364 research outputs found
Symplectic areas, quantization, and dynamics in electromagnetic fields
A gauge invariant quantization in a closed integral form is developed over a
linear phase space endowed with an inhomogeneous Faraday electromagnetic
tensor. An analog of the Groenewold product formula (corresponding to Weyl
ordering) is obtained via a membrane magnetic area, and extended to the product
of N symbols. The problem of ordering in quantization is related to different
configurations of membranes: a choice of configuration determines a phase
factor that fixes the ordering and controls a symplectic groupoid structure on
the secondary phase space. A gauge invariant solution of the quantum evolution
problem for a charged particle in an electromagnetic field is represented in an
exact continual form and in the semiclassical approximation via the area of
dynamical membranes.Comment: 39 pages, 17 figure
Magnetic Fourier Integral Operators
In some previous papers we have defined and studied a 'magnetic'
pseudodifferential calculus as a gauge covariant generalization of the Weyl
calculus when a magnetic field is present. In this paper we extend the standard
Fourier Integral Operators Theory to the case with a magnetic field, proving
composition theorems, continuity theorems in 'magnetic' Sobolev spaces and
Egorov type theorems. The main application is the representation of the
evolution group generated by a 1-st order 'magnetic' pseudodifferential
operator (in particular the relativistic Schr\"{o}dinger operator with magnetic
field) as such a 'magnetic' Fourier Integral Operator. As a consequence of this
representation we obtain some estimations for the distribution kernel of this
evolution group and a result on the propagation of singularities
Cotangent bundle quantization: Entangling of metric and magnetic field
For manifolds of noncompact type endowed with an affine connection
(for example, the Levi-Civita connection) and a closed 2-form (magnetic field)
we define a Hilbert algebra structure in the space and
construct an irreducible representation of this algebra in . This
algebra is automatically extended to polynomial in momenta functions and
distributions. Under some natural conditions this algebra is unique. The
non-commutative product over is given by an explicit integral
formula. This product is exact (not formal) and is expressed in invariant
geometrical terms. Our analysis reveals this product has a front, which is
described in terms of geodesic triangles in . The quantization of
-functions induces a family of symplectic reflections in
and generates a magneto-geodesic connection on . This
symplectic connection entangles, on the phase space level, the original affine
structure on and the magnetic field. In the classical approximation,
the -part of the quantum product contains the Ricci curvature of
and a magneto-geodesic coupling tensor.Comment: Latex, 38 pages, 5 figures, minor correction
Graphene as a quantum surface with curvature-strain preserving dynamics
We discuss how the curvature and the strain density of the atomic lattice
generate the quantization of graphene sheets as well as the dynamics of
geometric quasiparticles propagating along the constant curvature/strain
levels. The internal kinetic momentum of Riemannian oriented surface (a vector
field preserving the Gaussian curvature and the area) is determined.Comment: 13p, minor correction
Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model
General boundary conditions ("branes") for the Poisson sigma model are
studied. They turn out to be labeled by coisotropic submanifolds of the given
Poisson manifold. The role played by these boundary conditions both at the
classical and at the perturbative quantum level is discussed. It turns out to
be related at the classical level to the category of Poisson manifolds with
dual pairs as morphisms and at the perturbative quantum level to the category
of associative algebras (deforming algebras of functions on Poisson manifolds)
with bimodules as morphisms. Possibly singular Poisson manifolds arising from
reduction enter naturally into the picture and, in particular, the construction
yields (under certain assumptions) their deformation quantization.Comment: 21 pages, 2 figures; minor corrections, references updated; final
versio
Analogues of the central point theorem for families with -intersection property in
In this paper we consider families of compact convex sets in
such that any subfamily of size at most has a nonempty intersection. We
prove some analogues of the central point theorem and Tverberg's theorem for
such families
New Bryokhutuliinia species (bryophyta) with sporophytes from the upper jurassic of Transbaikalia
A new species of the moss genus Bryokhutuliinia, B. crassimarginata is described from the Upper Jurassic deposits from the Olov, Transbaikal Area of South Siberia. Its excellent preservation demonstra- tes that the leaves were not only complanate, but truly distichous. In addition to anatomically pre- served gametophytes, sporophytes on short lateral branches were found, although carbonized and not exhibiting structural details. Possible relationships with pleurocarpous mosses and with Fissidentaceae are discussedyesBelgorod State National Research Universit
Quantum Magnetic Algebra and Magnetic Curvature
The symplectic geometry of the phase space associated with a charged particle
is determined by the addition of the Faraday 2-form to the standard structure
on the Euclidean phase space. In this paper we describe the corresponding
algebra of Weyl-symmetrized functions in coordinate and momentum operators
satisfying nonlinear commutation relations. The multiplication in this algebra
generates an associative product of functions on the phase space. This product
is given by an integral kernel whose phase is the symplectic area of a
groupoid-consistent membrane. A symplectic phase space connection with
non-trivial curvature is extracted from the magnetic reflections associated
with the Stratonovich quantizer. Zero and constant curvature cases are
considered as examples. The quantization with both static and time dependent
electromagnetic fields is obtained. The expansion of the product by the
deformation parameter, written in the covariant form, is compared with the
known deformation quantization formulas.Comment: 23 page
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