A linear O(N) model: a functional renormalization group approach for flat and curved space

In questa tesi sono state applicate le tecniche del gruppo di rinormalizzazione funzionale allo studio della teoria quantistica di campo scalare con simmetria O(N) sia in uno spaziotempo piatto (Euclideo) che nel caso di accoppiamento ad un campo gravitazionale nel paradigma dell'asymptotic safety. Nel primo capitolo vengono esposti in breve alcuni concetti basilari della teoria dei campi in uno spazio euclideo a dimensione arbitraria. Nel secondo capitolo si discute estensivamente il metodo di rinormalizzazione funzionale ideato da Wetterich e si fornisce un primo semplice esempio di applicazione, il modello scalare. Nel terzo capitolo è stato studiato in dettaglio il modello O(N) in uno spaziotempo piatto, ricavando analiticamente le equazioni di evoluzione delle quantità rilevanti del modello. Quindi ci si è specializzati sul caso N infinito. Nel quarto capitolo viene iniziata l'analisi delle equazioni di punto fisso nel limite N infinito, a partire dal caso di dimensione anomala nulla e rinormalizzazione della funzione d'onda costante (approssimazione LPA), già studiato in letteratura. Viene poi considerato il caso NLO nella derivative expansion. Nel quinto capitolo si è introdotto l'accoppiamento non minimale con un campo gravitazionale, la cui natura quantistica è considerata a livello di QFT secondo il paradigma di rinormalizzabilità dell'asymptotic safety. Per questo modello si sono ricavate le equazioni di punto fisso per le principali osservabili e se ne è studiato il comportamento per diversi valori di N.

Renormalization group flows between non-unitary conformal models

Recentemente sono stati valutati come fisicamente consistenti diversi modelli non-hermitiani sia in meccanica quantistica che in teoria dei campi. La classe dei modelli pseudo-hermitiani, infatti, si adatta ad essere usata per la descrizione di sistemi fisici dal momento che, attraverso un opportuno operatore metrico, risulta possibile ristabilire una struttura hermitiana ed unitaria. I sistemi PT-simmetrici, poi, sono una categoria particolarmente studiata in letteratura. Gli esempi riportati sembrano suggerire che anche le cosiddette teorie conformi non-unitarie appartengano alla categoria dei modelli PT-simmetrici, e possano pertanto adattarsi alla descrizione di fenomeni fisici. In particolare, si tenta qui la costruzione di determinate lagrangiane Ginzburg-Landau per alcuni modelli minimali non-unitari, sulla base delle identificazioni esistenti per quanto riguarda i modelli minimali unitari. Infine, si suggerisce di estendere il dominio del noto teorema c alla classe delle teorie di campo PT-simmetriche, e si propongono alcune linee per una possibile dimostrazione dell'ipotizzato teorema c_{eff}.

Multi-scalar critical theories: first steps for a non perturbative functional renormalization group analysis

In this work the fundamental ideas to study properties of QFTs with the functional Renormalization Group are presented and some examples illustrated. First the Wetterich equation for the effective average action and its flow in the local potential approximation (LPA) for a single scalar field is derived. This case is considered to illustrate some techniques used to solve the RG fixed point equation and study the properties of the critical theories in D dimensions. In particular the shooting methods for the ODE equation for the fixed point potential as well as the approach which studies a polynomial truncation with a finite number of couplings, which is convenient to study the critical exponents. We then study novel cases related to multi field scalar theories, deriving the flow equations for the LPA truncation, both without assuming any global symmetry and also specialising to cases with a given symmetry, using truncations based on polynomials of the symmetry invariants. This is used to study possible non perturbative solutions of critical theories which are extensions of known perturbative results, obtained in the epsilon expansion below the upper critical dimension.

Substructuring approache in state space models for dynamic system parameters identification

In the recent decade, the request for structural health monitoring expertise increased exponentially in the United States. The aging issues that most of the transportation structures are experiencing can put in serious jeopardy the economic system of a region as well as of a country. At the same time, the monitoring of structures is a central topic of discussion in Europe, where the preservation of historical buildings has been addressed over the last four centuries. More recently, various concerns arose about security performance of civil structures after tragic events such the 9/11 or the 2011 Japan earthquake: engineers looks for a design able to resist exceptional loadings due to earthquakes, hurricanes and terrorist attacks. After events of such a kind, the assessment of the remaining life of the structure is at least as important as the initial performance design. Consequently, it appears very clear that the introduction of reliable and accessible damage assessment techniques is crucial for the localization of issues and for a correct and immediate rehabilitation. The System Identification is a branch of the more general Control Theory. In Civil Engineering, this field addresses the techniques needed to find mechanical characteristics as the stiffness or the mass starting from the signals captured by sensors. The objective of the Dynamic Structural Identification (DSI) is to define, starting from experimental measurements, the modal fundamental parameters of a generic structure in order to characterize, via a mathematical model, the dynamic behavior. The knowledge of these parameters is helpful in the Model Updating procedure, that permits to define corrected theoretical models through experimental validation. The main aim of this technique is to minimize the differences between the theoretical model results and in situ measurements of dynamic data. Therefore, the new model becomes a very effective control practice when it comes to rehabilitation of structures or damage assessment. The instrumentation of a whole structure is an unfeasible procedure sometimes because of the high cost involved or, sometimes, because it’s not possible to physically reach each point of the structure. Therefore, numerous scholars have been trying to address this problem. In general two are the main involved methods. Since the limited number of sensors, in a first case, it’s possible to gather time histories only for some locations, then to move the instruments to another location and replay the procedure. Otherwise, if the number of sensors is enough and the structure does not present a complicate geometry, it’s usually sufficient to detect only the principal first modes. This two problems are well presented in the works of Balsamo [1] for the application to a simple system and Jun [2] for the analysis of system with a limited number of sensors. Once the system identification has been carried, it is possible to access the actual system characteristics. A frequent practice is to create an updated FEM model and assess whether the structure fulfills or not the requested functions. Once again the objective of this work is to present a general methodology to analyze big structure using a limited number of instrumentation and at the same time, obtaining the most information about an identified structure without recalling methodologies of difficult interpretation. A general framework of the state space identification procedure via OKID/ERA algorithm is developed and implemented in Matlab. Then, some simple examples are proposed to highlight the principal characteristics and advantage of this methodology. A new algebraic manipulation for a prolific use of substructuring results is developed and implemented.