233 research outputs found
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
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