Jesús Clemente-Gallardo

Geometrization of quantum mechanics

We show that various descriptions of quantum mechanics can be represented in geometric terms. In particular, starting with the space of observables and using the momentum map associated with the unitary group, we give a unified geometric description of the different pictures of quantum mechanics. This construction is an alternative to the usual GNS construction for pure states.

Relating Lagrangian and Hamiltonian formalisms of LC circuits

The Lagrangian formalism defined by Scherpen et al. (2000) for (switching) electrical circuits, is adapted to the Lagrangian formalism defined on Lie algebroids. This allows us to define regular Lagrangians and consequently, well-defined Hamiltonian descriptions of arbitrary LC networks. The relation with other Hamiltonian approaches is also analyzed and interpreted.

BASICS OF QUANTUM MECHANICS, GEOMETRIZATION AND SOME APPLICATIONS TO QUANTUM INFORMATION

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schrodinger framework from this perspective and provide a description of the Weyl–Wigner construction. Finally, after reviewing the basics of the geometric formulation of quantum mechanics, we apply the methods presented to the most interesting cases of finite dimensional Hilbert spaces: those of two, three and four level systems (one qubit, one qutrit and two qubit systems). As a more practical application, we discuss the advantages that the geometric formulation of quantum mechanics can provide us with in the study of situations as the functional independence of entanglement witnesses.

CLASSICAL TENSORS AND QUANTUM ENTANGLEMENT I: PURE STATES

The geometrical description of a Hilbert space asociated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.

Comment on "Correlated electron-nuclear dynamics: exact factorization of the molecular wavefunction" [J. Chem. Phys. 137, 22A530 (2012)].

In spite of the relevance of the proposal introduced in the recent work by Abedi, Maitra, and Gross [J. Chem. Phys.137, 22A530 (Year: 2012)], there is an important ingredient which is missing. Namely, the proof that the norms of the electronic and nuclear wavefunctions which are the solutions to the nonlinear equations of motion are preserved by the evolution. To prove the conservation of these norms is precisely the objective of this Comment.

Statistics and Nosé formalism for Ehrenfest dynamics

Quantum dynamics (i.e. the Schr?dinger equation) and classical dynamics (i.e. Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper, we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well-defined Poisson bracket allows us to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of an Ehrenfest system, it is straightforward to extend the formalism of Nos? (invented to do constant temperature molecular dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.

Dirac structures for generalized Lie bialgebroids

We study Dirac structures for generalized Courant algebroids, which are doubles of generalized Lie bialgebroids. The cases investigated include graphs of bivector fields and characteristic pairs of some sub-bundles.

TOWARDS A DEFINITION OF QUANTUM INTEGRABILITY

We briefly review the most relevant aspects of complete integrability for classical systems and identify those aspects which should be present in a definition of quantum integrability. We show that a naive extension of classical concepts to the quantum framework would not work because all infinite dimensional Hilbert spaces are unitarilly isomorphic and, as a consequence, it would not be easy to define degrees of freedom. We argue that a geometrical formulation of quantum mechanics might provide a way out.

Discrete port-Hamiltonian systems

Either from a control theoretic viewpoint or from an analysis viewpoint it is necessary to convert smooth systems to discrete systems, which can then be implemented on computers for numerical simulations. Discrete models can be obtained either by discretizing a smooth model, or by directly modeling at the discrete level itself. One of the goals of this paper is to model port-Hamiltonian systems at the discrete level. We also show that the dynamics of the discrete models we obtain exactly correspond to the dynamics obtained via a usual discretization procedure. In this sense we offer an alternative to the usual procedure of modeling (at the smooth level) and discretization.

Reduction procedures in classical and quantum mechanics

We present, in a pedagogical style, many instances of reduction procedures appearing in a variety of physical situations, both classical and quantum. We concentrate on the essential aspects of any reduction procedure, both in the algebraic and geometrical setting, elucidating the analogies and the differences between the classical and the quantum situations.