G. Marmo

Geometric Hamilton-Jacobi Theory

The Hamilton–Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in order to avoid the bias of the existence of a natural symplectic structure on the cotangent bundle. First it is developed for systems described by regular Lagrangians and then extended to systems described by singular Lagrangians with no secondary constraints. We also consider the example of the free relativistic particle, the rigid body and the electron-monopole system.

The Feynman problem and the inverse problem for Poisson dynamics

Abstract We review the Feynman proof of the Lorentz force equations, as well as its generalization to the dynamics of particles with internal degrees of freedom. In addition, we discuss the inverse problem for Poisson dynamics and the inverse problem of the calculus of variations. It is proved that the only classical dynamics compatible with localizability and the existence of second order differential equations on tangent bundles over arbitrary configuration spaces is necessarily of the Lagrangian type. Furthermore, if the dynamics is independent of the velocity of test particles, it must correspond to that of a particle coupled to an electromagnetic field and/or a gravitational field. The same ideas are carried out for particles with internal degrees of freedom. In this case, if we insist on a weak localizability condition and the existence of a second order Hamiltonian differential equation, then the dynamics results from a singular Lagrangian. (Here we assume in addition that the dynamics satisfies a regularity condition.) These results extend those of Feynman and provide the conditions which guarantee the existence of a Lagrangian description. They are applied to systematically discuss Feynmans problem for systems possessing Lie groups as configuration spaces, with internal variables modeled on Lie algebras of groups. Finally, we illustrate what happens when the condition of localizability is dropped. In this regard, we obtain alternative Hamiltonian descriptions of standard dynamical systems. These non-standard solutions are discussed within the framework of Lie-Poisson structures.

Homodyne estimation of quantum state purity by exploiting the covariant uncertainty relation

We experimentally verify the uncertainty relations for the mixed states in tomographic representation by measuring the radiation field tomograms, i.e. homodyne distributions. Thermal states of a single-mode radiation field are discussed in detail as a paradigm of the mixed quantum state. On considering the connection between generalized uncertainty relations and optical tomograms, it is seen that the purity of the states can be retrieved by statistical analysis of the homodyne data. The purity parameter assumes a relevant role in quantum information where the effective fidelities of protocols depend critically on the purity of the information carrier states. In this context, the homodyne detector becomes an easy-to-handle purity-meter for the state on line with a running quantum information protocol.

Alternative structures and bi-Hamiltonian systems

In the study of bi-Hamiltonian systems (both classical and quantum) one starts with a given dynamics and looks for all alternative Hamiltonian descriptions it admits.In this paper we start with two compatible Hermitian structures (the quantum analog of two compatible classical Poisson brackets) and look for all the dynamical systems which turn out to be bi-Hamiltonian with respect to them.In the study of bi-Hamiltonian systems (both classical and quantum) one starts with a given dynamics and looks for all alternative Hamiltonian descriptions it admits. In this paper, we start with two compatible Hermitian structures (the quantum analogue of two compatible classical Poisson brackets) and look for all the dynamical systems which turn out to be bi-Hamiltonian with respect to them.

On the geometry of Lie algebras and Poisson tensors

A geometric programme to analyse the structure of Lie algebras is presented with special emphasis on the geometry of linear Poisson tensors. The notion of decomposable Poisson tensors is introduced and an algorithm to construct all solvable Lie algebras is presented. Poisson-Liouville structures are also introduced to discuss a new class of Lie algebras which include, as a subclass, semi-simple Lie algebras. A decomposition theorem for Poisson tensors is proved for a class of Poisson manifolds including linear ones. Simple Lie algebras are also discussed from this viewpoint and lower-dimensional real Lie algebras are analysed.

Isoperiodic classical systems and their quantum counterparts

Abstract One-dimensional isoperiodic classical systems have been first analyzed by Abel. Abel’s characterization can be extended for singular potentials and potentials which are not defined on the whole real line. The standard shear equivalence of isoperiodic potentials can also be extended by using reflection and inversion transformations. We provide a full characterization of isoperiodic rational potentials showing that they are connected by translations, reflections or Joukowski transformations. Upon quantization many of these isoperiodic systems fail to exhibit identical quantum energy spectra. This anomaly occurs at order O( ℏ 2 ) because semiclassical corrections of energy levels of order O( ℏ ) are identical for all isoperiodic systems. We analyze families of systems where this quantum anomaly occurs and some special systems where the spectral identity is preserved by quantization. Conversely, we point out the existence of isospectral quantum systems which do not correspond to isoperiodic classical systems.

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.

The Topology and Geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

The theory of self-adjoint extensions of first and second order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators. nThe theory is developed by exploiting the geometrical structures attached to them and, by using an adapted Cayley transform on each case, the space

Brownian motion on Lie groups and open quantum systems

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THE QUANTUM-CLASSICAL TRANSITION: THE FATE OF THE COMPLEX STRUCTURE

of such extensions is shown to have a canonical group composition law structure. nThe obtained results are compared with von Neumanns Theorem characterising the self-adjoint extensions of densely defined symmetric operators on Hilbert spaces. The 1D case is thoroughly investigated. nThe geometry of the submanifold of elliptic self-adjoint extensions