1,475 research outputs found
Spacetime could be simultaneously continuous and discrete in the same way that information can
There are competing schools of thought about the question of whether
spacetime is fundamentally either continuous or discrete. Here, we consider the
possibility that spacetime could be simultaneously continuous and discrete, in
the same mathematical way that information can be simultaneously continuous and
discrete. The equivalence of continuous and discrete information, which is of
key importance in information theory, is established by Shannon sampling
theory: of any bandlimited signal it suffices to record discrete samples to be
able to perfectly reconstruct it everywhere, if the samples are taken at a rate
of at least twice the bandlimit. It is known that physical fields on generic
curved spaces obey a sampling theorem if they possess an ultraviolet cutoff.
Most recently, methods of spectral geometry have been employed to show that
also the very shape of a curved space (i.e., of a Riemannian manifold) can be
discretely sampled and then reconstructed up to the cutoff scale. Here, we
develop these results further, and we here also consider the generalization to
curved spacetimes, i.e., to Lorentzian manifolds
Minimal Length Uncertainty Relation and Hydrogen Atom
We propose a new approach to calculate perturbatively the effects of a
particular deformed Heisenberg algebra on energy spectrum. We use this method
to calculate the harmonic oscillator spectrum and find that corrections are in
agreement with a previous calculation. Then, we apply this approach to obtain
the hydrogen atom spectrum and we find that splittings of degenerate energy
levels appear. Comparison with experimental data yields an interesting upper
bound for the deformation parameter of the Heisenberg algebra.Comment: 7 pages, REVTe
On Dirac theory in the space with deformed Heisenberg algebra. Exact solutions
The Dirac equation has been studied in which the Dirac matrices
\hat{\boldmath\alpha}, \hat\beta have space factors, respectively and
, dependent on the particle's space coordinates. The function deforms
Heisenberg algebra for the coordinates and momenta operators, the function
being treated as a dependence of the particle mass on its position. The
properties of these functions in the transition to the Schr\"odinger equation
are discussed. The exact solution of the Dirac equation for the particle motion
in the Coulomnb field with a linear dependence of the function on the
distance to the force centre and the inverse dependence on for the
function has been found.Comment: 13 page
Harmonic oscillator with nonzero minimal uncertainties in both position and momentum in a SUSYQM framework
In the context of a two-parameter deformation of the
canonical commutation relation leading to nonzero minimal uncertainties in both
position and momentum, the harmonic oscillator spectrum and eigenvectors are
determined by using techniques of supersymmetric quantum mechanics combined
with shape invariance under parameter scaling. The resulting supersymmetric
partner Hamiltonians correspond to different masses and frequencies. The
exponential spectrum is proved to reduce to a previously found quadratic
spectrum whenever one of the parameters , vanishes, in which
case shape invariance under parameter translation occurs. In the special case
where , the oscillator Hamiltonian is shown to coincide
with that of the q-deformed oscillator with and its eigenvectors are
therefore --boson states. In the general case where , the eigenvectors are constructed as linear combinations of
--boson states by resorting to a Bargmann representation of the latter
and to -differential calculus. They are finally expressed in terms of a
-exponential and little -Jacobi polynomials.Comment: LaTeX, 24 pages, no figure, minor changes, additional references,
final version to be published in JP
Asymptotically maximal families of hypersurfaces in toric varieties
A real algebraic variety is maximal (with respect to the Smith-Thom
inequality) if the sum of the Betti numbers (with coefficients)
of the real part of the variety is equal to the sum of Betti numbers of its
complex part. We prove that there exist polytopes that are not Newton polytopes
of any maximal hypersurface in the corresponding toric variety. On the other
hand we show that for any polytope there are families of hypersurfaces
with the Newton polytopes that are
asymptotically maximal when tends to infinity. We also show that
these results generalize to complete intersections.Comment: 18 pages, 1 figur
On the Space-Time Uncertainty Relations of Liouville Strings and D Branes
Within a Liouville approach to non-critical string theory, we argue for a
non-trivial commutation relation between space and time observables, leading to
a non-zero space-time uncertainty relation , which
vanishes in the limit of weak string coupling.Comment: 8 pages, LaTe
Kodaira-Spencer formality of products of complex manifolds
We shall say that a complex manifold is emph{Kodaira-Spencer formal} if its Kodaira-Spencer differential graded Lie algebra
is formal; if this happen, then the deformation theory of
is completely determined by the graded Lie algebra and the base space of the semiuniversal deformation is a quadratic singularity..
Determine when a complex manifold is Kodaira-Spencer formal is generally difficult and
we actually know only a limited class of cases where this happen. Among such examples we have
Riemann surfaces, projective spaces, holomorphic Poisson manifolds with surjective anchor map
and every compact K"{a}hler manifold with trivial or torsion canonical
bundle.
In this short note we investigate the behavior of this property under finite products. Let be compact complex manifolds; we prove that whenever and are
K"{a}hler, then is Kodaira-Spencer formal if and only if the same
holds for and . A revisit of a classical example by Douady shows that the above result fails if the K"{a}hler assumption is droppe
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