# Journal of Nonlinear Mathematical Physics

Volume 11, Issue Supplement 1, October 2004

### Geometrical Methods in Physics: Bialowieza XXI and XXII. Proceedings of the XXI and XXII Workshops on Geometrical Methods in Physics

June/July 2002 and 2003 Bialowieza, Poland

**Review Article**

## 3. So. . . what was the question?

Gérard G. Emch

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.

**Research Article**

## 4. Regularization and Renormalization of Quantum Field Theories on Noncommutative Spaces

Harald Grosse, Raimar Wulkenhaar

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...

**Research Article**

## 5. On the Resolution of Space-Time Singularities II

M. Maceda, J. Madore

Pages: 21 - 36

**Research Article**

## 6. On Pauli graded contractions of sl(3, C)

Miloslav Havlicek, Jiri Patera, Edita Pelantova, Jiri Tolar

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...

**Research Article**

## 7. On Poisson Realizations of Transitive Lie Algebroids

Yuri Vorobiev

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.

**Research Article**

## 8. Deformation quantization for almost-Kähler manifolds

Martin Schlichenmaier

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 supplies...

**Research Article**

## 9. New Geometrical Applications of the Elliptic Integrals: The Mylar Balloon

Ivaïlo M. Mladenov

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.

**Research Article**

## 10. von Neumann Quantization of Aharonov-Bohm Operator with Interaction: Scattering Theory, Spectral and Resonance Properties

Gilbert Honnouvo, Mahouton Norbert Hounkonnou, Gabriel Yves Hugues

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...

**Research Article**

## 11. Green function for Klein-Gordon-Dirac equation

Vasyl Kovalchuk

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...

**Research Article**

## 12. Reducible representations of CAR and CCR with possible applications to field quantization

Marek Czachor

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.

**Research Article**

## 13. Statistical mechanics of a Class of Anyonic Systems. The Rigorous Approach

Roman Gielerak, Robert Ralowski

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...

**Research Article**

## 14. When is a sum of projections equal to a scalar operator?

Yurii S. Samoǐlenko

Pages: 92 - 103

**Research Article**

## 15. EPR-B correlations: non-locality or geometry?

A.F. Kracklauer

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.

**Research Article**

## 16. Nonlinear Wave Equation in Special Coordinates

Alexander Shermenev

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....

**Research Article**

## 17. Limit Spectra of Random Gram Matrices

V.I. Serdobolskii

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...

**Research Article**

## 18. The Classification of the Bifurcations Emerging in the case of an Integrable Hamiltonian System with Two Degrees of Freedom when an Isoenergetic Surface is Non-Compact

Galina Goujvina

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....

**Research Article**

## 19. Geodetic Systems on Linear and Affine Groups. Classics and Quantization.

Jan J. Slawianowski

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....

**Research Article**

## 20. Action-Angle Analysis of Some Geometric Models of Internal Degrees of Freedom

Barbara Gołubowska

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 the...

**Research Article**

## 21. Hamiltonian dynamics of planar affinely-rigid body

Agnieszka Martens

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.

**Research Article**

## 22. Quantization of the planar affinely-rigid body

Agnieszka Martens

Pages: 151 - 156

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...

**Research Article**

## 23. Classical and Quantized Affine Physics: A Step towards it

Jan J. Slawianowski, Vasyl Kovalchuk

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...

**Research Article**

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

Agata Bezubik, Agata Dbrowska, Aleksander Strasburger

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 components...

**Research Article**

## 25. Wigner Quantization on the Circle and R+

G. Chadzitaskos, J. Tolar

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 the...

**Research Article**

## 26. Poisson configuration spaces, von Neumann algebras, and harmonic forms

Alexei Daletskii

Pages: 179 - 184

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

**Research Article**

## 27. Canonical Analysis of Symmetry Enhancement with Gauged Grassmannian Model

Sang-Ok Hahn, Phillial Oh, Cheonsoo Park, Sunyoung Shin

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 free...

**Short Communication**

## 28. On the complexified affine metaplectic representation

Ole Rask

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.

**Research Article**

## 29. Scattering and Spectral Singularities for some Dissipative Operators of Mathematical Physics

S.A. Stepin

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.

**Research Article**

## 30. Quadratic non-Riemannian Gravity

Dmitri Vassiliev

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...

**Research Article**

## 31. Geometry of differential operators, odd Laplacians, and homotopy algebras

Hovhannes Khudaverdian, Theodore Voronov

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.

**Research Article**

## 32. The Jungle Book updated

Bogdan Mielnik

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.