In dieser Arbeit werden die Grundlagen von zwei häufig auftretenden Merkmalen unserer Naturgesetze untersucht: Eichsymmetrien und Quantisierung. Durch die Betrachtung dieser Merkmale im mathematischen Rahmen von Homotopie-Algebren wollen wir neue Methoden zur Berechnung physikalischer Observablen beschreiben, insbesondere in der Kosmologie und der Quantenmechanik. Zunächst befassen wir uns mit dem Problem der Eichredundanzen, die es schwer machen zu erkennen, welche Größen eine physikalische Bedeutung haben. Im Jahr 1980 erreichte Bardeen dieses Ziel in der kosmologischen Störungstheorie zu erster Ordnung. Die Frage, ob dieses Verfahren auf die perturbative Expansion von Eichtheorien aller Ordnungen ausgedehnt werden kann, ist seitdem jedoch offen geblieben. Wir zeigen, dass die Umformulierung von Eichtheorien in eichinvariante Felder als ein Transfer von homotopie-algebraischer Strukturen verstanden werden kann. Unter Verwendung dieses mathematischen Rahmens erweitern wir dann die Gültigkeit der Bardeen-Variablen auf perturbative Eichtheorien zu allen Ordnungen. Nach der Einführung eines systematisches Verfahrens für die eichinvariante Störungstheorie betrachten wir die Berechnung von Observablen in der Doppelfeldtheorie um zeitabhängige Hintergründe. Indem wir die Doppelfeldtheorie um zeitabhängige Hintergründe quadratischer und kubischer Ordnung erweitern und die quadratische Wirkung in den eichinvarianten Variablen ausdrücken, schaffen wir eine Grundlage für zukünftige Berechnungen, insbesondere zur Untersuchung des Einflusses massiver Stringmoden in kosmologischen Hintergründen. Zum Schluss betrachten wir einen anderen Ansatz zur Berechnung von Erwartungswerten in der Quantenmechanik. Obwohl die Pfadintegralformulierung der Quantenmechanik für den Fortschritt der Quantentheorie von entscheidender Bedeutung war, fehlt ihr immer noch eine strenge mathematische Definition. Die Reduktion eines unendlich-dimensionalen Raums von klassisch erlaubten Trajektorien auf einen Erwartungswert, der lediglich eine Funktion der Anfangs- und Endrandbedingungen ist, hat jedoch eine homotopiealgebraische Interpretation. Mit Hilfe des Batalin-Vilkovisky-Formalismus, der eng mit Homotopie-Lie-Algebren verwandt ist, entwickeln wir einen homologischen Ansatz zur Berechnung von Quantenerwartungswerten. Als Beispiel betrachten wir den harmonischen Oszillator und zeigen, dass unsere Methode auch im Kontext der Quantenfeldtheorie in gekrümmter Raumzeit verwendet werden kann, indem wir den Unruh-Effekt berechnen. / This thesis examines the underpinnings of two frequently manifest features of our laws of nature: gauge symmetries and quantization. By studying these features through the mathematical framework of homotopy algebras, we aim to describe new methods towards the computation of physical observables, in particular for cosmology and quantum mechanics. First, we deal with the problem of gauge redundancies, which make it difficult to discern which quantities have physical meaning. In 1980, Bardeen introduced a procedure to achieve this goal in first order cosmological perturbation theory. However, the question whether this procedure can be extended to the perturbative expansion of gauge theories to all orders has remained open since then. We show that, in general, the reformulation of gauge theories in gauge invariant fields has the interpretation of transferring homotopy algebraic structure. Utilising this mathematical framework, we then generalize Bardeen’s procedure to perturbative expansions of gauge theories to all orders in perturbations. After establishing a systematic procedure for gauge invariant perturbation theory, we set up the stage for computing observables in double field theory around time-dependent backgrounds. Double field theory not only has T-duality as a manifest symmetry, which is expected to be important in string cosmology proposals, but is also (in its weakly constrained form) a description of massive string modes, and hence is a suitable arena to investigate the imprint of massive string modes in cosmological backgrounds. By expanding double field theory around time-dependent backgrounds to quadratic and cubic order and expressing the quadratic action in terms of gauge invariant variables, we provide a basis for future computations. Finally, we describe a different approach for computing expectation values in quantum mechanics. Though having been essential for the progress of quantum theory, the path integral formulation of quantum mechanics still lacks a rigorous mathematical definition. However, the act of reducing an infinite-dimensional space of classically allowed trajectories into an expectation value which is merely a function of the initial and final boundary conditions does have a homotopy algebraic interpretation. Through the Batalin-Vilkovisky formalism, which is closely related to homotopy Lie algebras, we build a homological approach for computing quantum expectation values. We demonstrate our method for the harmonic oscillator and we show that our method can also be used in the context of quantum field theory in curved spacetime by rederiving the Unruh effect.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-04-26T17:48:21Z No. of bitstreams: 1 brunoferreirarizzuti.pdf: 507162 bytes, checksum: 6ccfc7f9025a41c3c8bc647f63d413ea (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-05-13T12:04:02Z (GMT) No. of bitstreams: 1 brunoferreirarizzuti.pdf: 507162 bytes, checksum: 6ccfc7f9025a41c3c8bc647f63d413ea (MD5) / Made available in DSpace on 2017-05-13T12:04:02Z (GMT). No. of bitstreams: 1 brunoferreirarizzuti.pdf: 507162 bytes, checksum: 6ccfc7f9025a41c3c8bc647f63d413ea (MD5) Previous issue date: 2012-02-29 / Passados mais de 60 anos da sua formulação inicial, o método de Dirac-Bergmann para hamiltonização de sistemas lagrangianos singulares continua sendo uma ferramenta poderosa para análise e investigação de modelos atuais de física teórica. Como motivação, apresentaremos vários exemplos onde o método é utilizado e o descreveremos em detalhes em uma sequência de passos. O objetivo central deste trabalho será então apresentar uma série de aplicações distintas do método de Dirac, incluindo a busca de simetrias locais para teorias singulares, a construção da proposta de relatividade especial dupla de Magueijo-Smolin, a formulação da mecânica clássica com invariância de reparametrizações e sua quantização e por fim, discutiremos um modelo semiclássico mecânico que, quando quantizado, reproduz a equação de Dirac. / After more than 60 years of its initial development, the Dirac-Bergmann method for hamiltonization of constrained systems is still a powerful tool for analysis and investigation of modern theoretical models. As a motivation, we shall present several models where the method is applied, then we will describe it in details, with a sequence of steps. The main objective of this work is to provide distinct applications of the Dirac method, including the search for local symmetries of singular theories, the construction of the Magueijo-Smolin doubly special relativity proposal, the formulation of classical mechanics with reparametrization invariance and its quantization and finally, we discuss a semiclassical mechanical model that produces the Dirac equation through quantization.

Usando uma transformação de dualidade generalizada, considerações de simetria e supondo que as superfície críticas sejam contínuas, obtivemos o dia grama de fase para sistemas de spins Z (N) bidimensionais e sistemas com invariança de calibre Z (N) a quatro dimensões. Caracterizamos as diversas fases dos sistemas de spins pelo valor esperado das potências dos operadores de ordem e desordem. No sistema com invariança de calibre, por outro lado, estas fases caracterizadas pelo comportamento do valor esperado das potências das alças de Wilson e de \'t Hooft. Obtivemos para ambos os sistemas fases moles em que no caso de spins 2D (calibre 4D) todas as potências dos parâmetros de ordem e desordem ( todas as potências das alças de Wilson e \'t Hooft) são nulas (exibem decaimento com o perímetro da alça). Enquanto no sistema com invariança de calibre todas as combinações de decaimento (área ou perímetro) das alças de Wilson e \'t Hooft são permitidas, as relações de comutação no sistema de spins proíbe a existência de fases em que tanto o parâmetro de ordem como o de desordem são não nulos (exceto quando estes operadores comutam). Apresentamos por completeza as relações de dualidade para sistemas de calibre Z (N) com campos de Higgs a três dimensões. / Using a generalized duality transformation, symetry considerations and assuming that criticality is continuous in the system?s parameters, we obtain the phase diagram for two-dimensional Z (N) spins system?s and four-dimensional gauge Z (N) system\'s. For spins system we characterize the various phases by the expectation value of powers of the order and disorder operators. For gauge systems, on the other hand, the characterization is via decay law of powers of Wilson and \'t Hooft loops. We obtain soft phases for both systems, with the folowing, behaviour: for spins system all powers of order and disorder parameters vanish, whereas for gauge systems all powers of Wilson and \'t Hooft loops decay like the perimeter. Whereas all combinations of area and perimeter decay are allowed for Wilson\'s and \'t Hooft\'s loops, the Z (N) commutation relations for spin systems forbid the simultaneous non-vanishing of order and disorder parameters (except when these operators commute). For completeness we include the duality relations for three-dimensional gauge plus Higgs Z(N) systems.

Usando uma transformação de dualidade generalizada, considerações de simetria e supondo que as superfície críticas sejam contínuas, obtivemos o dia grama de fase para sistemas de spins Z (N) bidimensionais e sistemas com invariança de calibre Z (N) a quatro dimensões. Caracterizamos as diversas fases dos sistemas de spins pelo valor esperado das potências dos operadores de ordem e desordem. No sistema com invariança de calibre, por outro lado, estas fases caracterizadas pelo comportamento do valor esperado das potências das alças de Wilson e de \'t Hooft. Obtivemos para ambos os sistemas fases moles em que no caso de spins 2D (calibre 4D) todas as potências dos parâmetros de ordem e desordem ( todas as potências das alças de Wilson e \'t Hooft) são nulas (exibem decaimento com o perímetro da alça). Enquanto no sistema com invariança de calibre todas as combinações de decaimento (área ou perímetro) das alças de Wilson e \'t Hooft são permitidas, as relações de comutação no sistema de spins proíbe a existência de fases em que tanto o parâmetro de ordem como o de desordem são não nulos (exceto quando estes operadores comutam). Apresentamos por completeza as relações de dualidade para sistemas de calibre Z (N) com campos de Higgs a três dimensões. / Using a generalized duality transformation, symetry considerations and assuming that criticality is continuous in the system?s parameters, we obtain the phase diagram for two-dimensional Z (N) spins system?s and four-dimensional gauge Z (N) system\'s. For spins system we characterize the various phases by the expectation value of powers of the order and disorder operators. For gauge systems, on the other hand, the characterization is via decay law of powers of Wilson and \'t Hooft loops. We obtain soft phases for both systems, with the folowing, behaviour: for spins system all powers of order and disorder parameters vanish, whereas for gauge systems all powers of Wilson and \'t Hooft loops decay like the perimeter. Whereas all combinations of area and perimeter decay are allowed for Wilson\'s and \'t Hooft\'s loops, the Z (N) commutation relations for spin systems forbid the simultaneous non-vanishing of order and disorder parameters (except when these operators commute). For completeness we include the duality relations for three-dimensional gauge plus Higgs Z(N) systems.

This work is devoted to baryon and lepton number violation and flavour physics in theories beyond the Standard Model of particle interactions. First, the relations between the accidental and imposed symmetries are dis- cussed. For example, we argue that the Levi-Civita tensors in the color space may be understood as a unique spurion carrying the baryon number, and as such its absence in the Lagrangian of a specific model indicates baryon number conservation. Afterwards, the two minimal scenarios of quark-lepton unification are ana- lyzed in detail: the Minimal quark-lepton symmetry model and its extension by the inverse seesaw mechanism. We investigate if the observed anomalies in the B-meson decays could be to some extent accommodated within these models. Finally, possible low-energy effects of gauge leptoquarks are studied in the theory of quark-lepton unification and its simple extensions by vector-like leptons. Taking fully into account the freedom in the quark-lepton mixing, a catalogue of measurements currently forming the border of the excluded parameter space is found. We argue that, within the considered class of models, the gauge leptoquark can account for the discrepancies in the neutral-current B decays if and only if at least two extra generations of weak-isosinglet leptons exist. 1