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Lagrangians corresponding to some GUP models
Abstract
In this paper, we introduce the generalized Legendre transformation for the GUP Hamiltonian. From this, we define the non-canonical momentum. We interpret the momentum in GUP as the non-canonical momentum. We construct the GUP Lagrangian for some GUP models.
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Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman–von Neumann’s Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then, the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.
The Generalized Uncertainty Principle (GUP) has been directly applied to the motion of (macroscopic) test bodies on a given space-time in order to compute corrections to the classical orbits predicted in Newtonian Mechanics or General Relativity. These corrections generically violate the Equivalence Principle. The GUP has also been indirectly applied to the gravitational source by relating the GUP modified Hawking temperature to a deformation of the background metric. Such a deformed background metric determines new geodesic motions without violating the Equivalence Principle. We point out here that the two effects are mutually exclusive when compared with experimental bounds. Moreover, the former stems from modified Poisson brackets obtained from a wrong classical limit of the deformed canonical commutators.
In this paper we show that the position representation of the higher order generalized uncertainty principle (HGUP) is also well defined without approximation. We find the simpler form of this representation with a help of the signature of the discriminant of the cubic equation. Using this we discuss the HGUP derivative, HGUP exponential and HGUP trigonometric functions and HGUP-corrected Bohr atom model. Finally we discuss the D-dimensional HGUP algebra and compute the cosmological constant. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
In this paper we present a new higher order generalized (gravitational) uncertainty principle (GUP*) which has the maximal momentum as well as the minimal length. We discuss the position representation and momentum representation. We also discuss the position eigenfunction and maximal localization states. As examples we discuss one dimensional box problem and harmonic oscillator problem.
Based on the doubly special relativity we find a new type of generalized uncertainty principle (GUP) where the coordinate remains unaltered at the high energy while the momentum is deformed at the high energy so that it may be bounded from the above. For this GUP, we discuss some quantum mechanical problems in one dimension such as box problem, momentum wave function, and harmonic oscillator problem.
In this paper, using the new type of D-dimensional nonperturbative Generalized Uncertainty Principle (GUP) which has predicted both a minimal length uncertainty and
a maximal observable momentum, first, we obtain the maximally localized states and
express their connections to [P. Pedram, Phys. Lett. B 714, 317 (2012)]. Then, in the
context of our proposed GUP and using the generalized Schrodinger equation, we solve
some important problems including particle in a box and one-dimensional hydrogen
atom. Next, implying modified Bohr–Sommerfeld quantization, we obtain energy spectra of quantum harmonic oscillator and quantum bouncer. Finally, as an example, we
investigate some statistical properties of a free particle, including partition function and
internal energy, in the presence of the mentioned GUP.
In this letter, we present two new types of D-dimensional nonperturbative Generalized Uncertainty
Principle (GUP) which are predicted both a minimal length uncertainty and a maximal observable
momentum. Then, using these GUPs, we study the density of states for D-dimensional spherical
coordinate systems in the momentum space. Also, we investigate the cosmological constant in the
presence of these GUPs and finally, compare their massless type with the ones were predicted by
Kempf and Pedram in Refs. [Phys. Rev. D 52, 1108 (1995)] and [Phys. Lett. B 718, 638 (2012)].
Moreover, using a more general form of the higher order GUP, once again we compare the massless
cosmological constants.
In this paper, we study two generalized uncertainty principles (GUPs) including [X,P] = iℏ(1 + βP2j) and [X,P] = iℏ(1 + βP2 + kβ2P4) 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 existence of a minimum measurable length could deform not only the
standard quantum mechanics but also classical physics. The effects of the
minimal length on classical orbits of particles in a gravitation field have
been investigated before, using the deformed Poisson bracket or Schwarzschild
metric. In this paper, we use the Hamilton-Jacobi method to study motions of
particles in the context of deformed Newtonian mechanics and general
relativity. Specifically, the precession of planetary orbits, deflection of
light, and time delay in radar propagation are considered in this paper. We
also set limits on the deformation parameter by comparing our results with the
observational measurements. Finally, comparison with results from previous
papers is given at the end of this paper.
We consider quintessence scalar field cosmology in which the Lagrangian of the scalar field is modified by the generalized uncertainty principle. We show that the perturbation terms that arise from the deformed algebra are equivalent with the existence of a second scalar field, where the two fields interact in the kinetic part. Moreover, we consider a spatially flat Friedmann–Lemaître–Robertson–Walker spacetime, and we derive the gravitational field equations. We show that the modified equation of state parameter w
GUP can cross the phantom divide line; that is w
GUP < −1. Furthermore, we derive the field equations in the dimensionless parameters, the dynamical system that arises is a singular perturbation system in which we study the existence of the fixed points in the slow manifold. Finally, we perform numerical simulations for some well known models and we show that for these models with the specific initial conditions, the parameter w
GUP crosses the phantom barrier.
We compute the corrections to the Schwarzschild metric necessary to reproduce
the Hawking temperature derived from a Generalized Uncertainty Principle (GUP),
so that the GUP deformation parameter is directly linked to the deformation of
the metric. Using this modified Schwarzschild metric, we compute corrections to
the standard General Relativistic predictions for the light deflection and
perihelion precession, both for planets in the solar system and for binary
pulsars. This analysis allows us to set bounds for the GUP deformation
parameter from well-known astronomical measurements.
We calculate the orbits of a particle in Schwarzschild spacetime, assuming
that the dynamics is governed by a Snyder symplectic structure. With this
assumption, the perihelion shift of the planets acquires an additional
contribution with respect to the one predicted by general relativity. Moreover,
the equivalence principle is violated. If one assumes that Snyder mechanics is
valid also for macroscopic systems, these results impose strong constraints on
the value of the coupling parameter of the Snyder model.
We show that the Equivalence Principle is violated by Quantum Gravity
(QG) effects. The predicted violations are compared to experimental
observations for Gravitational Redshift, Law of Reciprocal Action and
Universality of Free Fall. This allows us to derive explicit bounds for
β—the QG scale. In our approach, there appears a deviation in
the geodesic motion of a particle. This deviation is induced by a
non-commutative spacetime, consistent with a Generalized Uncertainty
Principle (GUP). GUP admits the presence of a minimum length scale, that
is advocated by QG theories. Remarkably, the GUP induced corrections are
quite robust since the bound on β obtained by us, in General
Relativity scenario in an essentially classical setting of modified
geodesic motion, is closely comparable to similar bounds in recent
literature (Das and Vagenas 2008 Phys. Rev. Lett. 101 221301). The
latter are computed in purely quantum physics domain in flat spacetime.
We present a higher order generalized (gravitational) uncertainty principle
(GUP) in the form $[X,P]=i\hbar/(1-\beta P^2)$. This form of GUP 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. We show that the presence of the
maximal momentum results in an upper bound on the energy spectrum of the
momentum eigenstates and the harmonic oscillator.
In this paper we have constructed a coordinate space (or geometric) Lagrangian for a point particle that satisfies the exact doubly special relativity (DSR) dispersion relation in the Magueijo-Smolin framework. Next we demonstrate how a noncommutative phase space is needed to maintain Lorentz invariance for the DSR dispersion relation. Lastly we address the very important issue of velocity of this DSR particle. Exploiting the above noncommutative phase space algebra in a Hamiltonian framework, we show that the speed of massless particles is c and for massive particles the speed saturates at c when the particle energy reaches the maximum value κ, the Planck mass.
Studies in string theory and quantum gravity lead to the Generalized
Uncertainty Principle (GUP) and suggest the existence of a fundamental minimal
length which, as was established, can be obtained within the deformed
Heisenberg algebra. The first look on the classical motion of bodies in a space
with corresponding deformed Poisson brackets in a uniform gravitational field
can give an impression that bodies of different mass fall in different ways and
thus the equivalence principle is violated. Analyzing the kinetic energy of a
composite body we find that the motion of its center of mass in the deformed
space depends on some effective parameter of deformation. It gives a
possibility to recover the equivalence principle in the space with deformed
Poisson brackets. and thus GUP is reconciled with the equivalence principle. We
also show that the independence of kinetic energy on composition leads to the
recovering of the equivalence principle in the space with deformed Poisson
brackets.
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.
The scattering cross section of electrons in noble gas atoms exhibits a
minimum value at electron energies of approximately 1eV. This is the
Ramsauer-Townsend effect. In this letter, we study the Ramsauer-Townsend effect
in the framework of the Generalized Uncertainty Principle.
We review versions of the Generalized Uncertainty Principle (GUP) obtained in string theory and in gedanken experiments carried out in quantum gravity. We show how a GUP can be derived from a measure gedanken experiment involving micro-black holes at the Planck scale of spacetime. The model uses only Heisenberg principle and Schwarzschild radius and is independent from particular versions of Quantum Gravity.
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 PT-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.
We formulate generalized uncertainty relations in a crystal-like universe whose lattice spacing is of the order of Planck length -- "world crystal". In the particular case when energies lie near the border of the Brillouin zone, i.e., for Planckian energies, the uncertainty relation for position and momenta does not pose any lower bound on involved uncertainties. We apply our results to micro black holes physics, where we derive a new mass-temperature relation for Schwarzschild micro black holes. In contrast to standard results based on Heisenberg and stringy uncertainty relations, our mass-temperature formula predicts both a finite Hawking's temperature and a zero rest-mass remnant at the end of the micro black hole evaporation. We also briefly mention some connections of the world crystal paradigm with 't Hooft's quantization and double special relativity. Comment: 13 pages, 2 figures, latex4
By evaluating and resumming the large-s behaviour of string loops to all orders we are able to obtain an explicitly unitary operator for the light (closed) superstring S-matrix above planckian energies: it is dominated by graviton exchange at large impact parameters and by absorption at small impact parameters. In an intermediate, eikonal region a semiclassical description emerges as if, for small deflection angles at least, each string was moving in a static Schwarzschild metric.
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 \beta p^4 and \beta^2 p^6 respectively, where \beta \sim 1/(M_{Pl}c)^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. Comment: 7 pages. Based on talk by S. Das at Theory Canada 4, Montreal, 4 June, 2008. To be published in the Canadian Journal of Physics
The existence of a minimal observable length has long been suggested, in quantum gravity, as well as in string theory. In this context a generalized uncertainty relation has been derived which quantum theoretically describes the minimal length as a minimal uncertainty in position measurements. Here we study in full detail the quantum mechanical structure which underlies this uncertainty relation. Comment: 21 pages, latex, epsf, 4 figures, final version which appears in Physical Review D
In this paper we have studied the nature of kinematical and dynamical laws in $\kappa $-Minkowski spacetime from a new perspective: the canonical phase space approach. We discuss a particular form of $\kappa$-Minkowski phase space algebra that yields the $\kappa$-extended finite Lorentz transformations derived in \cite{kim}. This is a particular form of a Deformed Special Relativity model that admits a modified energy-momentum dispersion law as well as noncommutative $\kappa$-Minkowski phase space. We show that this system can be completely mapped to a set of phase space variables that obey canonical (and {\it{not}} $\kappa$-Minkowski) phase space algebra and Special Relativity Lorentz transformation (and {\it{not}} $\kappa$-extended Lorentz transformation). The complete set of deformed symmetry generators are constructed that obeys an unmodified closed algebra but induce deformations in the symmetry transformations of the physical $\kappa$-Minkowski phase space variables. Furthermore, we demonstrate the usefulness and simplicity of this approach through a number of phenomenological applications both in classical and quantum mechanics. We also construct a Lagrangian for the $\kappa$-particle.
We consider Uncertainty Principles which take into account the role of gravity and the possible existence of extra spatial dimensions. Explicit expressions for such Generalized Uncertainty Principles in 4+n dimensions are given and their holographic properties investigated. In particular, we show that the predicted number of degrees of freedom enclosed in a given spatial volume matches the holographic counting only for one of the available generalizations and without extra dimensions. Comment: LaTeX, 13 pages, accepted for publication in Class. Quantum Grav
We continue our investigation of the phenomenological implications of the "deformed" commutation relations [x_i,p_j]=i hbar[(1 + beta p^2) delta_{ij} + beta' p_i p_j]. These commutation relations are motivated by the fact that they lead to the minimal length uncertainty relation which appears in perturbative string theory. In this paper, we consider the effects of the deformation on the classical orbits of particles in a central force potential. Comparison with observation places severe constraints on the value of the minimum length.
I describe our understanding of physics near the Planck length, in particular the great progress of the last four years in string theory. The lectures were presented at the 1998 SLAC Summer Institute.
An analysis of the effect of gravitation on hypothetical experiments indicates that it is impossible to measure the position of a particle with error less than Deltax>~&surd;G=1.6×10-33 cm, where G is the gravitational constant in natural units. A similar limitation applies to the precise synchronization of clocks. It is possible that this result may aid in the solution of the divergence problems of field theory. An equivalence is established between the postulate of a fundamental length and a postulate about gravitational field fluctuations, and it is suggested that the formulation of a fundamental length theory which does not involve gravitational effects in an important way may be impossible.
In this Paper, we investigate the effects of space noncommutativity and the generalized uncertainty principle on the stability of circular orbits of particles in a central force potential. We show that, up to first order in noncommutativity parameter, an angular momentum dependent term will be appear in the equations of the particle orbits which affects the stability of circular orbits. In the case of high angular momentum, the condition for stability of circular orbits will change considerably relative to commutative case.
Studies in string theory and quantum gravity suggest the existence of a finite lower limit Δx0 to the possible resolution of distances, at the latest on the scale of the Planck length of 10-35 m. Within the framework of the Euclidean path integral we explicitly show ultraviolet regularization in field theory through this short distance structure. Both rotation and translation invariance can be preserved. An example is studied in detail.
It is usually assumed that space-time is a continuum. This assumption is
not required by Lorentz invariance. In this paper we give an example of
a Lorentz invariant discrete space-time.
The problem of the reduction of the wave function in quantum theory is treated from a new standpoint. First, by combining
Heisenberg's uncertainty relations with gravitation, quantitative limi tations on the sharpness of the structure of space-time
are derived. Second, the resulting uncertainty in space-time structure is incorporated into the equations for the propagation
of the quantum-mechanical wave amplitudes. In the resulting theory an initially pure wave function generally develops in time
into a mixture. A single pure wave function survives only as long as it corresponds to a sufficiently small spread in the
position of any massive part of the system under investigation. Quantitative relations between the mass and the maximum coherent
spread in the center-of-mass wave function of a single body are obtained.
Si tratta da un nuovo punto di vista il problema della riduzione della funzione d'onda nella teoria dei quanti. Dapprima,
combinando le relazioni di indeterminazione di Heisenberg con la gravitazione si deducono limitazioni quantitative alla chiarezza
della struttura dello spazio-tempo. Poi si incorpora la risultante indeterminazione dello spazio-tempo nelle equazioni della
progagazione delle ampiezze d'onda della meccanica quantistica. Nella teoria risultante una funzione d'onda inizialmente pura,
generalmente si sviluppa nel tempo in una mescolanza. Una singola funzione d'onda pura sopravvive solo finchè corrisponde
ad una dispersione sufficientemente piccola della posizione di ogni parte del sistema studiato dotata di massa. Si ottengono
relazioni quantitative fra la massa e la massima dispersione coerente della funzione d'onda del centro di massa di un singolo
corpo.
The large-energy, fixed-angle, behavior of string scattering amplitudes is considered to arbitrary order in perturbation theory. It is shown that the path integal over surfaces is dominated, in this limit, by a saddle point. The dominant saddle point is identified and its contribution to the amplitude is evaluated.
We investigate the effects to all orders in the Planck length from a generalized uncertainty principle (GUP) on black holes thermodynamics. We calculate the corrected Hawking temperature, entropy, and examine in details the Hawking evaporation process. As a result, the evaporation process is accelerated and the evaporation end-point is a zero entropy, zero heat capacity and finite non-zero temperature black hole remnant (BHR). In particular we obtain a drastic reduction of the decay time, in comparison with the results obtained in the Hawking semi classical picture and with the GUP to leading order in the Planck length.
A possible definition of path integrals for string theory is studied, based on a discretized version of Polyakov's generating functional. The finite resolution of string theory, as opposed to the infinite resolution in particle theory, clearly emerges from a renormalization group type analysis. We derive the existence of a minimum physical length (∼10−33cm) and generalized form of the uncertainty principle, and discuss some of their consequences.
We discuss a Gedanken experiment for the measurement of the area of the apparent horizon of a black hole in quantum gravity. Using rather general and model-independent considerations we find a generalized uncertainty principle which agrees with a similar result obtained in the framework of string theories. The result indicates that a minimum length of the order of the Planck length emerges naturally from any quantum theory of gravity, and that the concept of black hole is not operationally defined if the mass is smaller than the Planck mass.
Strings may be explored, through a scattering process at planckian energies, down to distances of the order of the string size. We show that this is possible if the energy is not so extreme to cause a gravitational instability and when the scattering angle approaches some critical value from below. Above this angle, the distance starts increasing, thus departing from the usual position-momentum uncertainty relation, and in no instance is the resolution smaller than the string length. This suggests that below the Planck scale the very concept of space-time changes meaning.
Generalized uncertainty principles are able to serve as useful descriptions
of some of the phenomenology of quantum gravity effects, providing an intuitive
grasp on non-trivial space-time structures such as a fundamental discreteness
of space, a universal bandlimit or an irreducible extendedness of elementary
particles. In this article, uncertainty relations are derived by a moment
expansion of states for quantum systems with a discrete coordinate, and
correspondingly a periodic momentum. Corrections to standard uncertainty
relations are found, with some similarities but also key differences to what is
often assumed in this context. The relations provided can be applied to
discrete models of matter or space-time, including loop quantum cosmology.
The Snyder model is an example of noncommutative spacetime admitting a
fundamental length scale $\beta$ and invariant under Lorentz transformations,
that can be interpreted as a realization of the doubly special relativity
axioms. Here, we consider its nonrelativistic counterpart, i.e. the Snyder
model restricted to three-dimensional Euclidean space. We discuss the classical
and the quantum mechanics of a free particle in this framework, and show that
they strongly depend on the sign of a coupling constant $\lambda$, appearing in
the fundamental commutators and proportional to $\beta^2$. For example, if
$\lambda$ is negative, momenta are bounded. On the contrary, for positive
$\lambda$, positions and areas are quantized. We also give the exact solution
of the harmonic oscillator equations both in the classical and the quantum
case, and show that its frequency is energy dependent.
A possible discrepancy has been found between the results of a neutron
interferometry experiment and Quantum Mechanics. This experiment suggests that
the weak equivalence principle is violated at small length scales, which
quantum mechanics cannot explain. In this paper, we investigated whether the
Generalized Uncertainty Principle (GUP), proposed by some approaches to quantum
gravity such as String Theory and Doubly Special Relativity Theories (DSR), can
explain the violation of the weak equivalence principle at small length scales.
We also investigated the consequences of the GUP on the Liouville theorem in
statistical mechanics. We have found a new form of invariant phase space in the
presence of GUP. This result should modify the density states and affect the
calculation of the entropy bound of local quantum field theory, the
cosmological constant, black body radiation, etc. Furthermore, such
modification may have observable consequences at length scales much larger than
the Planck scale. This modification leads to a \sqrt{A}-type correction to the
bound of the maximal entropy of a bosonic field which would definitely shed
some light on the holographic theory.
Recent Letters and Comments discuss the quantization of self-dual two-dimensional Lagrangeans which describe systems that are not explicitly canonical. We make some remarks on the most efficient method for exhibiting the canonical structure.
Heisenberg showed in the early days of quantum theory that the uncertainty principle follows as a direct consequence of the quantization of electromagnetic radiation in the form of photons. As we show here the gravitational interaction of the photon and the particle being observed modifies the uncertainty principle with an additional term. From the modified or gravitational uncertainty principle it follows that there is an absolute minimum uncertainty in the position of any particle, of order of the Planck length. A modified uncertainty relation of this form is a standard result of superstring theory, but the derivation given here is based on simpler and rather general considerations with either Newtonian gravitational theory or general relativity theory. Comment: Gravity Research Foundation essay. 14 pages Minor typos were corrected