No abstract

Starting from the famous Pauli problem on the possibility to associate quantum states with probabilities, the formulation of quantum mechanics in which quantum states are described by fair probability distributions (tomograms, i.e. tomographic probabilities) is reviewed in a pedagogical style. The relation between the quantum state description and the classical state description is elucidated. The difference of those sets of tomograms is described by inequalities equivalent to a complete set of uncertainty relations for the quantum domain and to nonnegativity of probability density on phase space in the classical domain. Intersection of such sets is studied. The mathematical mechanism which allows to construct different kinds of tomographic probabilities like symplectic tomograms, spin tomograms, photon number tomograms, etc., is clarified and a connection with abstract Hilbert space properties is established. Superposition rule and uncertainty relations in terms of probabilities as well as quantum basic equation like quantum evolution and energy spectra equations are given in explicit form. A method to check experimentally uncertainty relations is suggested using optical tomograms. Entanglement phenomena and the connection with semigroups acting on simplexes are studied in detail for spin states in the case of two qubits. The star-product formalism is associated with the tomographic probability formulation of quantum mechanics.

We analyze the global theory of boundary conditions for a constrained quantum system with classical configuration space a compact Riemannian manifold M with regular boundary Γ = ∂M . The space M of self-adjoint extensions of the covariant Laplacian on M is shown to have interesting geometrical and topological properties which are related to the different topological closures of M . In this sense, the change of topology of M is connected with the non-trivial structure of M. The space M itself can be identified with the unitary group U(L 2 (Γ, C N )) of the Hilbert space of boundary data L 2 (Γ, C N ). This description, is shown to be equivalent to the classical von Neumann's description in terms of deficiency index subspaces, but it is more efficient and explicit because it is given only in terms of the boundary data, which are the natural external inputs of the system. A particularly interesting family of boundary conditions, identified as the set of unitary operators which are singular under the Cayley transform, C − ∩ C + (the Cayley manifold), turns out to play a relevant role in topology change phenomena. The singularity of the Cayley transform implies that some energy levels, usually associated with edge states, acquire an infinity energy when by an adiabatic change the boundary conditions reaches the Cayley submanifold C − . In this sense topological transitions require an infinite amount of quantum energy to occur, although the description of the topological transition in the space M is smooth. This fact has relevant implications in string theory for possible scenarios with joint descriptions of open and closed strings. In the particular case of elliptic self-adjoint boundary conditions, the space C − can be identified with a Lagrangian submanifold of the infinite dimensional Grassmannian. The corresponding Cayley manifold C − is dual of the Maslov class of M. The phenomena are illustrated with some simple low dimensional examples.

A multisymplectic structure on a manifold is defined by a closed differential form with zero characteristic distribution. Starting from the linear case, some of the basic properties of multisymplectic structures are described. Various examples of multisymplectic manifolds are considered, and special attention is paid to the canonical multisymplectic structure living on a bundle of exterior fc-forms on a manifold. For a class of multisymplectic manifolds admitting a 'Lagrangian' fibration, a general structure theorem is given which, in particular, leads to a classification of these manifolds in terms of a prescribed family of cohomology classes.1991 Mathematics subject classification (Amer. Math. Soc): primary 53C15, 58Axx.

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