On the Tomographic Picture of Quantum Mechanics

We formulate necessary and sufficient conditions for a symplectic tomogram of a quantum state to determine the density state. We establish a connection between the (re)construction by means of symplectic tomograms with the construction by means of Naimark positive-definite functions on the Weyl-Heisenberg group. This connection is used to formulate properties which guarantee that tomographic probabilities describe quantum states in the probability representation of quantum mechanics.Comment: 10 pages,latex,submitted to Physics Letters

Three lectures on global boundary conditions and the theory of self--adjoint extensions of the covariant Laplace--Beltrami and Dirac operators on Riemannian manifolds with boundary

In these three lectures we will discuss some fundamental aspects of the theory of self-adjoint extensions of the covariant Laplace-Beltrami and Dirac operators on compact Riemannian manifolds with smooth boundary emphasizing the relation with the theory of global boundary conditions. Self-adjoint extensions of symmetric operators, specially of the Laplace-Beltrami and Dirac operators, are fundamental in Quantum Physics as they determine either the energy of quantum systems and/or their unitary evolution. The well-known von Neumann's theory of self-adjoint extensions of symmetric operators is not always easily applicable to differential operators, while the description of extensions in terms of boundary conditions constitutes a more natural approach. Thus an effort is done in offering a description of self-adjoint extensions in terms of global boundary conditions showing how an important family of self-adjoint extensions for the Laplace-Beltrami and Dirac operators are easily describable in this way. Moreover boundary conditions play in most cases an significant physical role and give rise to important physical phenomena like the Casimir effect. The geometrical and topological structure of the space of global boundary conditions determining regular self-adjoint extensions for these fundamental differential operators is described. It is shown that there is a natural homology class dual of the Maslov class of the space. A new feature of the theory that is succinctly presented here is the relation between topology change on the system and the topology of the space of self-adjoint extensions of its Hamiltonian. Some examples will be commented and the one-dimensional case will be thoroughly discussed.Comment: Proceedings of XXIFWGP 2012; Classifications: 02.30.Tb, 02.40.Vh, 03.65.-w, 03.65.D

On the tomographic description of classical fields

After a general description of the tomographic picture for classical systems, a tomographic description of free classical scalar fields is proposed both in a finite cavity and the continuum. The tomographic description is constructed in analogy with the classical tomographic picture of an ensemble of harmonic oscillators. The tomograms of a number of relevant states such as the canonical distribution, the classical counterpart of quantum coherent states and a new family of so called Gauss--Laguerre states, are discussed. Finally the Liouville equation for field states is described in the tomographic picture offering an alternative description of the dynamics of the system that can be extended naturally to other fields