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

The round table of Bialowieza forest

1. 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.

2. Regularization and Renormalization of Quantum Field Theories on Noncommutative Spaces

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...

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

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...

5. On Poisson Realizations of Transitive Lie Algebroids

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.

6. Deformation quantization for almost-Kähler manifolds

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...

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

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.

8. 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...

9. Green function for Klein-Gordon-Dirac equation

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...

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

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.

11. Statistical mechanics of a Class of Anyonic Systems. The Rigorous Approach

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...

13. EPR-B correlations: non-locality or geometry?

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.

14. Nonlinear Wave Equation in Special Coordinates

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....

15. Limit Spectra of Random Gram Matrices

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...

16. 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

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....

17. Geodetic Systems on Linear and Affine Groups. Classics and Quantization.

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....

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

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...

19. 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.

20. Quantization of the planar affinely-rigid body

Agnieszka MARTENS
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...

21. Classical and Quantized Affine Physics: A Step towards it

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 components...

23. Wigner Quantization on the Circle and R+

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...

24. Poisson configuration spaces, von Neumann algebras, and harmonic forms

Pages: 179 - 184
We give a short review of recent results on L2 -cohomology of infinite configuration spaces equipped with Poisson measures.

25. 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...

26. On the complexified affine metaplectic representation

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.

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

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.

28. Quadratic non-Riemannian Gravity

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...

29. Geometry of differential operators, odd Laplacians, and homotopy algebras

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.

30. The Jungle Book updated

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.