Pages: 1 - 8

An overview of the lectures at the 2002 Bialowiea Workshop is presented. The symbol*
after a proper name indicates that a copy of the corresponding contribution to the
proceedings was communicated to the author of this summary.

ISSN: 1402-9251

Volume 11, Issue Supplement 1, October 2004

June/July 2002 and 2003 Bialowieza, Poland

Preface

The round table of Bialowieza forest

Pages: 1 - 8

An overview of the lectures at the 2002 Bialowiea Workshop is presented. The symbol*
after a proper name indicates that a copy of the corresponding contribution to the
proceedings was communicated to the author of this summary.

Pages: 9 - 20

We first review regularization methods based on matrix geometry which provide an
ultraviolet cut-off for scalar fields respecting the symmetries. Sections of bundles over
the sphere can be quantized, too. This procedure even allows to regularize supesymmetry without violating it. Recently, this work...

Pages: 21 - 36

Pages: 37 - 42

We consider a special fine grading of sl(3, C), where the grading subspaces are geerated by 3 × 3 generalized Pauli matrices. This fine grading decomposes sl(3, C)
into eight onedimensional subspaces. Our aim is to find all contractions of sl(3, C)
which preserve this grading. We have found that the...

Pages: 43 - 48

We show that every transitive Lie algebroid over a connected symplectic manifold
comes from an intrinsic Lie algebroid of a symplectic leaf of a certain Poisson structure.
The reconstruction of the corresponding Poisson structures from the Lie algebroid is
given in terms of coupling tensors.

Pages: 49 - 54

On an arbitrary almost-Kähler manifold, starting from a natural affine connection
with nontrivial torsion which respects the almost-Kähler structure, in joint work with
A. Karabegov a Fedosov-type deformation quantization on this manifold was costructed. This contribution reports on the result and...

Pages: 55 - 65

An explicit parameterization in terms of elliptic integrals (functions) for the Mylar
balloon is found which then is used to calculate various geometric quantities as well
as to study all kinds of geodesics on this surface.

Pages: 66 - 71

Using the theory of self-adjoint extensions, we study the interaction model formally
given by the Hamiltonian H + V (r), where H is the Aharonov-Bohm Hamiltonian
and V (r) is the -type interaction potential on the cylinder of radius R . We give the
mathematical definition of the model, the self-adjointness...

Pages: 72 - 77

The Green function for Klein-Gordon-Dirac equation is obtained. The case with the
dominating Klein-Gordon term is considered. There seems to be a formal analogy
between our problem and a certain problem for a 4-dimensional particle moving in the
external field. The explicit relations between the wave...

Pages: 78 - 84

Reducible representations of CAR and CCR are applied to second quantization of
Dirac and Maxwell fields. The resulting field operators are indeed operators and not
operator-valued distributions. Examples show that the formalism may lead to a finite
quantum field theory.

Pages: 85 - 91

A class of involutive Wick algebras (called anyonic-type Wick algebras) is selected
and some its elementary properties are described. In particular, the Fock representtions of the selected anyonic-type commutation relations are described. For the class
of so-called r-yonic systems the question of the...

Pages: 92 - 103

Pages: 104 - 109

A photoelectron-by-photoelectron classical simulation of EPR-B correlations is dscribed. It is shown that this model can be made compatible with Bell's renowned
"no-go" theorem by restricting the source to that which produces only what is known
as paired photons.

Pages: 110 - 115

Some classical types of nonlinear periodic wave motion are studied in special coodinates. In the case of cylinder coordinates, the usual perturbation techniques leads
to the overdetermined systems of linear algebraic equations for unknown coefficients
whose compatibility is key step of the investigation....

Pages: 116 - 121

Solutions to basic non-linear limit spectral equation for matrices RT
R of increasing dmension are investigated, where R are rectangular random matrices with independent
normal entries. The analytical properties of limiting normed trace for the resolvent
of RT
R are investigated, boundaries of limit...

Pages: 122 - 129

On a symplectical manifold M4
consider a Hamiltonian system with two degrees of
freedom, integrable with the help of an additional integral f. According to the welknown Liouville theorem, non-singular level surfaces of the integrals H and f can be
represented as unions of tori, cylinders and planes....

Pages: 130 - 137

Described are classical and quantized systems on linear and affine groups. Unlike
the traditional models applied in astrophysics, nuclear physics, molecular vibrations
and elasticity, our models are not only kinematically ruled by the affine group, but
also their kinetic energies are affinely invariant....

Pages: 138 - 144

We derive and discuss equations of motion of infinitesimal affinely-rigid body moving
in Riemannian spaces. There is no concept of extended rigid and affinely rigid body in
a general Riemannian space. Therefore the gyroscopes with affine degrees of freedom
are described as moving bases attached to...

Pages: 145 - 150

We discuss the dynamics of an affinely-rigid body in two dimensions. Translational
degrees of freedom are neglected. The special stress is laid on completely integrable
models solvable in terms of the separation of variables method.

Pages: 145 - 150

This paper is a continuation of [1] where the classical model was analyzed. Discussed
are some quantization problems of two-dimensional affinely rigid body with the double
dynamical isotropy. Considered are highly symmetric models for which the variables
can be separated. Some explicit solutions are...

Pages: 157 - 166

The classical and quantum mechanics of systems on Lie groups and their homogeneous
spaces are described. The special stress is laid on the dynamics of deformable bodies
and the mutual coupling between rotations and deformations. Deformative modes are
discretized, i.e., it is assumed that the relevant...

22. # A new derivation of the plane wave expansion into spherical harmonics and related Fourier transforms

Pages: 167 - 173

This article summarizes a new, direct approach to the determination of the expansion
into spherical harmonics of the exponential ei(x|y)
with x, y Rd
. It is elementary in
the sense that it is based on direct computations with the canonical decomposition
of homogeneous polynomials into harmonic...

Pages: 174 - 178

We construct a deformation quantization for two cases of configuration spaces: the
multiplicative group of positive real numbers R+ and the circle S1
. In these cases we
define the momenta using the Fourier transform. Using the identification of symbols of
quantum observables -- real functions on...

Pages: 179 - 184

We give a short review of recent results on L2
-cohomology of infinite configuration
spaces equipped with Poisson measures.

Pages: 185 - 190

We study the Hamiltonian structure of the gauge symmetry breaking and enhancment. After giving a general discussion of these phenomena in terms of the constrained
phase space, we perform a canonical analysis of the Grassmannian nonlinear sigma
model coupled with Chern-Simons term, which contains a...

Pages: 191 - 193

We study exponentiability of the infinite polynomials with maximal degree 2 of cration and annihilation operators, which give a Fock Space-representation of the coplexification of the affine symplectic group.

Pages: 194 - 203

Analogies in the spectral study of dissipative Schrödinger operator and Boltmann transport operator are analyzed. Scattering theory technique together
with functional model approach are applied to construct spectral representtions for these operators.

Pages: 204 - 216

We consider spacetime to be a connected real 4-manifold equipped with a Lorentzian
metric and an affine connection. The 10 independent components of the (symmetric)
metric tensor and the 64 connection coefficients are the unknowns of our theory. We
introduce an action which is quadratic in curvature...

Pages: 217 - 227

We give a complete description of differential operators generating a given bracket. In
particular we consider the case of Jacobi-type identities for odd operators and brackets.
This is related with homotopy algebras using the derived bracket construction.

Pages: 228 - 236

Data from many sources indicate that the Earth ecological crisis might not wait till
distant future. To avert it, some difficult truth must be accepted and adequate steps
taken. One of them is the strict protection of the world forests, even at the cost of
the short term economic growth.