Classical limit of the Nelson model

Since the development of quantum mechanics it has been natural to analyze the connection between classical and quantum mechanical descriptions of physical systems. In particular one should expect that in some sense when quantum mechanical effects becomes negligible the system will behave like it is dictated by classical mechanics. One famous relation between classical and quantum theory is due to Ehrenfest. This result was later developed and put on firm mathematical foundations by Hepp. He proved that matrix elements of bounded functions of quantum observables between suitable coherents states (that depend on Planck's constant h) converge to classical values evolving according to the expected classical equations when h goes to zero. His results were later generalized by Ginibre and Velo to bosonic systems with infinite degrees of freedom and scattering theory. In this thesis we study the classical limit of Nelson model, that describes non relativistic particles, whose evolution is dictated by Schrödinger equation, interacting with a scalar relativistic field, whose evolution is dictated by Klein-Gordon equation, by means of a Yukawa-type potential. The classical limit is a mean field and weak coupling limit. We proved that the transition amplitude of a creation or annihilation operator, between suitable coherent states, converges in the classical limit to the solution of the system of differential equations that describes the classical evolution of the theory. The quantum evolution operator converges to the evolution operator of fluctuations around the classical solution. Transition amplitudes of normal ordered products of creation and annihilation operators between coherent states converge to suitable products of the classical solutions. Transition amplitudes of normal ordered products of creation and annihilation operators between fixed particle states converge to an average of products of classical solutions, corresponding to different initial conditions.

Regularity of solutions of the p-Laplace equation in the Heisenberg group

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An infinite level atom coupled to a heat bath

Wir untersuchen die Mathematik endlicher, an ein Wärmebad gekoppelter Teilchensysteme. Das Standard-Modell der Quantenelektrodynamik für Temperatur Null liefert einen Hamilton-Operator H, der die Energie von Teilchen beschreibt, welche mit Photonen wechselwirken. Im Heisenbergbild ist die Zeitevolution des physikalischen Systems durch die Wirkung einer Ein-Parameter-Gruppe auf eine Menge von Observablen A gegeben: Diese steht im Zusammenhang mit der Lösung der Schrödinger-Gleichung für H. Um Zustände von A, welche das physikalische System in der Nähe des thermischen Gleichgewichts zur Temperatur T darstellen, zu beschreiben, folgen wir dem Ansatz von Jaksic und Pillet, eine Darstellung von A zu konstruieren. Die Vektoren in dieser Darstellung definieren die Zustände, die Zeitentwicklung wird mit Hilfe des Standard Liouville-Operators L beschrieben. In dieser Doktorarbeit werden folgende Resultate bewiesen bzw. hergeleitet: - die Konstuktion einer Darstellung - die Selbstadjungiertheit des Standard Liouville-Operators - die Existenz eines Gleichgewichtszustandes in dieser Darstellung - der Limes des physikalischen Systems für große Zeiten.

Analysis of the Kohn Laplacian on the Heisenberg Group and on Cauchy-Riemann Manifolds

The purpose of this study is to analyse the regularity of a differential operator, the Kohn Laplacian, in two settings: the Heisenberg group and the strongly pseudoconvex CR manifolds. The Heisenberg group is defined as a space of dimension 2n+1 with a product. It can be seen in two different ways: as a Lie group and as the boundary of the Siegel UpperHalf Space. On the Heisenberg group there exists the tangential CR complex. From this we define its adjoint and the Kohn-Laplacian. Then we obtain estimates for the Kohn-Laplacian and find its solvability and hypoellipticity. For stating L^p and Holder estimates, we talk about homogeneous distributions. In the second part we start working with a manifold M of real dimension 2n+1. We say that M is a CR manifold if some properties are satisfied. More, we say that a CR manifold M is strongly pseudoconvex if the Levi form defined on M is positive defined. Since we will show that the Heisenberg group is a model for the strongly pseudo-convex CR manifolds, we look for an osculating Heisenberg structure in a neighborhood of a point in M, and we want this structure to change smoothly from a point to another. For that, we define Normal Coordinates and we study their properties. We also examinate different Normal Coordinates in the case of a real hypersurface with an induced CR structure. Finally, we define again the CR complex, its adjoint and the Laplacian operator on M. We study these new operators showing subelliptic estimates. For that, we don't need M to be pseudo-complex but we ask less, that is, the Z(q) and the Y(q) conditions. This provides local regularity theorems for Laplacian and show its hypoellipticity on M.

Dynamics of the Heisenberg XXZ model across the ferromagnetic transition

In questa tesi viene presentata un'analisi numerica dell'evoluzione dinamica del modello di Heisenberg XXZ, la cui simulazione è stata effettuata utilizzando l'algoritmo che va sotto il nome di DMRG. La transizione di fase presa in esame è quella dalla fase paramagnetica alla ferromagnetica: essa viene simulata in una catena di 12 siti per vari tempi di quench. In questo modo si sono potuti esplorare diversi regimi di transizione, da quello istantaneo al quasi-adiabatico. Come osservabili sono stati scelti l'entropia di entanglement, la magnetizzazione di mezza catena e lo spettro dell'entanglement, particolarmente adatti per caratterizzare la fisica non all'equilibrio di questo tipo di sistemi. Lo scopo dell'analisi è tentare una descrizione della dinamica fuori dall'equilibrio del modello per mezzo del meccanismo di Kibble-Zurek, che mette in relazione la sviluppo di una fase ordinata nel sistema che effettua la transizione quantistica alla densità di difetti topologici, la cui legge di scala è predicibile e legata agli esponenti critici universali caratterizzanti la transizione.

Perinatal care at the limit of viability between 22 and 26 completed weeks of gestation in Switzerland. 2011 revision of the Swiss recommendations

Perinatal care of pregnant women at high risk for preterm delivery and of preterm infants born at the limit of viability (22-26 completed weeks of gestation) requires a multidisciplinary approach by an experienced perinatal team. Limited precision in the determination of both gestational age and foetal weight, as well as biological variability may significantly affect the course of action chosen in individual cases. The decisions that must be taken with the pregnant women and on behalf of the preterm infant in this context are complex and have far-reaching consequences. When counselling pregnant women and their partners, neonatologists and obstetricians should provide them with comprehensive information in a sensitive and supportive way to build a basis of trust. The decisions are developed in a continuing dialogue between all parties involved (physicians, midwives, nursing staff and parents) with the principal aim to find solutions that are in the infant's and pregnant woman's best interest. Knowledge of current gestational age-specific mortality and morbidity rates and how they are modified by prenatally known prognostic factors (estimated foetal weight, sex, exposure or nonexposure to antenatal corticosteroids, single or multiple births) as well as the application of accepted ethical principles form the basis for responsible decision-making. Communication between all parties involved plays a central role. The members of the interdisciplinary working group suggest that the care of preterm infants with a gestational age between 22 0/7 and 23 6/7 weeks should generally be limited to palliative care. Obstetric interventions for foetal indications such as Caesarean section delivery are usually not indicated. In selected cases, for example, after 23 weeks of pregnancy have been completed and several of the above mentioned prenatally known prognostic factors are favourable or well informed parents insist on the initiation of life-sustaining therapies, active obstetric interventions for foetal indications and provisional intensive care of the neonate may be reasonable. In preterm infants with a gestational age between 24 0/7 and 24 6/7 weeks, it can be difficult to determine whether the burden of obstetric interventions and neonatal intensive care is justified given the limited chances of success of such a therapy. In such cases, the individual constellation of prenatally known factors which impact on prognosis can be helpful in the decision making process with the parents. In preterm infants with a gestational age between 25 0/7 and 25 6/7 weeks, foetal surveillance, obstetric interventions for foetal indications and neonatal intensive care measures are generally indicated. However, if several prenatally known prognostic factors are unfavourable and the parents agree, primary non-intervention and neonatal palliative care can be considered. All pregnant women with threatening preterm delivery or premature rupture of membranes at the limit of viability must be transferred to a perinatal centre with a level III neonatal intensive care unit no later than 23 0/7 weeks of gestation, unless emergency delivery is indicated. An experienced neonatology team should be involved in all deliveries that take place after 23 0/7 weeks of gestation to help to decide together with the parents if the initiation of intensive care measures appears to be appropriate or if preference should be given to palliative care (i.e., primary non-intervention). In doubtful situations, it can be reasonable to initiate intensive care and to admit the preterm infant to a neonatal intensive care unit (i.e., provisional intensive care). The infant's clinical evolution and additional discussions with the parents will help to clarify whether the life-sustaining therapies should be continued or withdrawn. Life support is continued as long as there is reasonable hope for survival and the infant's burden of intensive care is acceptable. If, on the other hand, the health car...