2,956 research outputs found
Quantum Field Theory with Nonzero Minimal Uncertainties in Positions and Momenta
A noncommutative geometric generalisation of the quantum field theoretical
framework is developed by generalising the Heisenberg commutation relations.
There appear nonzero minimal uncertainties in positions and in momenta. As the
main result it is shown with the example of a quadratically ultraviolet
divergent graph in theory that nonzero minimal uncertainties in
positions do have the power to regularise. These studies are motivated with the
ansatz that nonzero minimal uncertainties in positions and in momenta arise
from gravity. Algebraic techniques are used that have been developed in the
field of quantum groups.Comment: 52 pages LATEX, DAMTP/93-33. Revised version now includes a chapter
on the Poincare algebra and curvature as noncommutativity of momentum spac
Hydrogen atom as an eigenvalue problem in 3D spaces of constant curvature and minimal length
An old result of A.F. Stevenson [Phys. Rev.} 59, 842 (1941)] concerning the
Kepler-Coulomb quantum problem on the three-dimensional (3D) hypersphere is
considered from the perspective of the radial Schr\"odinger equations on 3D
spaces of any (either positive, zero or negative) constant curvature. Further
to Stevenson, we show in detail how to get the hypergeometric wavefunction for
the hydrogen atom case. Finally, we make a comparison between the ``space
curvature" effects and minimal length effects for the hydrogen spectrumComment: 6 pages, v
Unsharp Degrees of Freedom and the Generating of Symmetries
In quantum theory, real degrees of freedom are usually described by operators
which are self-adjoint. There are, however, exceptions to the rule. This is
because, in infinite dimensional Hilbert spaces, an operator is not necessarily
self-adjoint even if its expectation values are real. Instead, the operator may
be merely symmetric. Such operators are not diagonalizable - and as a
consequence they describe real degrees of freedom which display a form of
"unsharpness" or "fuzzyness". For example, there are indications that this type
of operators could arise with the description of space-time at the string or at
the Planck scale, where some form of unsharpness or fuzzyness has long been
conjectured.
A priori, however, a potential problem with merely symmetric operators is the
fact that, unlike self-adjoint operators, they do not generate unitaries - at
least not straightforwardly. Here, we show for a large class of these operators
that they do generate unitaries in a well defined way, and that these operators
even generate the entire unitary group of the Hilbert space. This shows that
merely symmetric operators, in addition to describing unsharp physical
entities, may indeed also play a r{\^o}le in the generation of symmetries, e.g.
within a fundamental theory of quantum gravity.Comment: 23 pages, LaTe
Maximally localized states and causality in non commutative quantum theories
We give simple representations for quantum theories in which the position
commutators are non vanishing constants. A particular representation reproduces
results found using the Moyal star product. The notion of exact localization
being meaningless in these theories, we adapt the notion of ``maximally
localized states'' developed in another context . We find that gaussian
functions play this role in a 2+1 dimensional model in which the non
commutation relations concern positions only. An interpretation of the wave
function in this non commutative geometry is suggested. We also analyze higher
dimensional cases. A possible incidence on the causality issue for a Q.F.T with
a non commuting time is sketched.Comment: 11 pages, Revtex. The presentation has been improved, the subsection
on high dimensions has been modified. This version will appear in PR
On Fields with Finite Information Density
The existence of a natural ultraviolet cutoff at the Planck scale is widely
expected. In a previous Letter, it has been proposed to model this cutoff as an
information density bound by utilizing suitably generalized methods from the
mathematical theory of communication. Here, we prove the mathematical
conjectures that were made in this Letter.Comment: 31 pages, to appear in Phys.Rev.
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
Vacuum entanglement enhancement by a weak gravitational field
Separate regions in space are generally entangled, even in the vacuum state.
It is known that this entanglement can be swapped to separated Unruh-DeWitt
detectors, i.e., that the vacuum can serve as a source of entanglement. Here,
we demonstrate that, in the presence of curvature, the amount of entanglement
that Unruh-DeWitt detectors can extract from the vacuum can be increased.Comment: 6 pages, 1 figur
Quantum gravity effects on statistics and compact star configurations
The thermodynamics of classical and quantum ideal gases based on the
Generalized uncertainty principle (GUP) are investigated. At low temperatures,
we calculate corrections to the energy and entropy. The equations of state
receive small modifications. We study a system comprised of a zero temperature
ultra-relativistic Fermi gas. It turns out that at low Fermi energy
, the degenerate pressure and energy are lifted. The
Chandrasekhar limit receives a small positive correction. We discuss the
applications on configurations of compact stars. As increases,
the radius, total number of fermions and mass first reach their nonvanishing
minima and then diverge. Beyond a critical Fermi energy, the radius of a
compact star becomes smaller than the Schwarzschild one. The stability of the
configurations is also addressed. We find that beyond another critical value of
the Fermi energy, the configurations are stable. At large radius, the increment
of the degenerate pressure is accelerated at a rate proportional to the radius.Comment: V2. discussions on the stability of star configurations added, 17
pages, 2 figures, typos corrected, version to appear in JHE
Liquidity and its impact on bond prices
In this paper, we propose a theoretical continuous-time model to analyze the impact of liquidity on bond prices. This model prices illiquid bonds relative to liquid bonds and provides a testable theory of illiquidity induced price discounts. The model is tested using 1992-1994 data from bonds issued by the german government.These bonds define a market segment that is homogeneous in bankruptcy risk, taxes, age, and coupons, but the bonds differ with respect to their liquidity. The empirical findings suggest that bond prices not only depend on the dynamics of interest rates, but also on the liquidity of bonds. Therefore, bond liquidity should be used as an additional pricing factor. The findings of the out-of-sample test demonstrate the superiority of the model in comparison with traditional pricing models
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