272,300 research outputs found
Cosmological Horizons, Uncertainty Principle and Maximum Length Quantum Mechanics
The cosmological particle horizon is the maximum measurable length in the
Universe. The existence of such a maximum observable length scale implies a
modification of the quantum uncertainty principle. Thus due to non-locality of
quantum mechanics, the global properties of the Universe could produce a
signature on the behaviour of local quantum systems. A Generalized Uncertainty
Principle (GUP) that is consistent with the existence of such a maximum
observable length scale is where ( is the Hubble parameter and is the
speed of light). In addition to the existence of a maximum measurable length
, this form of GUP implies also the existence
of a minimum measurable momentum . Using appropriate representation of the position and momentum
quantum operators we show that the spectrum of the one dimensional harmonic
oscillator becomes where
is the dimensionless properly
normalized energy level, is a dimensionless parameter
with and for
(we show the full form of in the text). For a typical
vibrating diatomic molecule and we find and therefore for such a system, this effect is beyond reach of
current experiments. However, this effect could be more important in the early
universe and could produce signatures in the primordial perturbation spectrum
induced by quantum fluctuations of the inflaton field.Comment: 11 pages, 7 Figures. The Mathematica file that was used for the
production of the Figures may be downloaded from
http://leandros.physics.uoi.gr/maxlenqm
Entropy Bound with Generalized Uncertainty Principle in General Dimensions
In this letter, the entropy bound for local quantum field theories (LQFT) is
studies in a class of models of the generalized uncertainty principle(GUP)
which predicts a minimal length as a reflection of the quantum gravity effects.
Both bosonic and fermionic fields confined in arbitrary spatial dimension
ball are investigated. It is found that the GUP leads
to the same scaling correction to the entropy bound for
bosons and fermions, although the coefficients of this correction are different
for each case. Based on our calculation, we conclude that the GUP effects can
become manifest at the short distance scale. Some further implications and
speculations of our results are also discussed.Comment: 8 pages, topos corrected and references adde
Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter's great red spot
We introduce a new set of generalized Fokker-Planck equations that conserve
energy and mass and increase a generalized entropy until a maximum entropy
state is reached. The concept of generalized entropies is rigorously justified
for continuous Hamiltonian systems undergoing violent relaxation. Tsallis
entropies are just a special case of this generalized thermodynamics.
Application of these results to stellar dynamics, vortex dynamics and Jupiter's
great red spot are proposed. Our prime result is a novel relaxation equation
that should offer an easily implementable parametrization of geophysical
turbulence. This relaxation equation depends on a single key parameter related
to the skewness of the fine-grained vorticity distribution. Usual
parametrizations (including a single turbulent viscosity) correspond to the
infinite temperature limit of our model. They forget a fundamental systematic
drift that acts against diffusion as in Brownian theory. Our generalized
Fokker-Planck equations may have applications in other fields of physics such
as chemotaxis for bacterial populations. We propose the idea of a
classification of generalized entropies in classes of equivalence and provide
an aesthetic connexion between topics (vortices, stars, bacteries,...) which
were previously disconnected.Comment: Submitted to Phys. Rev.
Self-Completeness and the Generalized Uncertainty Principle
The generalized uncertainty principle discloses a self-complete
characteristic of gravity, namely the possibility of masking any curvature
singularity behind an event horizon as a result of matter compression at the
Planck scale. In this paper we extend the above reasoning in order to overcome
some current limitations to the framework, including the absence of a
consistent metric describing such Planck-scale black holes. We implement a
minimum-size black hole in terms of the extremal configuration of a neutral
non-rotating metric, which we derived by mimicking the effects of the
generalized uncertainty principle via a short scale modified version of
Einstein gravity. In such a way, we find a self-consistent scenario that
reconciles the self-complete character of gravity and the generalized
uncertainty principle.Comment: 20 pages, 6 figures, v2: additional references, version in press on
JHE
Geometric Approach to Pontryagin's Maximum Principle
Since the second half of the 20th century, Pontryagin's Maximum Principle has
been widely discussed and used as a method to solve optimal control problems in
medicine, robotics, finance, engineering, astronomy. Here, we focus on the
proof and on the understanding of this Principle, using as much geometric ideas
and geometric tools as possible. This approach provides a better and clearer
understanding of the Principle and, in particular, of the role of the abnormal
extremals. These extremals are interesting because they do not depend on the
cost function, but only on the control system. Moreover, they were discarded as
solutions until the nineties, when examples of strict abnormal optimal curves
were found. In order to give a detailed exposition of the proof, the paper is
mostly self\textendash{}contained, which forces us to consider different areas
in mathematics such as algebra, analysis, geometry.Comment: Final version. Minors changes have been made. 56 page
Entropy bound of local quantum field theory with generalized uncertainty principle
We study the entropy bound for local quantum field theory (LQFT) with
generalized uncertainty principle. The generalized uncertainty principle
provides naturally a UV cutoff to the LQFT as gravity effects.
Imposing the non-gravitational collapse condition as the UV-IR relation, we
find that the maximal entropy of a bosonic field is limited by the entropy
bound rather than with the boundary area.Comment: 11 pages, 1figure, version to appear in PL
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