Lie antialgebras: cohomology and representations

We describe the main algebraic and geometric properties of the class of algebras introduced in [arXiv:0705.1629]. We discuss their origins in symplectic geometry and associative algebra, and the notions of cohomology and representations. We formulate classification theorems and give a number of examples.Comment: 11 pages, XXVII Workshop, Geometrical Methods in Mathematical Physics., Bialowieza : Pologne (2008

Lorentz Covariant Distributions with the Spectral Conditions

The Lorentz covariant tempered disributions with the supports in the product of the closed upper light cones are described.Comment: 13 page

Topological spectrum of classical configurations

For any classical field configuration or mechanical system with a finite number of degrees of freedom we introduce the concept of topological spectrum. It is based upon the assumption that for any classical configuration there exists a principle fiber bundle that contains all the physical and geometric information of the configuration. The topological spectrum follows from the investigation of the corresponding topological invariants. Examples are given which illustrate the procedure and the significance of the topological spectrum as a discretization relationship among the parameters that determine the physical meaning of classical configurations

Time Asymmetric Quantum Mechanics

The meaning of time asymmetry in quantum physics is discussed. On the basis of a mathematical theorem, the Stone--von Neumann theorem, the solutions of the dynamical equations, the Schr\"odinger equation (1) for states or the Heisenberg equation (6a) for observables are given by a unitary group. Dirac kets require the concept of a RHS (rigged Hilbert space) of Schwartz functions; for this kind of RHS a mathematical theorem also leads to time symmetric group evolution. Scattering theory suggests to distinguish mathematically between states (defined by a preparation apparatus) and observables (defined by a registration apparatus (detector)). If one requires that scattering resonances of width $\Gamma$ and exponentially decaying states of lifetime $\tau=\frac{\hbar}{\Gamma}$ should be the same physical entities (for which there is sufficient evidence) one is led to a pair of RHS's of Hardy functions and connected with it, to a semigroup time evolution $t_{0}\leq t<\infty$, with the puzzling result that there is a quantum mechanical beginning of time, just like the big bang time for the universe, when it was a quantum system. The decay of quasi-stable particles is used to illustrate this quantum mechanical time asymmetry. From the analysis of these processes, we show that the properties of rigged Hilbert spaces of Hardy functions are suitable for a formulation of time asymmetry in quantum mechanics

The Jacobi group and the squeezed states - some comments

The generalized coherent states attached to the Jacobi group realize the squeezed states. Imposing hermitian conjugacy to the generators of the Jacobi algebra, we find out the form of the weight function appearing in the scalar product. We show effectively the orthonormality of the base functions with respect to the scalar product. From the explicit form of the reproducing kernel, we find out the expression of the multiplier in a holomorphic representation of the Jacobi group.Comment: 9 pages, to appear in AIP Conference Proceedings, Geometric Methods In Physics, Bialowieza (Poland), June 28 -July 4 2008, Editor(s): P. Kielanowski, S. T. Ali, A. Odzijewicz, M. Schlichenmaier, Th. Vorono

Relativistic Equations for Spin Particles: What Can We Learn From Noncommutativity?

We derive relativistic equations for charged and neutral spin particles. The approach for higher-spin particles is based on generalizations of the Bargmann-Wigner formalism. Next, we study, what new physical information can the introduction of non-commutativity give us. Additional non-commutative parameters can provide a suitable basis for explanation of the origin of mass.Comment: 5 pages, aipproc.cls, no figures, presented at the XXVIII WGMP09, Bialowieza, Poland, June 28-July 4, 2009. The extended version is contributed to the 12th International Workshop "What Comes Beyond the Standard Models?", July 14- 24, 2009, Bled, Sloveni

Noncommutative complex Grosse-Wulkenhaar model

This paper stands for an application of the noncommutative (NC) Noether theorem, given in our previous work [AIP Proc 956 (2007) 55-60], for the NC complex Grosse-Wulkenhaar model. It provides with an extension of a recent work [Physics Letters B 653 (2007) 343-345]. The local conservation of energy-momentum tensors (EMTs) is recovered using improvement procedures based on Moyal algebraic techniques. Broken dilatation symmetry is discussed. NC gauge currents are also explicitly computed.Comment: 8 page