788 research outputs found
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
Mode Generating Mechanism in Inflation with Cutoff
In most inflationary models, space-time inflated to the extent that modes of
cosmological size originated as modes of wavelengths at least several orders of
magnitude smaller than the Planck length. Recent studies confirmed that,
therefore, inflationary predictions for the cosmic microwave background
perturbations are generally sensitive to what is assumed about the Planck
scale. Here, we propose a framework for field theories on curved backgrounds
with a plausible type of ultraviolet cutoff. We find an explicit mechanism by
which during cosmic expansion new (comoving) modes are generated continuously.
Our results allow the numerical calculation of a prediction for the CMB
perturbation spectrum.Comment: 9 pages, LaTe
Coherent States for the Non-Linear Harmonic Oscillator
Wave packets for the Quantum Non-Linear Oscillator are considered in the
Generalized Coherent State framerwork. To first order in the non-linearity
parameter the Coherent State behaves very similarly to its classical
counterpart. The position expectation value oscillates in a simple harmonic
manner. The energy-momentum uncertainty relation is time independent as in a
harmonic oscillator. Various features, (such as the Squeezed State nature), of
the Coherent State have been discussed
Resolution of null fiber and conormal bundles on the Lagrangian Grassmannian
We study the null fiber of a moment map related to dual pairs. We construct
an equivariant resolution of singularities of the null fiber, and get conormal
bundles of closed -orbits in the Lagrangian Grassmannian as the
categorical quotient. The conormal bundles thus obtained turn out to be a
resolution of singularities of the closure of nilpotent -orbits, which
is a "quotient" of the resolution of the null fiber.Comment: 17 pages; completely revised and add reference
Operator identities in q-deformed Clifford analysis
In this paper, we define a q-deformation of the Dirac operator as a generalization of the one dimensional q-derivative. This is done in the abstract setting of radial algebra. This leads to a q-Dirac operator in Clifford analysis. The q-integration on R(m), for which the q-Dirac operator satisfies Stokes' formula, is defined. The orthogonal q-Clifford-Hermite polynomials for this integration are briefly studied
Deformed algebras, position-dependent effective masses and curved spaces: An exactly solvable Coulomb problem
We show that there exist some intimate connections between three
unconventional Schr\"odinger equations based on the use of deformed canonical
commutation relations, of a position-dependent effective mass or of a curved
space, respectively. This occurs whenever a specific relation between the
deforming function, the position-dependent mass and the (diagonal) metric
tensor holds true. We illustrate these three equivalent approaches by
considering a new Coulomb problem and solving it by means of supersymmetric
quantum mechanical and shape invariance techniques. We show that in contrast
with the conventional Coulomb problem, the new one gives rise to only a finite
number of bound states.Comment: 22 pages, no figure. Archive version is already official. Published
by JPA at http://stacks.iop.org/0305-4470/37/426
Twisted Classical Poincar\'{e} Algebras
We consider the twisting of Hopf structure for classical enveloping algebra
, where is the inhomogenous rotations algebra, with
explicite formulae given for Poincar\'{e} algebra
The comultiplications of twisted are obtained by conjugating
primitive classical coproducts by where
denotes any Abelian subalgebra of , and the universal
matrices for are triangular. As an example we show that
the quantum deformation of Poincar\'{e} algebra recently proposed by Chaichian
and Demiczev is a twisted classical Poincar\'{e} algebra. The interpretation of
twisted Poincar\'{e} algebra as describing relativistic symmetries with
clustered 2-particle states is proposed.Comment: \Large \bf 19 pages, Bonn University preprint, November 199
Matrix oscillator and Calogero-type models
We study a single matrix oscillator with the quadratic Hamiltonian and
deformed commutation relations. It is equivalent to the multispecies Calogero
model in one dimension, with inverse-square two-body and three-body
interactions. Specially, we have constructed a new matrix realization of the
Calogero model for identical particles, without using exchange operators. The
critical points at which singular behaviour occurs are briefly discussed.Comment: Accepted for publication in Phys.Lett.
Motivic Serre invariants, ramification, and the analytic Milnor fiber
We show how formal and rigid geometry can be used in the theory of complex
singularities, and in particular in the study of the Milnor fibration and the
motivic zeta function. We introduce the so-called analytic Milnor fiber
associated to the germ of a morphism f from a smooth complex algebraic variety
X to the affine line. This analytic Milnor fiber is a smooth rigid variety over
the field of Laurent series C((t)). Its etale cohomology coincides with the
singular cohomology of the classical topological Milnor fiber of f; the
monodromy transformation is given by the Galois action. Moreover, the points on
the analytic Milnor fiber are closely related to the motivic zeta function of
f, and the arc space of X.
We show how the motivic zeta function can be recovered as some kind of Weil
zeta function of the formal completion of X along the special fiber of f, and
we establish a corresponding Grothendieck trace formula, which relates, in
particular, the rational points on the analytic Milnor fiber over finite
extensions of C((t)), to the Galois action on its etale cohomology.
The general observation is that the arithmetic properties of the analytic
Milnor fiber reflect the structure of the singularity of the germ f.Comment: Some minor errors corrected. The original publication is available at
http://www.springerlink.co
The 10 micron amorphous silicate feature of fractal aggregates and compact particles with complex shapes
We model the 10 micron absorption spectra of nonspherical particles composed
of amorphous silicate. We consider two classes of particles, compact ones and
fractal aggregates composed of homogeneous spheres. For the compact particles
we consider Gaussian random spheres with various degrees of non-sphericity. For
the fractal aggregates we compute the absorption spectra for various fractal
dimensions. The 10 micron spectra are computed for ensembles of these particles
in random orientation using the well-known Discrete Dipole Approximation. We
compare our results to spectra obtained when using volume equivalent
homogeneous spheres and to those computed using a porous sphere approximation.
We conclude that, in general, nonspherical particles show a spectral signature
that is similar to that of homogeneous spheres with a smaller material volume.
This effect is overestimated when approximating the particles by porous spheres
with the same volume filling fraction. For aggregates with fractal dimensions
typically predicted for cosmic dust, we show that the spectral signature
characteristic of very small homogeneous spheres (with a volume equivalent
radius r_V<0.5 micron) can be detected even in very large particles. We
conclude that particle sizes are underestimated when using homogeneous spheres
to model the emission spectra of astronomical sources. In contrast, the
particle sizes are severely overestimated when using equivalent porous spheres
to fit observations of 10 micron silicate emission.Comment: Accepted for publication in A&
- …