Introduction to Quantum Mechanics and the Quantum-Classical transition

In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schroedinger and Heisenberg frameworks from this perspective and discuss how the momentum map associated to the action of the unitary group on the Hilbert space allows to relate both approaches. We also study Weyl-Wigner approach to Quantum Mechanics and discuss the implications of bi-Hamiltonian structures at the quantum level.Comment: Survey paper based on the lectures delivered at the XV International Workshop on Geometry and Physics Puerto de la Cruz, Tenerife, Canary Islands, Spain September 11-16, 2006. To appear in Publ. de la RSM

Energy and technological refurbishment of the School of Architecture Valle Giulia, Rome

Modern architecture built in historical urban contexts represents a demanding issue when its energy efficiency should be improved. Indeed, the strongest efforts have to be made to maintain the architectural identity and its harmony with the surrounding cultural heritage. This study deals with the main building of the School of Architecture Valle Giulia in Rome, designed by Enrico Del Debbio in the 30’s. Further constraints are related to several interventions of airspace expansion starting from 1958 which involved the building starting from 1958. So, preservation would mean highlighting its historic change but, adapting the built environment to the contemporary users’ needs. As above-mentioned, the building belongs to the Valle delle Accademie, within the historic park of Villa Borghese, so that to acquire landscaping values. Those latter ones call for ulterior requirements when any new design process is conceived. The study provides a global renewal of the building accounting for the current low Indoor Environmental Quality in both summer and winter seasons and the lack of suitability to the contemporary University student’s needs. The interaction between building performance and HVAC systems was studied by collecting data and architectural surveys conducted by all the architects who modified the building. This procedure was chosen since thermo-physical investigations are considered destructive due to required perforations to identify the actual wall layers. Moreover, thermographic surveys were carried out to validate the modelled building response. The result of the study is the identification of viable interventions to improve the accessibility and fruition of the building as well as its energy performance. A specific cost-benefit analysis was done to prioritize the design options along with considering the measures needed to preserve all the architectural features and values

Tensorial characterization and quantum estimation of weakly entangled qubits

In the case of two qubits, standard entanglement monotones like the linear entropy fail to provide an efficient quantum estimation in the regime of weak entanglement. In this paper, a more efficient entanglement estimation, by means of a novel class of entanglement monotones, is proposed. Following an approach based on the geometric formulation of quantum mechanics, these entanglement monotones are defined by inner products on invariant tensor fields on bipartite qubit orbits of the group SU(2)xSU(2).Comment: 23 pages, 3 figure

The Future Impacts of ESL Events in Euro-Mediterranean Coastal Cities: The Coast-RiskBySea Model to Assess the Potential Economic Damages in Naples, Marseille and Barcelona

In coastal cities, the effects of climate change will cause an increase in the intensity and frequency of extreme sea level (ESL). In this scenario, the application of the Coast-RiskBySea model is proposed to assess the economic impacts of ESL on the built environment in three Euro-Mediterranean coastal cities: Naples, Barcelona, and Marseille. The risk (land use-based) is assessed in the GIS environment as a function of the potential direct and tangible economic damages. The results highlight risk scenarios in all three cities with significant economic damages expected, requiring the implementation of climate mitigation and adaptation measures to reduce the current impacts and limit future ones. The simulations highlight the potential of both remote sensing data and GIS systems to carry out homogeneous environmental analyses over wide areas. The results that were obtained are compared with existing works to verify the reliability of the Coast-RiskBySea model

Tensorial dynamics on the space of quantum states

A geometric description of the space of states of a finite-dimensional quantum system and of the Markovian evolution associated with the Kossakowski-Lindblad operator is presented. This geometric setting is based on two composition laws on the space of observables defined by a pair of contravariant tensor fields. The first one is a Poisson tensor field that encodes the commutator product and allows us to develop a Hamiltonian mechanics. The other tensor field is symmetric, encodes the Jordan product and provides the variances and covariances of measures associated with the observables. This tensorial formulation of quantum systems is able to describe, in a natural way, the Markovian dynamical evolution as a vector field on the space of states. Therefore, it is possible to consider dynamical effects on non-linear physical quantities, such as entropies, purity and concurrence. In particular, in this work the tensorial formulation is used to consider the dynamical evolution of the symmetric and skew-symmetric tensors and to read off the corresponding limits as giving rise to a contraction of the initial Jordan and Lie products.Comment: 31 pages, 2 figures. Minor correction

Tangent bundle geometry from dynamics: application to the Kepler problem

In this paper we consider a manifold with a dynamical vector field and inquire about the possible tangent bundle structures which would turn the starting vector field into a second order one. The analysis is restricted to manifolds which are diffeomorphic with affine spaces. In particular, we consider the problem in connection with conformal vector fields of second order and apply the procedure to vector fields conformally related with the harmonic oscillator (f-oscillators) . We select one which covers the vector field describing the Kepler problem.Comment: 17 pages, 2 figure

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: 16 pages, 1 figure, to appear in Int. J. Geom. Meth. Mod. Phy