Vector Fields in a Rindler Space

In questo lavoro ci si propone di studiare la quantizzazione del campo vettoriale, massivo e non massivo, in uno spazio-tempo di Rindler, considerando in particolare i gauge di Feynman e assiale. Le equazioni del moto vengono risolte esplicitamente in entrambi i casi; sotto opportune condizioni, è stato inoltre possibile trovare una base completa e ortonormale di soluzioni delle equazioni di campo in termini di modi normali di Fulling. Si è poi analizzata la quantizzazione dei campi vettoriali espressi in questa base.

Measurement of CP asymmetries in $\lambda^0_b \to pk^-$ and $\lambda^0_b \to p \pi^-$ decays at LHCb

The LHCb experiment has been designed to perform precision measurements in the flavour physics sector at the Large Hadron Collider (LHC) located at CERN. After the recent observation of CP violation in the decay of the Bs0 meson to a charged pion-kaon pair at LHCb, it is interesting to see whether the same quark-level transition in Λ0b baryon decays gives rise to large CP-violating effects. Such decay processes involve both tree and penguin Feynman diagrams and could be sensitive probes for physics beyond the Standard Model. The measurement of the CP-violating observable defined as ∆ACP = ACP(Λ0b → pK−)−ACP(Λ0b →pπ−),where ACP(Λ0b →pK−) and ACP(Λ0b →pπ−) are the direct CP asymmetries in Λ0b → pK− and Λ0b → pπ− decays, is presented for the first time using LHCb data. The procedure followed to optimize the event selection, to calibrate particle identification, to parametrise the various components of the invariant mass spectra, and to compute corrections due to the production asymmetry of the initial state and the detection asymmetries of the final states, is discussed in detail. Using the full 2011 and 2012 data sets of pp collisions collected with the LHCb detector, corresponding to an integrated luminosity of about 3 fb−1, the value ∆ACP = (0.8 ± 2.1 ± 0.2)% is obtained. The first uncertainty is statistical and the second corresponds to one of the dominant systematic effects. As the result is compatible with zero, no evidence of CP violation is found. This is the most precise measurement of CP violation in the decays of baryons containing the b quark to date. Once the analysis will be completed with an exhaustive study of systematic uncertainties, the results will be published by the LHCb Collaboration.

The Feynman-Kac formula and applications to PDEs

The thesis presents a probabilistic approach to the theory of semigroups of operators, with particular attention to the Markov and Feller semigroups. The first goal of this work is the proof of the fundamental Feynman-Kac formula, which gives the solution of certain parabolic Cauchy problems, in terms of the expected value of the initial condition computed at the associated stochastic diffusion processes. The second target is the characterization of the principal eigenvalue of the generator of a semigroup with Markov transition probability function and of second order elliptic operators with real coefficients not necessarily self-adjoint. The thesis is divided into three chapters. In the first chapter we study the Brownian motion and some of its main properties, the stochastic processes, the stochastic integral and the Itô formula in order to finally arrive, in the last section, at the proof of the Feynman-Kac formula. The second chapter is devoted to the probabilistic approach to the semigroups theory and it is here that we introduce Markov and Feller semigroups. Special emphasis is given to the Feller semigroup associated with the Brownian motion. The third and last chapter is divided into two sections. In the first one we present the abstract characterization of the principal eigenvalue of the infinitesimal generator of a semigroup of operators acting on continuous functions over a compact metric space. In the second section this approach is used to study the principal eigenvalue of elliptic partial differential operators with real coefficients. At the end, in the appendix, we gather some of the technical results used in the thesis in more details. Appendix A is devoted to the Sion minimax theorem, while in appendix B we prove the Chernoff product formula for not necessarily self-adjoint operators.