264 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
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
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
Tverberg-type theorems for intersecting by rays
In this paper we consider some results on intersection between rays and a
given family of convex, compact sets. These results are similar to the center
point theorem, and Tverberg's theorem on partitions of a point set
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
Method for the rapid temperature correction of a transmission in an inhomogeneous atmosphere
The value of the transmission function in a heterogeneous atmosphere is determined by iterative correction of values in particular layers. The iterative equation and a set of absolute values of errors is presented in two tables
Infinitesimal deformations of a formal symplectic groupoid
Given a formal symplectic groupoid over a Poisson manifold ,
we define a new object, an infinitesimal deformation of , which can be
thought of as a formal symplectic groupoid over the manifold equipped with
an infinitesimal deformation of the Poisson bivector
field . The source and target mappings of a deformation of are
deformations of the source and target mappings of . To any pair of natural
star products having the same formal symplectic groupoid
we relate an infinitesimal deformation of . We call it the deformation
groupoid of the pair . We give explicit formulas for the
source and target mappings of the deformation groupoid of a pair of star
products with separation of variables on a Kaehler- Poisson manifold. Finally,
we give an algorithm for calculating the principal symbols of the components of
the logarithm of a formal Berezin transform of a star product with separation
of variables. This algorithm is based upon some deformation groupoid.Comment: 22 pages, the paper is reworked, new proofs are adde
A note on Makeev's conjectures
A counterexample is given for the Knaster-like conjecture of Makeev for
functions on . Some particular cases of another conjecture of Makeev, on
inscribing a quadrangle into a smooth simple closed curve, are solved
positively
Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics
The knowledge of quantum phase flow induced under the Weyl's association rule
by the evolution of Heisenberg operators of canonical coordinates and momenta
allows to find the evolution of symbols of generic Heisenberg operators. The
quantum phase flow curves obey the quantum Hamilton's equations and play the
role of characteristics. At any fixed level of accuracy of semiclassical
expansion, quantum characteristics can be constructed by solving a coupled
system of first-order ordinary differential equations for quantum trajectories
and generalized Jacobi fields. Classical and quantum constraint systems are
discussed. The phase-space analytic geometry based on the star-product
operation can hardly be visualized. The statement "quantum trajectory belongs
to a constraint submanifold" can be changed e.g. to the opposite by a unitary
transformation. Some of relations among quantum objects in phase space are,
however, left invariant by unitary transformations and support partly geometric
relations of belonging and intersection. Quantum phase flow satisfies the
star-composition law and preserves hamiltonian and constraint star-functions.Comment: 27 pages REVTeX, 6 EPS Figures. New references added. Accepted for
publication to JM
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