Aspects of Integrability in Gauge/String Correspondence

In this thesis, we present our work about some generalisations of ideas, techniques and physical interpretations typical for integrable models to one of the most outstanding advances in theoretical physics of nowadays: the AdS/CFT correspondences. We have undertaken the problem of testing this conjectured duality under various points of view, but with a clear starting point - the integrability - and with a clear ambitious task in mind: to study the finite-size effects in the energy spectrum of certain string solutions on a side and in the anomalous dimensions of the gauge theory on the other. Of course, the final desire woul be the exact comparison between these two faces of the gauge/string duality. In few words, the original part of this work consists in application of well known integrability technologies, in large parte borrowed by the study of relativistic (1+1)-dimensional integrable quantum field theories, to the highly non-relativisic and much complicated case of the thoeries involved in the recent conjectures of AdS5/CFT4 and AdS4/CFT3 corrspondences. In details, exploiting the spin chain nature of the dilatation operator of N = 4 Super-Yang-Mills theory, we concentrated our attention on one of the most important sector, namely the SL(2) sector - which is also very intersting for the QCD understanding - by formulating a new type of nonlinear integral equation (NLIE) based on a previously guessed asymptotic Bethe Ansatz. The solutions of this Bethe Ansatz are characterised by the length L of the correspondent spin chain and by the number s of its excitations. A NLIE allows one, at least in principle, to make analytical and numerical calculations for arbitrary values of these parameters. The results have been rather exciting. In the important regime of high Lorentz spin, the NLIE clarifies how it reduces to a linear integral equations which governs the subleading order in s, o(s0). This also holds in the regime with L ! 1, L/ ln s finite (long operators case). This region of parameters has been particularly investigated in literature especially because of an intriguing limit into the O(6) sigma model defined on the string side. One of the most powerful methods to keep under control the finite-size spectrum of an integrable relativistic theory is the so called thermodynamic Bethe Ansatz (TBA). We proposed a highly non-trivial generalisation of this technique to the non-relativistic case of AdS5/CFT4 and made the first steps in order to determine its full spectrum - of energies for the AdS side, of anomalous dimensions for the CFT one - at any values of the coupling constant and of the size. At the leading order in the size parameter, the calculation of the finite-size corrections is much simpler and does not necessitate the TBA. It consists in deriving for a nonrelativistc case a method, invented for the first time by L¨uscher to compute the finite-size effects on the mass spectrum of relativisic theories. So, we have formulated a new version of this approach to adapt it to the case of recently found classical string solutions on AdS4 × CP3, inside the new conjecture of an AdS4/CFT3 correspondence. Our results in part confirm the string and algebraic curve calculations, in part are completely new and then could be better understood by the rapidly evolving developments of this extremely exciting research field.

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.

Complex higher spins, Weyl invariance and tractors

In this thesis work I analyze higher spin field theories from a first quantized perspective, finding in particular new equations describing complex higher spin fields on Kaehler manifolds. They are studied by means of worldline path integrals and canonical quantization, in the framework of supersymmetric spinning particle theories, in order to investigate their quantum properties both in flat and curved backgrounds. For instance, by quantizing a spinning particle with one complex extended supersymmetry, I describe quantum massless (p,0)-forms and find a worldline representation for their effective action on a Kaehler background, as well as exact duality relations. Interesting results are found also in the definition of the functional integral for the so called O(N) spinning particles, that will allow to study real higher spins on curved spaces. In the second part, I study Weyl invariant field theories by using a particular mathematical framework known as tractor calculus, that enable to maintain at each step manifest Weyl covariance.

Strongly correlated quantum gases in one dimension

In this work, we discuss some theoretical topics related to many-body physics in ultracold atomic and molecular gases. First, we present a comparison between experimental data and theoretical predictions in the context of quantum emulator of quantum field theories, finding good results which supports the efficiency of such simulators. In the second and third parts, we investigate several many-body properties of atomic and molecular gases confined in one dimension.

Master Equation: Biological Applications and Thermodynamic Description

It is well known that many realistic mathematical models of biological systems, such as cell growth, cellular development and differentiation, gene expression, gene regulatory networks, enzyme cascades, synaptic plasticity, aging and population growth need to include stochasticity. These systems are not isolated, but rather subject to intrinsic and extrinsic fluctuations, which leads to a quasi equilibrium state (homeostasis). The natural framework is provided by Markov processes and the Master equation (ME) describes the temporal evolution of the probability of each state, specified by the number of units of each species. The ME is a relevant tool for modeling realistic biological systems and allow also to explore the behavior of open systems. These systems may exhibit not only the classical thermodynamic equilibrium states but also the nonequilibrium steady states (NESS). This thesis deals with biological problems that can be treat with the Master equation and also with its thermodynamic consequences. It is organized into six chapters with four new scientific works, which are grouped in two parts: (1) Biological applications of the Master equation: deals with the stochastic properties of a toggle switch, involving a protein compound and a miRNA cluster, known to control the eukaryotic cell cycle and possibly involved in oncogenesis and with the propose of a one parameter family of master equations for the evolution of a population having the logistic equation as mean field limit. (2) Nonequilibrium thermodynamics in terms of the Master equation: where we study the dynamical role of chemical fluxes that characterize the NESS of a chemical network and we propose a one parameter parametrization of BCM learning, that was originally proposed to describe plasticity processes, to study the differences between systems in DB and NESS.

La filosofia di Hans-Georg Gadamer e il problema del disagio della modernità. Ermeneutica, estetica, etica e politica

L’ermeneutica filosofica di Hans-Georg Gadamer – indubbiamente uno dei capisaldi del pensiero novecentesco – rappresenta una filosofia molto composita, sfaccettata e articolata, per così dire formata da una molteplicità di dimensioni diverse che si intrecciano l’una con l’altra. Ciò risulta evidente già da un semplice sguardo alla composizione interna della sua opera principale, Wahrheit und Methode (1960), nella quale si presenta una teoria del comprendere che prende in esame tre differenti dimensioni dell’esperienza umana – arte, storia e linguaggio – ovviamente concepite come fondamentalmente correlate tra loro. Ma questo quadro d’insieme si complica notevolmente non appena si prendano in esame perlomeno alcuni dei numerosi contributi che Gadamer ha scritto e pubblicato prima e dopo il suo opus magnum: contributi che testimoniano l’importante presenza nel suo pensiero di altre tematiche. Di tale complessità, però, non sempre gli interpreti di Gadamer hanno tenuto pienamente conto, visto che una gran parte dei contributi esegetici sul suo pensiero risultano essenzialmente incentrati sul capolavoro del 1960 (ed in particolare sui problemi della legittimazione delle Geisteswissenschaften), dedicando invece minore attenzione agli altri percorsi che egli ha seguito e, in particolare, alla dimensione propriamente etica e politica della sua filosofia ermeneutica. Inoltre, mi sembra che non sempre si sia prestata la giusta attenzione alla fondamentale unitarietà – da non confondere con una presunta “sistematicità”, da Gadamer esplicitamente respinta – che a dispetto dell’indubbia molteplicità ed eterogeneità del pensiero gadameriano comunque vige al suo interno. La mia tesi, dunque, è che estetica e scienze umane, filosofia del linguaggio e filosofia morale, dialogo con i Greci e confronto critico col pensiero moderno, considerazioni su problematiche antropologiche e riflessioni sulla nostra attualità sociopolitica e tecnoscientifica, rappresentino le diverse dimensioni di un solo pensiero, le quali in qualche modo vengono a convergere verso un unico centro. Un centro “unificante” che, a mio avviso, va individuato in quello che potremmo chiamare il disagio della modernità. In altre parole, mi sembra cioè che tutta la riflessione filosofica di Gadamer, in fondo, scaturisca dalla presa d’atto di una situazione di crisi o disagio nella quale si troverebbero oggi il nostro mondo e la nostra civiltà. Una crisi che, data la sua profondità e complessità, si è per così dire “ramificata” in molteplici direzioni, andando ad investire svariati ambiti dell’esistenza umana. Ambiti che pertanto vengono analizzati e indagati da Gadamer con occhio critico, cercando di far emergere i principali nodi problematici e, alla luce di ciò, di avanzare proposte alternative, rimedi, “correttivi” e possibili soluzioni. A partire da una tale comprensione di fondo, la mia ricerca si articola allora in tre grandi sezioni dedicate rispettivamente alla pars destruens dell’ermeneutica gadameriana (prima e seconda sezione) ed alla sua pars costruens (terza sezione). Nella prima sezione – intitolata Una fenomenologia della modernità: i molteplici sintomi della crisi – dopo aver evidenziato come buona parte della filosofia del Novecento sia stata dominata dall’idea di una crisi in cui verserebbe attualmente la civiltà occidentale, e come anche l’ermeneutica di Gadamer possa essere fatta rientrare in questo discorso filosofico di fondo, cerco di illustrare uno per volta quelli che, agli occhi del filosofo di Verità e metodo, rappresentano i principali sintomi della crisi attuale. Tali sintomi includono: le patologie socioeconomiche del nostro mondo “amministrato” e burocratizzato; l’indiscriminata espansione planetaria dello stile di vita occidentale a danno di altre culture; la crisi dei valori e delle certezze, con la concomitante diffusione di relativismo, scetticismo e nichilismo; la crescente incapacità a relazionarsi in maniera adeguata e significativa all’arte, alla poesia e alla cultura, sempre più degradate a mero entertainment; infine, le problematiche legate alla diffusione di armi di distruzione di massa, alla concreta possibilità di una catastrofe ecologica ed alle inquietanti prospettive dischiuse da alcune recenti scoperte scientifiche (soprattutto nell’ambito della genetica). Una volta delineato il profilo generale che Gadamer fornisce della nostra epoca, nella seconda sezione – intitolata Una diagnosi del disagio della modernità: il dilagare della razionalità strumentale tecnico-scientifica – cerco di mostrare come alla base di tutti questi fenomeni egli scorga fondamentalmente un’unica radice, coincidente peraltro a suo giudizio con l’origine stessa della modernità. Ossia, la nascita della scienza moderna ed il suo intrinseco legame con la tecnica e con una specifica forma di razionalità che Gadamer – facendo evidentemente riferimento a categorie interpretative elaborate da Max Weber, Martin Heidegger e dalla Scuola di Francoforte – definisce anche «razionalità strumentale» o «pensiero calcolante». A partire da una tale visione di fondo, cerco quindi di fornire un’analisi della concezione gadameriana della tecnoscienza, evidenziando al contempo alcuni aspetti, e cioè: primo, come l’ermeneutica filosofica di Gadamer non vada interpretata come una filosofia unilateralmente antiscientifica, bensì piuttosto come una filosofia antiscientista (il che naturalmente è qualcosa di ben diverso); secondo, come la sua ricostruzione della crisi della modernità non sfoci mai in una critica “totalizzante” della ragione, né in una filosofia della storia pessimistico-negativa incentrata sull’idea di un corso ineluttabile degli eventi guidato da una razionalità “irrazionale” e contaminata dalla brama di potere e di dominio; terzo, infine, come la filosofia di Gadamer – a dispetto delle inveterate interpretazioni che sono solite scorgervi un pensiero tradizionalista, autoritario e radicalmente anti-illuminista – non intenda affatto respingere l’illuminismo scientifico moderno tout court, né rinnegarne le più importanti conquiste, ma più semplicemente “correggerne” alcune tendenze e recuperare una nozione più ampia e comprensiva di ragione, in grado di render conto anche di quegli aspetti dell’esperienza umana che, agli occhi di una razionalità “limitata” come quella scientista, non possono che apparire come meri residui di irrazionalità. Dopo aver così esaminato nelle prime due sezioni quella che possiamo definire la pars destruens della filosofia di Gadamer, nella terza ed ultima sezione – intitolata Una terapia per la crisi della modernità: la riscoperta dell’esperienza e del sapere pratico – passo quindi ad esaminare la sua pars costruens, consistente a mio giudizio in un recupero critico di quello che egli chiama «un altro tipo di sapere». Ossia, in un tentativo di riabilitazione di tutte quelle forme pre- ed extra-scientifiche di sapere e di esperienza che Gadamer considera costitutive della «dimensione ermeneutica» dell’esistenza umana. La mia analisi della concezione gadameriana del Verstehen e dell’Erfahrung – in quanto forme di un «sapere pratico (praktisches Wissen)» differente in linea di principio da quello teorico e tecnico – conduce quindi ad un’interpretazione complessiva dell’ermeneutica filosofica come vera e propria filosofia pratica. Cioè, come uno sforzo di chiarificazione filosofica di quel sapere prescientifico, intersoggettivo e “di senso comune” effettivamente vigente nella sfera della nostra Lebenswelt e della nostra esistenza pratica. Ciò, infine, conduce anche inevitabilmente ad un’accentuazione dei risvolti etico-politici dell’ermeneutica di Gadamer. In particolare, cerco di esaminare la concezione gadameriana dell’etica – tenendo conto dei suoi rapporti con le dottrine morali di Platone, Aristotele, Kant e Hegel – e di delineare alla fine un profilo della sua ermeneutica filosofica come filosofia del dialogo, della solidarietà e della libertà.

Cosmological perturbations in generalized theories of gravity

The first part of the thesis concerns the study of inflation in the context of a theory of gravity called "Induced Gravity" in which the gravitational coupling varies in time according to the dynamics of the very same scalar field (the "inflaton") driving inflation, while taking on the value measured today since the end of inflation. Through the analytical and numerical analysis of scalar and tensor cosmological perturbations we show that the model leads to consistent predictions for a broad variety of symmetry-breaking inflaton's potentials, once that a dimensionless parameter entering into the action is properly constrained. We also discuss the average expansion of the Universe after inflation (when the inflaton undergoes coherent oscillations about the minimum of its potential) and determine the effective equation of state. Finally, we analyze the resonant and perturbative decay of the inflaton during (p)reheating. The second part is devoted to the study of a proposal for a quantum theory of gravity dubbed "Horava-Lifshitz (HL) Gravity" which relies on power-counting renormalizability while explicitly breaking Lorentz invariance. We test a pair of variants of the theory ("projectable" and "non-projectable") on a cosmological background and with the inclusion of scalar field matter. By inspecting the quadratic action for the linear scalar cosmological perturbations we determine the actual number of propagating degrees of freedom and realize that the theory, being endowed with less symmetries than General Relativity, does admit an extra gravitational degree of freedom which is potentially unstable. More specifically, we conclude that in the case of projectable HL Gravity the extra mode is either a ghost or a tachyon, whereas in the case of non-projectable HL Gravity the extra mode can be made well-behaved for suitable choices of a pair of free dimensionless parameters and, moreover, turns out to decouple from the low-energy Physics.

Accretion disk winds in active galactic nuclei: an X-ray view

This Thesis focuses on the X-ray study of the inner regions of Active Galactic Nuclei, in particular on the formation of high velocity winds by the accretion disk itself. Constraining AGN winds physical parameters is of paramount importance both for understanding the physics of the accretion/ejection flow onto supermassive black holes, and for quantifying the amount of feedback between the SMBH and its environment across the cosmic time. The sources selected for the present study are BAL, mini-BAL, and NAL QSOs, known to host high-velocity winds associated to the AGN nuclear regions. Observationally, a three-fold strategy has been adopted: - substantial samples of distant sources have been analyzed through spectral, photometric, and statistical techniques, to gain insights into their mean properties as a population; - a moderately sized sample of bright sources has been studied through detailed X-ray spectral analysis, to give a first flavor of the general spectral properties of these sources, also from a temporally resolved point of view; - the best nearby candidate has been thoroughly studied using the most sophisticated spectral analysis techniques applied to a large dataset with a high S/N ratio, to understand the details of the physics of its accretion/ejection flow. There are three main channels through which this Thesis has been developed: - [Archival Studies]: the XMM-Newton public archival data has been extensively used to analyze both a large sample of distant BAL QSOs, and several individual bright sources, either BAL, mini-BAL, or NAL QSOs. - [New Observational Campaign]: I proposed and was awarded with new X-ray pointings of the mini-BAL QSOs PG 1126-041 and PG 1351+640 during the XMM-Newton AO-7 and AO-8. These produced the biggest X-ray observational campaign ever made on a mini-BAL QSO (PG 1126-041), including the longest exposure so far. Thanks to the exceptional dataset, a whealth of informations have been obtained on both the intrinsic continuum and on the complex reprocessing media that happen to be in the inner regions of this AGN. Furthermore, the temporally resolved X-ray spectral analysis field has been finally opened for mini-BAL QSOs. - [Theoretical Studies]: some issues about the connection between theories and observations of AGN accretion disk winds have been investigated, through theoretical arguments and synthetic absorption line profiles studies.