oscillator in terms of generalized trigonometric functions. Also, for the latter GUP, we obtain quantum mechanical solutions of a particle in a box and harmonic oscillator. Finally we investigate the statistical properties of the harmonic oscillator including partition function, internal energy, and heat capacity in the context of the first GUP.

The properties of the time-independent Schrodinger equation are introduced step by step, starting from a short discussion about its boundary conditions. Considering that the equation is seldom amenable to analytical solutions, two simple cases are examined first: that of a free particle and that of a particle in a box. The determination of the lower energy bound follows, introducing more general issues that build up the mathematical frame of the theory: norm of a function, scalar product of functions, Hermitean operators, eigenfunctions and eigenvalues of operators, orthogonal functions, and completeness of a set of functions. This chapter is concluded with the important examples of the Hamiltonian operator and momentum operator. The complements provide examples of Hermitean operators, a collection of operators’ definitions and properties, examples of commuting operators, and a further discussion about the free-particle case.

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.

One can use the generalized uncertainty principle (GUP) to incorporate the minimum measurable length in quantum gravity. It may be interesting to have a minimal time interval as well as the minimal length in the relativistic version of quantum mechanics in the presence of the gravitational effects. In this paper, we considered a covariant version of the GUP to investigate the effects of both the minimal time and the minimal length. Using the covariant GUP, the energy–momentum dispersion relation was modified. Starting with the modified dispersion relation, the corrections to the wave function were obtained and the problems of the particle in a box and the hydrogen atom were revisited. Considering the minimal time may be an interesting new topic, which is not widely studied before. It may lead to new results that can open a new window into quantum gravity.

Theoretical formulation of finite-dimensional discrete phase spaces: I. Algebraic structures and uncertainty principles

A class of highly symmetric ,nonrelativistic, Euclidean invariant, model scalar field theories are examined assuming the existence of field and momentum operators that satisfy the canonical commutation relations (CCR). The high degree of symmetry that we assume permits explicit determination of every relevant CCR representation. These consist of a two-parameter family of unitarily inequivalent representations, some of which are irreducible while the others are reducible. It is demonstrated that only those models that are analogs of the free field can be encompassed within the irreducible representations. Hence every model with interaction—including an analog of the relativistic λφ4 theory—requires a reducible CCR representation. For the reducible representations, we determine every relevant Hamiltonian operator possessing the required high degree of symmetry. These Hamiltonians, as well as the generators of space translations, cannot be expressed (solely) as functions of the field and momentum operators, which is characteristic of any system with a unique ground state and reducible CCR representation. Nevertheless, it is demonstrated that these Hamiltonians, as well as the generators of space translations, fulfill the ``weak correspondence principle,'' in which the expectation value of a quantum generator, such as the Hamiltonian operator, in a suitably labeled overcomplete family of states is identified with the associated classical generator, such as the classical Hamiltonian. Our principal results depend on the existence and make extensive use of the countably infinite number of degrees of freedom existing in a field theory. Entirely analogous results apply to related models defined in a finite spacial volume since they still have an infinite number of degrees of freedom.

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.

Quantum mechanics of a particle in an infinite square well under the influence of a time-dependent electric field is reconsidered. In some gauge, the Hamiltonian depends linearly on the momentum operator, which is symmetric but not self-adjoint when defined on a finite interval. In spite of this symmetric part, the Hamiltonian operator is shown to be self-adjoint. This follows from a theorem by Kato and Rellich which guarantees the stability of a self-adjoint operator under certain symmetric perturbations. The result, which has been assumed tacitly by other authors, is important in order to establish the equivalence of different Hamiltonian operators related to each other by quantum gauge transformations.

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

In this paper we revisit the formulation of scalar field theories on de Sitter backgrounds subject to the generalized uncertainty principle (GUP). The GUP arises in several contexts in string theory, but is most readily thought of as resulting from using strings as effective probes of geometry, which suggests an uncertainty relation incorporating the string scale $l_s$. After reviewing the string theoretic case for the GUP, which implies a minimum length scale $l_s$, we follow in the footsteps of Kempf and concern ourselves with how one might write down field theories which respect the GUP. We uncover a new representation of the GUP, which unlike previous studies, readily permits exact analytical solutions for the mode functions of a scalar field on de Sitter backgrounds. We find that scalar fields cannot be quantized on inflationary backgrounds with a Hubble radius $H^{-1}$ smaller than the string scale, implying a sensibly stringy (as opposed to Planckian) cutoff on the scale of inflation resulting from the GUP. We also compute $(H l_s)^2$ corrections to the two point correlation function analytically and comment on the future prospects of observing such corrections in the fortunate circumstance our universe is described by a very weakly coupled string theory.

Possible corrections to the thermodynamic quantities of a 4D Schwarzschild–anti–de Sitter (SAdS) black hole are investigated by considering the generalized uncertainty principle (GUP) and the modified dispersion relation (MDR) separately. The quantum gravitational corrections to the Hawking temperature, energy and entropy of the black hole are calculated based on both the GUP and the MDR analysis. The explicit form of the corrections is worked out up to the sixth power of the Planck length. The quantum-corrected thermodynamic quantities due to GUP and MDR are used separately to obtain the quantum-gravitational corrections to the Cardy–Verlinde (C–V) formula. It is found that the C–V formula receives some new corrections in either of approaches. Through comparison of the corrections obtained from GUP and MDR approaches, it is shown that the results of these two alternative approaches should be identical if one uses the suitable expansion coefficients. Finally, the renormalized form of the C–V formula is introduced by redefining the Virasoro operator and the central charge within both the GUP and the MDR.

We discuss one-dimensional Dirac oscillator, by using the concept doubly special relativity. We calculate the energy spectrum by using the concept doubly special relativity. Then, we derive another representation that the coordinate operator remains unchanged at the high energy while the momentum operator is deformed at the high energy so that it may be bounded from the above. Actually, we study the Dirac oscillator by using of the generalized uncertainty principle version and the concept doubly special relativity.

Self-adjoint extensions for linear Hamiltonian systems with two singular endpoints

Spectral analysis of dissipative Dirac operators with general boundary conditions

Various theories of quantum gravity argue that near the Planck scale, the Heisenberg uncertainty principle should be replaced by the so called generalized uncertainty principle (GUP). We show that the GUP gives rise to two additional terms in any quantum mechanical Hamiltonian, proportional to βp4 and β2p6, respectively, where β ∼ 1/(MPlc)2 is the GUP parameter. These terms become important at or above the Planck energy. Considering only the first of these and treating it as a perturbation, we show that the GUP affects the Lamb shift, Landau levels, reflection and transmission coefficients of a potential step and potential barrier, and the current in a scanning tunnel microscope (STM). Although these are too small to be measurable at present, we speculate on the possibility of extracting measurable predictions in the future.