Symmetry reduction, integrability and reconstruction in k-symplectic field theory

We investigate the reduction process of a k-symplectic field theory whose Lagrangian is invariant under a symmetry group. We give explicit coordinate expressions of the resulting reduced partial differential equations, the so-called Lagrange-Poincare field equations. We discuss two issues about reconstructing a solution from a given solution of the reduced equations. The first one is an interpretation of the integrability conditions, in terms of the curvatures of some connections. The second includes the introduction of the concept of a k-connection to provide a reconstruction method. We show that an invariant Lagrangian, under suitable regularity conditions, defines a `mechanical' k-connection.Comment: 37 page

QSES's and the Quantum Jump

The stochastic methods in Hilbert space have been used both from a fundamental and a practical point of view. The result we report here concerns only the idea of applying these methods to model the evolution of quantum systems and does not enter into the question of their fundamental or practical status. It can be easily stated as follows: Once a quantum stochastic evolution scheme is assumed, the incompatibility between the Markov property and the notion of quantum jump is rapidly established.Comment: LaTeX2e, 3 pages, no figures. Included in the Proceedings of the 3rd Workshop on Mysteries, Puzzles and Paradoxes in Quantum Mechanics, Gargnano, Italy, September 17-23, 200