Dissertations / Theses on the topic 'Quantum field theory; Curved spacetime'

Unruh-DeWitt particle detector models are studied in a variety of time-dependent and time-independent settings. We work within the framework of first-order perturbation theory and couple the detector to a massless scalar field. The necessity of switching on (off) the detector smoothly is emphasised throughout, and the transition rate is found by taking the sharp-switching limit of the regulator-free and finite response function. The detector is analysed on a variety of spacetimes: d-dimensional Minkowski, the Banados-Teitelboim-Zanelli (BTZ) black hole, the two-dimensional Minkowski half-plane, two-dimensional Minkowski with a receding mirror, and the two- and four-dimensional Schwarzschild black holes. In d-dimensional Minkowski spacetime, the transition rate is found to be finite up to dimension five. In dimension six, the transition rate diverges unless the detector is on a trajectory of constant proper acceleration, and the implications of this divergence to the global embedding spacetime (GEMS) methods are studied. In three-dimensional curved spacetime, the transition rate for the scalar field in an arbitrary Hadamard state is found to be finite and regulator-free. Then on the Banados-Teitelboim-Zanelli (BTZ) black hole spacetime, we analyse the detector coupled to the field in the Hartle-Hawking vacua, under both transparent and reflective boundary conditions at infinity. Results are presented for the co-rotating detector, which responds thermally, and for the radially-infalling detector. Finally, detectors on the Schwarzschild black hole are considered. We begin in two dimensions, in an attempt to gain insight by exploiting the conformal triviality, and where we apply a temporal cut-off to regulate the infrared divergence. In four-dimensional Schwarzschild spacetime, we proceed numerically, and the Hartle-Hawking, Boulware and Unruh vacua rates are compared. Results are presented for the case of the static detectors, which respond thermally, and also for the case of co-rotating detectors.

One of the fundamental questions in quantum field theory in curved spacetime concerns the nature of the vacuum state. This is the subject that we address in this thesis. A generalised Schrödinger wave functional equation is constructed, and a general framework is introduced to solve the Schrödinger equation for a general curved spacetime. Vacuum wave functional solutions are given for a wide class of spacetimes; namely static, dynamic and conformally static. These include the physically important Robertson-Walker, Bianchi type I, de Sitter, Rindler and Schwarzschild spacetimes. The nature of the vacuum state in these spacetimes is extensively discussed. Matrix elements and two-point functions are then calculated in this general framework. This formalism is used to study particle creation and boundary effects on curved spacetime. This is carried out first for cosmological models, which exhibit particle creation as a result of the expansion of the universe. Secondly, quantum field theory from the view point of an accelerating observer is investigated. The Minkowski vacuum state is shown to be a thermal state with respect to the accelerating observers ground state. The application of this formalism to black hole spacetimes is also investigated.

This work discusses aspects of a fundamental problem in the theory of quantum fields on curved spacetimes - that of giving physical meaning to the particle representations of the theory. In particular, the response of model particle detectors is analysed in detail. Unruh (1976) first introduced the idea of a model particle detector in order to give an operational definition to particles. He found that even in flat spacetime, the excitation of a particle detector does not necessarily correspond to the presence of an energy carrier - an accelerating detector will excite in response to the zero-energy state of the Minkowski vacuum. The central question I consider in this work Is - where does the energy for the excitation of the accelerating detector come from? The accepted response has been that the accelerating force provides the energy. Evaluating the energy carried by the (conformally-invariant massless scalar) field after the Interaction with the detector, however, I find that the detector excitation is compensated by an equal but opposite emission of negative energy. This result suggests that there may be states of lesser energy than that of the Minkowski vacuum. To resolve this paradox, I argue that the emission of a detector following a more realistic trajectory than that of constant acceleration - one that starts and finishes in inertial motion - will in total be positive, although during periods of constant acceleration the detector will still emit negative energy. The Minkowski vacuum retains its status as the field state of lowest energy. The second question I consider is' the response of Unruh's detector in curved spacetime - is it possible to use such a detector to measure the energy carried by the field? In the particular case of a detector following a Killing trajectory, I find that there is a response to the energy of the field, but that there is also an inherent 'noise'. In a two dimensional model spacetime, I show that this 'noise' depends on the detector's acceleration and on the curvature of the spacetime, thereby encompassing previous results of Unruh (1976) and of Gibbons & Hawking (1977).
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This thesis is concerned with aspects of quantum theory of fields in flat and curved spacetimes of arbitrary dimensions along with detecting bosons and fermions on these spacetimes. The thesis is divided into two main parts. In the first part, we analyse an Unruh-DeWitt particle detector that is coupled linearly to the scalar density of a massless Dirac field (neutrino field) in Minkowski spacetimes of dimension d ≥ 2 and on the two-dimensional static Minkowski cylinder, allowing the detector’s motion to remain arbitrary and working to leading order in perturbation theory. In d-dimensional Minkowski spacetime, with the field in the usual Fock vacuum, we show that the detector’s response is identical to that of a detector coupled linearly to a massless scalar field in 2d-dimensional Minkowski. In the special case of uniform linear acceleration, the detector’s response hence exhibits the Unruh effect with a Planckian factor in both even and odd dimensions, in contrast to the Rindler power spectrum of the Dirac field, which has a Planckian factor for odd d but a Fermi-Dirac factor for even d. On the two-dimensional cylinder, we set the oscillator modes in the usual Fock vacuum but allow an arbitrary state for the zero mode of the periodic spinor. We show that the detector’s response distinguishes the periodic and antiperiodic spin structures, and the zero mode of the periodic spinor contributes to the response by a state-dependent but well defined amount. Explicit analytic and numerical results on the cylinder are obtained for inertial and uniformly accelerated trajectories, recovering the d = 2 Minkowski results in the limit of large circumference. The detector’s response has no infrared ambiguity for d = 2, neither in Minkowski nor on the cylinder. In the second part, firstly, we give a thorough discussion for the Bogolubov transformation for Dirac field, and discuss pair creation in a non-stationary spacetime. Secondly, we derive the in and out vacua Wightman two-point functions for the Dirac field and the Klein-Gordon field for certain class of spatially flat Friedmann-Robertson-Walker (FRW) cosmological spacetimes wherein the two-point functions have the Hadamard form. We then establish the equivalence between the adiabatic vacuum of infinite order and the conformal vacuum in the massless limit. With the field in the conformal Fock vacuum, we then show that the detector’s response to an UDW particle detector coupled linearly to the scalar density of a massless Dirac field in the spatially flat FRW spacetimes in d-dimensions is identical to the response of a detector coupled to the massless scalar field in the spatially flat FRW spacetimes in 2d-dimensions. Lastly, we discuss a massive scalar field in the spatially compactified (1 + 1)-dimensional FRW spacetime. There, the issue of the conformal zero momentum mode arises. To resolve this issue, we develop a new scheme for quantizing the conformal zero-mode. This new quantization scheme introduces a family of two real parameters for every zero-momentum mode with an associated two-real-parameter set of in/out vacua. We then show that the zero momentum initial state’s wave functional corresponds to a two-real parameter set of Gaussian wave packets. For applications, we examine the finite-time detector’s response to a massive scalar field in the (1 + 1)-dimensional, spatially compactified Milne spacetime. Explicit analytic results are obtained for the comoving and inertially non-comoving trajectories. Numerical results are provided for the comoving trajectory. The numerical results suggest that when the in-vacuum is chosen to be very far from the conventional Minkowski vacuum state, then it contains particles. As result, spontaneous excitation of the comoving detector occurs.

This thesis has two main themes: on the one hand, in Chapters 3 and 5 we study some effects of the presence of timelike boundaries on linear classical and quantum field theories; the second theme is the analysis of technical issues with the paper B.S. Kay and R.M. Wald, Phys. Rep. 207, 49-136 (1991), which is carried out in parts of Chapter 2 and in Chapter 4. Chapter 2 contains a novel result on the characteristic initial value problem on globally hyperbolic spacetimes. In Chapter 3, we conjecture that (when the notion of a Hadamard state is suitably adapted to spacetimes with timelike boundaries) there is no isometry-invariant Hadamard state for the Klein-Gordon equation defined on the region of the Kruskal spacetime 'to the left of' a surface of constant Schwarzschild radius in the right Schwarzschild wedge, if Dirichlet boundary conditions are imposed there. We also prove that, under a suitable notion for 'boost-invariant Hadamard state' which also takes into account the special infra-red pathology of massless fields in 1+1 dimensions, there is no such state for the massless 1+1 wave equation on the region of Minkowski space to the left of an eternally uniformly accelerating mirror – with Dirichlet boundary conditions at the mirror. Chapter 5 collects and extends results of Solis about the causal structure of spacetimes with timelike boundaries, and deals with algebraic aspects of the interplay between Green hyperbolicity and boundary conditions in classical field theory. It also outlines a plan for generalizing the established work on wave-like equations from globally hyperbolic spacetimes to 'globally hyperbolic spacetimes-with-timelike-boundaries'. Appendix B contains a non-existence result for boost-invariant Hadamard states of a massless scalar field in (1+1)-dimensional Minkowski spacetime.

In this work the extension of the criterion for local thermal equilibrium by Buchholz, Ojima and Roos to curved spacetime as introduced by Schlemmer is investigated. Several problems are identified and especially the instability under time evolution which was already observed by Schlemmer is inspected. An alternative approach to local thermal equilibrium in quantum field theories on curved spacetimes is presented and discussed. In the following the dynamic system of the linear field and matter perturbations in the generic model of inflation is studied in the view of ambiguity of quantisation. In the last part the compatibility of the temperature fluctuations of the cosmic microwave background radiation with local thermal equilibrium is investigated
In dieser Arbeit wird die von Schlemmer eingeführte Erweiterung des Kriteriums für lokales thermisches Gleichgewicht in Quantenfeldtheorien von Buchholz, Ojima und Roos auf gekrümmte Raumzeiten untersucht. Dabei werden verschiedene Probleme identifiziert und insbesondere die bereits von Schlemmer gezeigte Instabilität unter Zeitentwicklung untersucht. Es wird eine alternative Herangehensweise an lokales thermisches Gleichgewicht in Quantenfeldtheorien auf gekrümmten Raumzeiten vorgestellt und deren Probleme diskutiert. Es wird dann eine Untersuchung des dynamischen Systems der linearen Feld- und Metrikstörungen im üblichen Inflationsmodell mit Blick auf Uneindeutigkeit der Quantisierung durchgeführt. Zuletzt werden die Temperaturfluktuationen der kosmischen Hintergrundstrahlung auf Kompatibilität mit lokalem thermalem Gleichgewicht überprüft

The aim of this work is present the phenomenon denoted entanglement harvesting. We begin by introducing entanglement historically. Following, we go beyond the one particle theory in flat spacetime and introduce Quantum Field Theory in Curved Spacetime, showing two famous consequences: the Unruh effect and the Hawking radiation. Finally, we analyze entanglement harvesting for two Unruh-deWitt detectors. In the fisrt example, we see that there is a \"sudden death\" point of entanglement harvesting when the detectors are near the BTZ black hole event horizon, due to redshift effect and Hawking radiation. Then, we compare the phenomenon for different scenarios, and find out that it is sensitive to the structure of spacetime. Finally, we see how detectors\' parameters affect it and find out that the smoothness of the switching of the detectors\' coupling to the field is extremely relevant. We also see how the parameters can be used to optimize entanglement harvested.
O objetivo desse trabalho é apresentar o fenômeno denotado colheita de emaranhamento. Primeiramente fazemos uma introdução histórica de emaranhamento de estados quânticos. Em seguida, introduzimos a Teoria Quântica de Campos no Espaço-tempo Curvo, como um passo além da teoria quântica de uma partícula no espaço-tempo plano, e demonstramos dois resultados famosos da teoria: o efeito Unruh e a radiação Hawking. Por fim, fazemos uma analise do fenômeno de colheita de emaranhamento para dois detectores Unruh-deWitt. Nosso primeiro exemplo mostra que há um ponto de \"morte súbita\" do fenômeno quando os detectores se aproximam do horizonte de eventos de um buraco negro de BTZ, que é uma consequência do efeito de redshift e da radiação Hawking. Em seguida, comparamos o fenômeno em cenários diferentes, e observamos que a colheita de emaranhamento é sensível à estrutura do espaço-tempo. Por último, analisamos como os parâmetros dos detectores afetam a colheita de emaranhamento, e vemos que a suavidade em que o acoplamento dos detectores com o campo é \"ligado\" e \"desligado\" é extremamente relevante. Também analisamos como podemos usar os parâmetros dos detectores para otimizar a quantidade de emaranhamento colhida.

We employ the formalism of the Horizon Quantum Mechanics to describe the gravitational radius of compact sources by means of an operator and derive a Horizon Wave Function which will allow us to estimate the probability of formation of black holes in scattering processes. If the Planck scale is kept at its standard value, however, it will be impossible to test that regime with any foreseeable technology. We then review how the introduction of extra dimensions can potentially lower the Planck scale down to the TeV range in an attempt to solve the hierarchy problem. In this context, we proceed by studying black holes described by a generalisation of the Kerr metric for higher dimensional spacetime known as the Myers-Perry metric. Our computation of the probability that a rotating source in higher dimensions is a black hole suggests that, even if the fundamental Planck scale is as low as a few TeV's, we should not be able to detect any black hole in colliders as is indeed the case.

Depuis 1916, l'étude des Trous Noirs a soulevé des questions intrigantes. Seulement certaines ont été résolues. En effet, nous faisons face à des régimes où s’entremêlent la théorie quantique et l'espace-temps. Les TN comme porte d'entrée pour comprendre la nature quantique de la gravité. Ma thèse a été entièrement dédiée à ce domaine central de la physique théorique, avec pour but la compréhension la plus large possible des débats autour de ces questions. C'est ainsi qu'ont été produits des résultats originaux qui constituent le cœur de ce manuscrit. 1-Les surfaces de volume maximal des TN sont étudiées. Un TN astrophysique terminera sa vie avec une aire planckienne de $10^{-70} m^2$ dissimulant $10^5$ fois le volume de l'univers observable. Ceci peut avoir des conséquences sur la viabilité du "remnant scenario" comme solution au paradoxe de l'information. 2-Le scénario "trou-noir-trou-blanc" est fortement instable. Une modification minimale est proposée pour résoudre ce problème. 3-Une généralisation des quatre lois de la thermodynamique des TN est démontrée pour des cônes de lumière s'intersectant dans un espace de Minkowski. 4-On étudie des espaces conformellement plats où de telles lois acquièrent une interprétation thermodynamique standard. Le plus simple est l'espace-temps de Bertotti-Robinson, connu pour encoder la géométrie proche de l'horizon d'un TN chargé. 5-Pour peu que le bon tenseur énergie-impulsion soit identifié, les équations du champ Einstein-Cartan peuvent être retrouvées comme l'équation d'état d'un équilibre thermodynamique, comme dans le cas original de la RG. Ces résultats contribuent au débat intense sur les questions cruciales posées ci-dessus
Since 1916 intriguing questions have arisen from the study of Black Holes (BH). Only some of them have been resolved. Indeed, we are faced with regimes where the yet unknown interplay between quantum theory and spacetime unveils. BH physics is a gateway to the quantum nature of gravity. My thesis has been completely devoted to this central domain of theoretical physics, with the guiding aim of understanding in the widest possible manner the debate around those questions. The process has produced original results that constitute the main core of the manuscript. 1- The maximal volume surfaces of evaporating BHs are studied. An astrophysical BH will end its life with an external planckian area $10^{-70} m^2$ hiding $10^5$ times the volume of our observable Universe. This can have consequences on the viability of the “remnant scenario” as solution to the BH information paradox. 2- The “black-hole-to-white-hole scenario” is analyzed. The model is shown to be strongly unstable, and a minimal resolutive modification is proposed. 3- A generalisation of the four laws of BH thermodynamics is proven for intersecting light cones in Minkowski spacetime. 4- Conformally flat spaces where such laws acquire the standard thermodynamical interpretation are studied. The simplest one is the Bertotti-Robinson spacetime, known to encode the near-horizon geometry of a charged BH. 5- It is shown that, if the correct energy-momentum tensor is identified, the Eintein-Cartan’s field equations can be recovered as a thermodynamical equilibrium equation of state just like in the GR original case. Such results contribute to the intense debate on the opening crucial questions

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
O campo espinorial ELKO associado às partículas de matéria de spin 1/2 e dimensão de massa 1, construídos em um conjunto completo de autoespinores de helicidade dual do operador conjugação de carga, é um canditado a descrever a matéria escura. Devido a sua natureza, possui diversas aplicações cosmológicas. Neste trabalho, o estudo de tais espinores é realizado sob universo de Friedmann-Robertson-Walker plano, homogênio e isotrópico. À luz da Teoria Quântica de Campos em Espaços-Tempo Curvos, com as transformações de Bogoliubov, obteve-se a densidade de partículas criadas em uma métrica modelo de regime assintóticamente plano no passado e no futuro, permitindo-se comparar com soluções exatas presentes na literatura para produção de partículas escalares e férmions de Dirac. Discute- se, também, a compatibilidade do fenômeno de criação de partículas provenientes da Teoria Quântica de Campos em Espaço Curvos com os estudos de Prigogine para descrição deste fenômeno à escala cosmológica, considerando-se um universo termodinamicamente aberto.
The ELKO spinor field associated with spin 1/2 particles and mass dimension 1, con- structed on a complete set of dual helicity eigenspinors of the charge conjugation operator, is a candidate for describing dark matter. Due to its nature, it has several cosmological applications. In this work, the study of such spinors is performed under a homogenous and isotropic Friedmann-Robertson-Walker universe. In the light of the Quantum Field Theory in Curved Space-Time, with the Bogoliubov transformations, we obtained the density of particles created in a model of asymptotically plane metric in the past and in the future, allowing to compare with exact solutions present in the literature for the production of scalar particles and Dirac fermions. It is also discussed the compatibility of the phenomenon of particle creation from the Quantum Field Theory in Curved Space with the studies of Prigogine to describe this phenomenon at the cosmological scale, considering a thermodynamically open universe.

This thesis considers the problem of calculating and observing the mixing of modes of positive and negative frequency in inhomogeneous, dispersive media. Scattering of vacuum modes of the electromagnetic field at a moving interface in the refractive index of a dielectric medium is discussed. Kinematics arguments are used to demonstrate that this interface may, in a regime of linear dispersion, act as the analogue of the event horizon of a black hole to modes of the field. Furthermore, a study of the dispersion of the dielectric shows that five distinct configurations of modes of the inhomogeneous medium at the interface exist as a function of frequency. Thus it is shown that the interface is simultaneously a black- and white-hole horizon-like and horizonless emitter. The role, and importance, of negative-frequency modes of the field in mode conversion at the horizon is established and yields a calculation of the spontaneous photonic flux at the interface. An algorithm to calculate the scattering of vacuum modes at the interface is introduced. Spectra of the photonic flux in the moving and laboratory frame, for all modes and all realisable increase in the refractive index at the interface are computed. As a result of the various mode configurations, the spectra are highly structured in intervals with black-hole, white-hole and no horizon. The spectra are dominated by a negative-frequency mode, which is the partner in any Hawking-type emission. An experiment in which an incoming positive-frequency wave is populated with photons is assembled to observe the transfer of energy to outgoing waves of positive and negative frequency at the horizon. The effect of mode conversion at the interface is clearly shown to be a feature of horizon physics. This is a classical version of the quantum experiment that aims at validating the mechanism of Hawking radiation.

Orientador: Prof. Dr. André Gustavo Scagliusi Landulfo
Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, 2017.
A descoberta do comportamento termodinâmico de buracos negros no contexto da teoria quântica de campos em espaços-tempos curvos deu origem a diversas questões, dentre elas onde estão e quais são os graus de liberdade associados a entropia dos mesmos. Nessa dissertação estudamos esse tópico propondo que, se não toda, alguma parte desta entropia é devida ao emaranhamento entre os graus de liberdade, internos e externos ao buraco, dos campos quânticos presentes no espaço-tempo.
The discovery of thermodynamic behavior of black holes in the context of quantum field theory in curved spacetimes has given rise to several issues, among them where and what are the degrees of freedom associated to their entropy. In this dissertation we study this topic proposing that, if not all, some of this entropy is due to the entanglement between degrees of freedom, internal and external to the hole, of quantum fields present in the spacetime.

Elaboramos um estudo detalhado de alguns aspectos d(e uma versão d)a correspondência AdS/CFT, conjeturada por Maldacena e Witten, entre teorias quânticas de campo num fundo gravitacional dado por um espaço-tempo assintoticamente anti-de Sitter (AAdS), e teorias quânticas de campos conformalmente covariantes no infinito conforme (no sentido de Penrose) deste espaço-tempo, aspectos estes: (a) independentes d(o par d)e modelos específicos em Teoria Quântica de Campos, e (b) suscetíveis a uma reformulação em moldes matematicamente rigorosos. Adotamos como ponto de partida o teorema demonstrado por Rehren no contexto da Física Quântica Local (também conhecida como Teoria Quântica de Campos Algébrica) em espaços-tempos anti-de Sitter (AdS), denominado holografia algébrica ou dualidade de Rehren. O corpo do presente trabalho consiste em estender o resultado de Rehren para uma classe razoavelmente geral de espaços-tempos AAdS d-dimensionais (d>3), escrutinar como as propriedades desta extensão são enfraquecidas e/ou modificadas em relação ao espaço-tempo AdS, e como efeitos gravitacionais não-triviais se manifestam na teoria quântica no infinito conforme. Dentre os resultados obtidos, citamos: condições razoavelmente gerais sobre geodésicas nulas no interior (cuja plausibilidade justificamos por meio de resultados de rigidez geométrica) não só garantem que a nossa generalização é geometricamente consistente com causalidade, como também permite uma reconstrução ``holográfica\'\' da topologia do interior na ausência de horizontes e singularidades; a implementação das simetrias conformes na fronteira, que associamos explicitamente a uma família de isometrias assintóticas do interior construída de maneira intrínseca, ocorre num caráter puramente assintótico e é atingida dinamicamente por um processo de retorno ao equilíbrio, mediante condições de contorno adequadas no infinito; efeitos gravitacionais podem eventualmente causar obstruções à reconstrução da teoria quântica no interior, ou por torná-la trivial em regiões suficientemente pequenas ou devido à existência de múltiplos vácuos inequivalentes, que por sua vez levam à existência de excitações solitônicas localizadas ao redor de paredes de domínio no interior, similares a D-branas. As demonstrações fazem uso extensivo de geometria Lorentziana global. A linguagem empregada para as teorias quânticas relevantes para nossa generalização da dualidade de Rehren segue a formulação funtorial de Brunetti, Fredenhagen e Verch para a Física Quântica Local, estendida posteriormente por Sommer para incorporar condições de contorno.
We elaborate a detailed study of certain aspects of (a version of) the AdS/CFT correspondence, conjectured by Maldacena and Witten, between quantum field theories in a gravitational background given by an asymptotically anti-de Sitter (AAdS) spacetime, and conformally covariant quantum field theories in the latter\'s conformal infinity (in the sense of Penrose), aspects such that: (a) are independent from (the pair of) specific models in Quantum Field Theory, and (b) susceptible to a recast in a mathematically rigorous mould. We adopt as a starting point the theorem demonstrated by Rehren in the context of Local Quantum Physics (also known as Algebraic Quantum Field Theory) in anti-de Sitter (AdS) spacetimes, called algebraic holography or Rehren duality. The main body of the present work consists in extending Rehren\'s result to a reasonably general class of d-dimensional AAdS spacetimes (d>3), scrutinizing how the properties of such an extension are weakened and/or modified as compared to AdS spacetime, and probing how non-trivial gravitational effects manifest themselves in the conformal infinity\'s quantum theory. Among the obtained results, we quote: not only does the imposition of reasonably general conditions on bulk null geodesics (whose plausibility we justify through geometrical rigidity techniques) guarantee that our generalization is geometrically consistent with causality, but it also allows a ``holographic\'\' reconstruction of the bulk topology in the absence of horizons and singularities; the implementation of conformal symmetries in the boundary, which we explicitly associate to an intrinsically constructed family of bulk asymptotic isometries, have a purely asymptotic character and is dynamically attained through a process of return to equilibrium, given suitable boundary conditions at infinity; gravitational effects may cause obstructions to the reconstruction of the bulk quantum theory, either by making the latter trivial in sufficiently small regions or due to the existence of multiple inequivalent vacua, which on their turn lead to the existence of solitonic excitations localized around domain walls, similar to D-branes. The proofs make extensive use of global Lorentzian geometry. The language employed for the quantum theories relevant for our generalization of Rehren duality follows the functorial formulation of Local Quantum Physics due to Brunetti, Fredenhagen and Verch, extended afterwards by Sommer in order to incorporate boundary conditions. (An English translation of the full text can be found at arXiv:0712.0401)

Orientador: Prof. Dr. André Gustavo Scagliusi Landulfo
Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, Santo André, 2018.
Nas últimas décadas, pesquisas na área de física de buracos negros mostraram que existe uma relação profunda entre buracos negros, termodinâmica e mecânica quântica. Em particular, buracos negros em um contexto semiclássico parecem possuir uma temperatura TH=k/2p e uma entropia Sbh =A/4, proporcionais à sua gravidade superficial k e à sua área A, respectivamente. Essa surpreendente conexão é reforçada ao analisarmos a validade da chamada Segunda Lei Generalizada da Termodiâmica (SLG), que afirma que a entropia da matéria fora do buraco negro somada à entropia do buraco negro nunca decresce com o tempo. Este trabalho investiga provas gerais da validade da Segunda Lei Generalizada, já que tem um papel central na interpretação de buracos negros como entidadestermodinâmicas. Emseguida, é analisado como podemos usar a SLG para extender um teorema clássico de singularidade para um contexto semiclássico.
In the past few decades, research in black hole physics have shown a deep relation between black holes, thermodynamics, and quantum mechanics. In particular, blackholes in a semiclassical context appear to have a temperatureTH =k/2p and an entropy Sbh = A/4 proportional to its surface gravity k and area A,respectively. This striking connection is strengthened when one analyses the validity of the so called Generalized Second Law of Thermodynamics (GSL), which states that the entropy of matter outside the black hole plus the black hole¿s entropy never decrease with time. This work investigate general proofsofthevalidityofGeneralizedSecondLaw,sinceitplaysacentralroleininterpreting black holes as thermodynamical entities. In addition, it is analyzed how the GSL can be used to extend a classical singularity theorem to a semiclassical context.

Famously, Einstein read off the geometry of spacetime from Maxwell's equations. Today, we take this geometry that serious that our fundamental theory of matter, the standard model of particle physics, is based on it. However, it seems that there is a gap in our understanding if it comes to the physics outside of the solar system. Independent surveys show that we need concepts like dark matter and dark energy to make our models fit with the observations. But these concepts do not fit in the standard model of particle physics. To overcome this problem, at least, we have to be open to matter fields with kinematics and dynamics beyond the standard model. But these matter fields might then very well correspond to different spacetime geometries. This is the basis of this thesis: it studies the underlying spacetime geometries and ventures into the quantization of those matter fields independently of any background geometry. In the first part of this thesis, conditions are identified that a general tensorial geometry must fulfill to serve as a viable spacetime structure. Kinematics of massless and massive point particles on such geometries are introduced and the physical implications are investigated. Additionally, field equations for massive matter fields are constructed like for example a modified Dirac equation. In the second part, a background independent formulation of quantum field theory, the general boundary formulation, is reviewed. The general boundary formulation is then applied to the Unruh effect as a testing ground and first attempts are made to quantize massive matter fields on tensorial spacetimes.
Bekanntermaßen hat Albert Einstein die Geometrie der Raumzeit an den Maxwell-Gleichungen abgelesen. Heutzutage nehmen wie diese Geometrie so ernst, dass unsere fundamentale Materietheorie, das Standardmodell der Teilchenphysik, darauf beruht. Sobald es jedoch um die Physik außerhalb des Sonnensystems geht, scheinen einige Dinge unverstanden zu sein. Unabhängige Beobachtungsreihen zeigen, dass wir Konzepte wie dunkle Materie und dunkle Energie brauchen um unsere Modelle mit den Beobachtungen in Einklang zu bringen. Diese Konzepte passen aber nicht in das Standardmodell der Teilchenphysik. Um dieses Problem zu überwinden, müssen wir zumindest offen sein für Materiefelder mit Kinematiken und Dynamiken die über das Standardmodell hinaus gehen. Diese Materiefelder könnten dann aber auch durchaus zu anderen Raumzeitgeometrien gehören. Das ist die Grundlage dieser Arbeit: sie untersucht die zugehörigen Raumzeitgeometrien und beschäftigt sich mit der Quantisierung solcher Materiefelder unabhängig von jeder Hintergrundgeometrie. Im ersten Teil dieser Arbeit werden Bedingungen identifiziert, die eine allgemeine tensorielle Geometrie erfüllen muss um als sinnvolle Raumzeitgeometrie dienen zu können. Die Kinematik masseloser und massiver Punktteilchen auf solchen Raumzeitgeometrien werden eingeführt und die physikalischen Implikationen werden untersucht. Zusätzlich werden Feldgleichungen für massive Materiefelder konstruiert, wie zum Beispiel eine modifizierte Dirac-Gleichung. Im zweiten Teil wird eine hintergrundunabhängige Formulierung der Quantenfeldtheorie, die General Boundary Formulation, betrachtet. Die General Boundary Formulation wird dann auf den Unruh-Effekt angewendet und erste Versuche werden unternommen massive Materiefelder auf tensoriellen Raumzeiten zu quantisieren.

Thesis (Ph. D.) -- University of Maryland, College Park, 2007.
Thesis research directed by: Dept. of Physics. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.

La thèse contient une étude de la structure et les symétries de l'espace-temps présentée de quelques perspectives différentes. Géodésiques de particules sont analysées dans les champs gravitationnels forts. Les orbites circulaires et l'existence de la sphère des photons sont considerés dans la géométrie de Schwarzschild et des espaces-temps « Schwarzschild-like », résultants de la gravité de Hořava-Lifshitz. Communication au moyen de signaux électromagnétiques radiaux est étudié dans la proximité de l'horizon des événements dans l'espace-temps de Schwarzschild, où Kruskal-Szekeres système de coordonnées est employé. Dans le cas de l'espace-temps plat de Minkowski la brisure spontanée des symétries relativistes est examinée dans le cadre de la théorie quantique des champs. Violation de l'invariance de Lorentz est la conséquence de la présence de dérivés d'ordre supérieur dans l’action. La cohérence des théories effectives avec des dérivés d'ordre supérieur est établie par la régularisation sur réseau, en utilisant la propriété de la positivité de réflexion. Le modèle étendu de l'électrodynamique quantique scalaire, où le vide brise la symétrie de Lorentz, est construit et ses principales caractéristiques, y compris le spectre des quasi-particules, sont décrits
The thesis contains a study of spacetime structure and symmetries presented from a few different perspectives. Geodesics of massless and massive particles are investigated in strong gravitational fields. Circular orbits and the existence of the so-called photon sphere are considered in Schwarzschild geometry and in Schwarzschild-like spacetimes arising from Hořava-Lifshitz gravity. Communication by means of radial electromagnetic signals is studied in the vicinity of the event horizon in Schwarzschild spacetime where Kruskal-Szekeres coordinate system is employed. In the case of flat Minkowski spacetime the spontaneous breakdown of relativistic symmetries is analyzed in the framework of quantum field theory. Lorentz invariance violation occurs due to the presence of higher order derivative terms in the action. The consistency of effective theories with derivatives of higher order is established in lattice regularization, using the property of reflection positivity. The extended model of scalar QED, where vacuum violates Lorentz symmetry, is constructed and its main features, including quasi-particle content, are described

Le sujet de la thèse est la construction d'une théorie perturbative des champs quantiques en interaction sur un espace-temps courbe, suivant un point de vue conçu par Stueckelberg et Bogoliubov et developpé par Epstein-Glaser sur l'espace de Minkowski plat. En 2000 un progrès important fut réalisé par Brunetti et Fredenhagen qui réussirent à étendre la théorie d'Epstein-Glaser en exploitant le point de vue développé par Radzikowski pour definir les états quantiques sur un espace-temps courbe en terme d'ensembles de front d'onde. Ces résultats furent ultérieurement généralisés par Fredenhagen, Brunetti, Hollands, Wald, Rejzner, etc. Aux théories de Yang-Mills et de la gravitation. Cependant, même pour des théories sans invariance de jauge, de nombreux détails mathématiques sont restés inexplorés et parfois sans vérification. Dans cette thèse, on construit d'une façon totalement rigoureuse cette théorie dans le cas des champs sans invariance de jauge. Dans mon travail, j' ai revisité complètement cette théorie, résolvant au passage plusieurs questions laissées en suspens, incorporant de nombreux résultats nouveaux autour de ce programme et, le cas échéant, apportant des détails beaucoup plus précis sur les contre-termes dans le processus de renormalisation, une compréhension plus approfondie des ambiguïtés et une description géométrique des ensembles de front d'onde. L'ensemble de la thèse utilise un large éventail de techniques mathématiques : de la géométrie différentielle et pseudo riemannienne, des techniques d'analyse micro-locale et de géométrie symplectique pour les fronts d'onde, de l'analyse fonctionnelle, des résultats fins de la théorie des distributions
The subject of the thesis is the construction of a perturbative quantum theory of interacting fields on a curved space-time, following a point of view pioneered by Stueckelberg and Bogoliubov and developed by Epstein-Glaser on the flat Minkowski space-time. In 2000 a breakthrough was done by Brunetti and Fredenhagen who were able to extend the Epstein-Glaser theory by exploiting the point of view developed by Radzikowski to define quantum states on a curved space-time in terms of wave-front sets. These results were further extended by Fredenhagen, Brunetti, Hollands, Wald, Rejzner, etc. To Yang-Mills fields and the gravitation. However, even for theories without gauge invariance, many mathematical details were left unexplored and unquestioned. The task of Viet was precisely to derive fully rigorously this theory in the case there is no gauge invariance. In my work, I propose a complete review of the result, solving numerous questions, adding many new results around this program and, eventually, giving more precise details on the counterterms and ambiguities in the renormalization process, and a deeper understanding of the geometry of the wave front set of the n-point functions. All this thesis uses various mathematical techniques: differential and pseudo Riemannian geometry, microlocal analysis and the symplectic geometry of wavefront sets, functional analysis, fine results from the theory of distributions, Hopf algebras, etc

Quantum field theory is studied within the semi-classical gravity approximation. The quantum correction to the propagation of both photons and gravitons in a general curved space background is calculated showing a non-trivial spacetime refractive index as well as a dynamical dressing (or undressing) of the particle state. The initial interacting particle's 'dressing', the cloud of virtual particles that surrounds it, may receive corrections from an infinite number of modes even for flat space. When gravitational tidal effects remove this dressing, squeezing it back into the bare particle, this leads to an amplification in a way consistent with unitarity. There is a possible shift discovered in the graviton wavefront velocity related to higher order curvature couplings, although in this calculation there is also a logarithmic divergence at high frequencies, leading to a breakdown of the perturbative approximation. Next we consider initial value problems and the stability of de Sitter space. Here the self decay of a massive scalar in de Sitter space is proposed to lead to a particle explosion where divergent growth of the field expectation value is observed. Directly investigating this divergent field expectation value a one loop calculation is completed for a massive scalar particle in 3-dimensional de Sitter space. This result has characteristic secular growth that can be summed into a rapidly decaying exponential by using the dynamical renormalisation group. Finally the evolution of two point functions is studied, by numerically solving their equations of motion using the Kaydanoff-Baym equations in 2-dimensional de Sitter space. Here we see a decay of the vacuum state due to the coupling. This appears to be related to the choice of initial conditions be chosen to match the free field vacuum plus non-interacting particles. This choice is made inappropriate by the dynamical dressing of the bare particle states.

The time coordinate is a common obstacle in the theory of non-commutative (nc.) spacetimes. Despite that, this work shows how the interplay between quantum fields and an underlying nc. spacetime can still be analyzed, even for the case of nc. time. This is done for the example of a general Moyal-type external potential scattering of the Dirac field in Moyal-Minkowski spacetime. The spacetime is a rare example of a Lorentzian non-compact nc. geometry. Elements of the associated spectral function algebra are shown to be operationally involved at the level of quantum field operators by Bogoliubovs formula. Furthermore, a similar task is attacked in the case of locally nc. spacetimes. An explicit star-product is constructed by a method of Kontsevich. It implements a decay of non-commutativity with increasing distance. This behavior should benefit the technical side - diverse interesting formal attempts are discussed. It is striven for unification of several toy models of nc. spacetimes and a general strategy to define quantum field operators. Within the latter one has to implement the usual quantum behavior as well as a new kind of spacetime behavior. It is shown how this two-fold character causes key difficulties in understanding.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
We start this work studying the appearance of geometric quantum phases as in the relativistic as in the non-relativistic quantum dynamics of a neutral particle with permanent magnetic and electric dipole moment which interacts with external electric and magnetic fields in the presence of linear topological defects. We describe the linear topological defects using the approach proposed by Katanaev and Volovich, where the topological defects in solids are described by line elements which are solutions of the Einstein's equations in the context of general relativity. We also analyze the in uence of non-inertial effects in the quantum dynamics of a neutral particle using two distinct reference frames for the observers: one is the Fermi-Walker reference frame and another is a rotating frame. As a result, we shall see that the difference between these two reference frames is in the presence/absence of dragging effects of the spacetime which makes its in uence on the phase shift of the wave function of the neutral particle. In the following, we shall use our study of geometric quantum phases to make an application on the Holonomic Quantum Computation, where we shall show a new approach to implement the Holonomic Quantum Computation via the interaction between the dipole moments of the neutral particle and external fields and the presence of linear topological defects. Another applications for the Holonomic Quantum Computation is based in the structure of the topological defects in graphene layers. In the presence of topological defects, a graphene layer shows two distinct phase shifts: one comes from the mix of Fermi points while the other phase shift comes from the topology of the defect. To provide a geometric description for each phase shift in the graphene layer, we use the Kaluza-Klein theory where we establish that the extra dimension describes the Fermi points in the graphene layer. Hence, we can implement the Holonomic Quantum Computation through the possibility to build cones and anticones of graphite in such way we can control the quantum uxes in graphene layers. In the last part of this work, we study the Landau quantization for neutral particles as in the relativistic dynamics and non-relativistic dynamics. In the non-relativistic dynamics, we study the Landau quantization in the presence of topological defects as in an inertial as in a non-inertial reference frame. In the relativistic quantum dynamics, we start our study with the Landau quantization in the Minkowisky considering two different gauge fields. At the end, we study the relativistic Landau quantization for neutral particles in the Cosmic Dislocation spacetime.
Neste trabalho estudamos inicialmente o surgimento de fases geometricas nas dinâmicas quânticas relativística e não-relativística de uma partícula neutra que possui momento de dipolo magnético e elétrico permanente interagindo com campos elétricos e magnéticos externos na presença de defeitos topológicos lineares. Para descrevermos defeitos topológicos lineares usamos a aproximação proposta por Katanaev e Volovich, onde defeitos lineares em sólidos são descritos por elementos de linha que são soluções das equações de Einstein no contexto da relatividade geral. Analisamos também a inuência de efeitos não-inerciais na dinâmica quântica de uma partícula neutra em dois tipos distintos de referenciais para os observadores: um é o referencial de Fermi-Walker e outro é um referencial girante. Vemos que a diferença entre dois referenciais está na presença/ausência de efeitos de arrasto do espaço-tempo que irá influenciar diretamente na mudança de fase na funçãao de onda da partícula neutra. Em seguida, usamos nosso estudo de fases geométricas para fazer aplicações na Computação Quântica Holonômica onde mostramos uma nova maneira de implementar a Computação Quântica Holonômica através da interação entre momentos de dipolo e campos externos e pela presença de defeitos topológicos lineares. Outra aplicação para a Computação Quântica Holonômica está baseada na estrutura de defeitos topológicos em um material chamado grafeno. Na presença de defeitos topológicos lineares, esse material apresenta duas fases quânticas de origens distintas: uma da mistura dos pontos de Fermi e outra da topologia do defeito. Para dar uma descrição geométrica para a origem de cada fase no grafeno usamos a Teoria de Kaluza-Klein, onde a dimensão extra sugerida por esta teoria descreve os pontos de Fermi no grafeno. Portanto, a implementação da Computação Quântica Holonômica no grafeno está baseada na possibilidade de construir cones e anticones de grafite de tal maneira que se possa controlar os fluxos quânticos no grafeno. Na última parte deste trabalho estudamos a quantização de Landau para partículas neutras tanto na dinâmica não-relativística quanto na dinâmica relativística. Na dinâmica não-relativítica, estudamos a quantização de Landau na presença de defeitos em um referecial inercial e, em seguida, em um referencial nãoo-inercial. Na dinâmica relativística, estudamos inicialmente a quantização de Landau no espaço-tempo plano em duas configurações de campos diferentes. Por fim, estudamos a quantização de Landau relativística para partículas neutras no espaço-tempo da deslocação cósmica.

De nombreuses approches à la gravité quantique suggèrent que la description usuelle de l’espace-temps ne serait pas adaptée à la description des phénomènes physiques impliquant à la fois des processus gravitationnels et quantiques. Une meilleure description pourrait consister à munir l’espace-temps d’une structure non-commutative en remplaçant les coordonnées locales sur la variété par des opérateurs ne commutant pas deux-à-deux. Il s’ensuit que le comportement des théories de champs construites sur de tels espaces diffère en général de celui des théories de champs ordinaires. L’étude de ces possibles nouvelles propriétés est l’objet de la théorie non-commutative des champs (TNCC) dont nous étudions certains des aspects.Dans le présent mémoire, nous considérons deux familles d’espaces quantiques dont l’algèbres de coordonnées admet une structure d’algèbre de Lie. La première famille est caractérisée par l’algèbre su(2) et apparait dans le cadre de modèle de gravité quantique en 3 dimensions, ainsi que dans certains modèles de « brane » et de « group field theory ». La seconde famille d’espaces quantiques est connue sous le nom de kappa-Minkowski. L’intérêt de cet espace réside dans le fait qu’il est défini comme l’espace homogène associé à l’algèbre de Hopf de kappa-Poincaré. Cette dernière définit une déformation, à l’échelle de Planck, de l’algèbre de Poincaré et s’avère être étroitement liée à certains modèles de gravité quantique.Afin d’étudier les TNCC, il est commode de représenter l’espace quantique comme une algèbre non-commutative de fonctions munie d’un produit déformé appelé « star-product ». Une façon canonique de construire un tel produit consiste à se servir d’outils d’analyse harmonique et à adapter le schéma de quantification de Weyl (originellement introduit dans le cadre de la mécanique quantique) à l’algèbre considérée. Les expressions de star-product associé aux espaces susmentionnés sont dérivées de manière explicite. Nous montrons en particulier que des familles de star-product inéquivalents peuvent être classifiées par des considérations cohomologiques. Nous étudions enfin les propriétés quantiques de différents modèles de TNCC scalaire quartique construits à l’aide de ces star-product. Dans le cas où l’espace quantique est caractérisé par l’algèbre su(2), nous trouvons que la fonction 2-point est fini à l’ordre une boucle, le paramètre de déformation jouant le rôle d’une coupure ultraviolette et infrarouge. Dans le cas de kappa-Minkowski, nous insistons sur l’invariance sous kappa-Poincaré de l’action fonctionnelle et montrons que certains modèles de TNCC scalaire quartique divergent moins que dans le cas commutatif. Par ailleurs, la fonction 4-point est trouvée finie à l’ordre une boucle. Nos résultats, ainsi que leurs conséquences, sont finalement discutés
As many theoretical studies point out, the classical description of spacetime, as a continuum, might be no longer adequate to reconcile gravity with quantum mechanics at very high energy (the relevant energy scale being often regarded as the Planck scale). Instead, a more appropriate description could be provided by the data of a noncommutative algebra of coordinate operators replacing the usual commutative local coordinates on smooth manifold. Once the noncommutative nature of spacetime is assumed, it is to expect that the (classical and quantum) properties of field theories on noncommutative background differ from the ones of field theories on classical background. This is the aim of Non-Commutative Field Theory (NCFT) to explore and study these new properties.In the present dissertation, we consider two families of quantum spacetimes of Lie algebra type noncommutativity. The first family is characterised by su(2) noncommutativity and appears in the description of some models of quantum gravity in 3-dimensions. The other family of quantum spacetimes is known in the physics literature as the 4-d kappa-Minkowski space. The importance of this quantum spacetime lies into the fact that its symmetries are provided by the (quantum) kappa-Poincaré algebra (a deformation of the classical Poincaré algebra) together with the fact that the deformation parameter 'kappa', which is of mass dimension, provides a natural energy scale at which the quantum gravity effects may be relevant (and is often regarded as being related to the Planck scale). For these reasons, the kappa-Minkowski space appears as a good candidate for a spacetime to be involved in the description of Doubly Special Relativity and Relative Locality models.To study NCFT it is often convenient to introduce a star product characterising the (noncommutative) C*-algebra of fields modelling the quantum spacetime under consideration. We emphasise that a canonical star product can be obtained by using the group algebraic structures underlying the construction of such Lie algebra type quantum spaces, namely by making use of harmonic analysis on the corresponding Lie group together with the Weyl quantisation scheme. The explicit derivation of such star product for kappa-Minkowski is given. In addition, we show that su(2) Lie algebras of coordinate operators related to quantum spaces with su(2) noncommutativity can be conveniently represented by SO(3)-equivariant poly-differential involutive representations and show that the quantized plane waves obtained from the quantization map action on the usual exponential functions are determined by polar decomposition of operators combined with constraint stemming from the Wigner theorem for SU(2). We finally indicate a convenient way to extend this construction to other semi-simple but non simply connected Lie groups by making use of results from group cohomology with value in an abelian group that would replace the constraints stemming from the simple Wigner theorem.Then, we investigate the quantum properties of various models of interacting scalar field theory on noncommutative background making use of the aforementioned star product formalism to construct physically reasonable expressions for the action functional. Considering quantum spacetime with su(2) noncommutativity, we find that the one-loop 2-point function for complex scalar field theories with quartic interactions is finite, the deformation parameter playing the role of a natural UV cut-off. Special attention is paid to the derivation of the one-loop corrections to both the 2-point and 4-point functions for various models of kappa-Poincaré invariant scalar field theory with quartic interactions. In that case, we show that for some models the 2-point function divergences linearly thus slightly milder than their commutative counterpart, while the one-loop 4-point function is shown to be finite. The results we obtained together with their consequences are finally discussed

In Part I we are dealing with effective description of Yang-Mills theories based on gauge-invarint variables. For pure Yang-Mills we study the spin-charge separation varibles. The dynamics in these variables resembles the Skyrme-Faddeev model. Thus the spin-charge separation is an important intermediate step between the fundamental Yang-Mills theory and the low-energy effective models, used to model the low-energy dynamics of gluons. Similar methods may be useful for describing the Electroweak sector of the Standard Model in terms of gauge-invariant field variables called supercurrents. We study the geometric structure of spin-charge separation in 4D Euclidean space (paper III) and elaborate onconnection with gravity toy model. Such reinterpretation gives a way to see how effective flat background metric is created in toy gravity model by studying the appearance of dimension-2 condensate in the Yang-Mills (paper IV). For Electroweak theory we derive the effective gauge-invariant Lagrangian by doing the Kaluza-Klein reduction of higher-dimensional gravity with 3-brane, thus making explicit the geometric interpretation for gauge-invariant supercurrents. The analogy is then made more precise in the framework of exact supergravity solutions. Thus, we interpret the Higgs effect as spontaneous breaking of Kaluza-Klein gauge symmetry and this leads to interpretation of Higgs field as a dilaton (papers I and II). In Part II of the thesis we study rather simple field theories, called “geometric” or “instantonic”. Their defining property is exact localization on finite-dimensional spaces – the moduli spaces of instantons. These theories allow to account exactly for non-linearity of space of fields, in this respect they go beyond the standard Gaussian perturbation theory. In paper V we show how to construct a geometric theory of chiral boson by embedding it into the geometric field theory. In Paper VI we elaborate on the simplest geometric field theory – the supersymmetric Quantum Mechanics and construct new non-perturbative topological observables that have a transparent meaning both in geometric and in the Hamiltonian formalisms. In Paper VII we are motivated by making perturbations away from the simple instantonic limit. For that we need to carefully define the observables that are quadratic in momenta and develop the way to compute them in geometric framework. These correspond geometrically to bivector fields (or, in general, the polyvector fields). We investigate the local limit of polyvector fields and compare the geometric calculation with free-field approach.

Lo scopo primario di questa tesi e' l’analisi di una nuova procedura di regolarizzazione di path integral su spazi curvi, presentata inizialmente dal fisico J. Guven e applicata al caso di una teoria di campo scalare , ma mai utilizzata per svolgere ulteriori calcoli espliciti. Questa procedura, se corretta, permetterebbe di utilizzare il formalismo di path integral su spazi piatti anche nel caso in cui la varieta' di background risulti localmente curva. Tale procedura trasforma di fatto un modello sigma non lineare in un modello efficace lineare, permettando pertanto di aggirare le usuali complicazioni dovute alla generalizzazione di path integral. Una prova diretta della correttezza della procedura di Guven sembra mancare in letteratura: per questo motivo in questa tesi verranno eseguiti vari test volti a tale verifica. Alcuni errori sono stati riscontrati nella proposta iniziale, tra i quali un termine di potenziale che risulta essere non corretto. Ad ogni modo siamo stati in grado di identificare un potenziale che permetta di riprodurre correttamente i primi due coefficienti dell’espansione in serie dell’heat kernel. Utilizzando lo stesso metodo abbiamo poi cercato di ottenere il successivo coefficiente dell’espansione (cubico in termini di curvatura): il risultato ottenuto non risulta essere corretto, cosa che segnala il fallimento di tale metodo ad ordini superiori. Visti tali risultati preliminari, siamo stati indotti a considerare una classe speciale di spazi curvi, quella degli spazi massimamente simmetrici, trovando invece che su tali spazi la procedura di Guven riproduce i risultati corretti. Come verifica abbiamo ottenuto la parte diagonale dell’heat kernel, che ́ stata poi utilizzata per riprodurre l’anomalia di traccia di tipo A per campi scalari in dimensioni arbitrarie fino a D = 12. Questi risultati sono in accordo con quelli attesi. Viene pertanto fornita una prova della validita' di tale procedura su questi spazi.

As propriedades termodinâmicas relacionadas a um gás composto por partículas bosônicas em geometrias esfericamente simétricas são apresentadas ao longo deste trabalho. Utilizando o formalismo proposto por Ishibashi e Kodama, verificou-se que as equações de movimento associadas aos campos escalar e eletromagnético podem ser reduzidas a uma equação tipo Schrödinger. Ao considerarmos algumas soluções esfericamente simétricas observou-se que o espectro de energia associado às partículas é discretizado. Em particular, no estudo das lightspheres, superfícies onde os fótons estão confinados em órbitas fechadas, propomos um mecanismo de quantização para as partículas bosônicas. Estudamos algumas propriedades termodinâmicas e dentro deste tratamento, é apresentada uma expressão para a densidade de energia espectral da radiação emitida. Nossos resultados sugerem que as lightspheres quando termalizadas com o seu ambiente, possuem propriedades termodinâmicas não-usuais. Ao levarmos em consideração a presença da constante cosmológica negativa, constatou-se que além de um espectro de energia discretizado, a geometria adS possui um comportamento confinante, podendo ser interpretada como uma caixa de tamanho conhecido. Considerando um gás de partículas bosônicas confinadas na geometria anti-de Sitter, obtivemos as grandezas termodinâmicas associadas tais como energia interna, entropia e pressão. Para a energia interna observamos um comportamento diferente do usual para o limite de baixas temperaturas.
The thermodynamic properties related to a gas composed of bosonic particles in spherically symmetrical geometries are presented in this work. Using the formalism proposed by Ishibashi and Kodama, we have seen that the equations of motion associated to the scalar and electromagnetic fields can be reduced to a Schrödinger-like equation. For some spherically symmetrical solutions it has been demonstrated that the energy spectrum associated with the particles is discretized. In particular, when we considered lightspheres, surfaces where photons are confined in closed orbits, we propose a quantization procedure for the bosonics particles. In this treatment, it is presented an expression for the spectral energy density of the emmited radiation. Our results suggest that lightspheres thermalized with its environment, have unusual thermodynamical properties. When taken into account the presence of the negative cosmological constant, it has been shown that, besides a discretized energy spectrum, the geometry has a confining behavior and can be interpreted as a finite size box. Considering a gas of bosonic particles confined in the anti-Sitter geometry, we obtained the associated thermodynamics quantities such as internal energy, entropy and pressure. For the internal energy, in the low temperatures limit, we observe a different behavior from the usual one.

O objeto de estudo desta dissertação é o efeito de criação de partículas pela curvatura sob o escopo de uma teoria de espalhamento, discutindo quando que a interpretação a partir de uma matriz S é tangível e obtendo sua expressão nesses casos. O capítulo de introdução aborda superficialmente conceitos de relatividade geral e de teoria quântica de campos em espaços planos e curvos, necessários para a construção da matriz S. O conteúdo deste capítulo segue as apresentações feitas por Wald, Parker e Birrell em geral, tendo como guia as obras de Bar, Wald e Hawking no que se trata especificamente de relatividade geral, e de Penrose e Rindler no que se trata da estrutura espinorial. A construção da matriz S se dá no capítulo 2, tendo como guia o trabalho de Wald. O capítulo 3 apresenta exemplos que permitem a contextualização da criação de partículas em casos específicos de espaços-tempos em expansão. Este estudo nos permite verificar que as condições que precisam ser satisfeitas em um espaço-tempo globalmente hiperbólico e assintoticamente estacionário para que a formulação da matriz S possa ser feita são que as teorias no passado e futuro distantes devem ser unitariamente equivalentes, que a relação entre as regiões se dá através de transformações de Bogolyubov dadas por operadores limitados definidos em toda a parte e que tais operadores satisfaçam a condição de Hilbert-Schmidt. Nestes casos obtemos uma expressão para a matriz $S$ que descreve a criação de partículas pela curvatura do espaço-tempo para o campo de Klein-Gordon e de Dirac, além de outras relações úteis, como número médio de partículas criadas e probabilidade de se encontrar partículas em determinado modo, o que permite uma analogia com a radiação de corpo negro, passo fundamental para se entender fenômenos de grande interesse na física, como a radiação de Hawking e a criação de partículas no período inflacionário.
This master\'s thesis deals with the effect of particle creation by the curvature of space-time according to the point of view of scattering theory, discussing when such interpretation is possible by means of an S-matrix and obtaining its expression in those cases. The first chapter treats, superficially, some concepts of general relativity and quantum field theory in plane and curved space-times that are imperative to understand the construction of the S-matrix. The subject of this chapter is covered in the work of Wald, Parker, and Birrell, and follows closely the work of Bar, Wald and Hawking, when treats concepts specifically from general relativity, and from Penrose and Rindler, when talking about the spinor structure of space-time. The construction of the S-matrix is made in the second chapter, along the lines of the work of Wald. The third chapter presents some examples that bring some light on the creation of particles in specific cases of expanding space-times. This study let us verify that an S-matrix formulation is tenable, on globally hyperbolic asymptotic stationary curved space-times, if both quantum theories in the distant past and distant future are unitary equivalent, the relation of both regions is made by Bogolyubov transformations by means of everywhere defined bounded operators and that those operators satisfy the Hilbert-Schmidt condition. In those cases we derive the expression of the S-matrix for the Klein-Gordon and Dirac fields. Also we obtain the number of particles created and the probability of find particles in a particular mode, with let one make an analogy with the black body radiation, which is a fundamental step in the direction of understanding interesting phenomena in quantum field theory in curved space-times, like the Hawking radiation and particle creation in the early universe.

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In this work, we consider a charged massive fermionic quantum field in the idealized cosmic string spacetime and in the presence of a magnetic field confined in a cylindrical tube of finite radius. Three distinct configurations for the magnetic fields are taken into account: (i) a cylindrical shell of radius a, (ii) a magnetic field proportional to 1/r and (iii) a constant magnetic field. In these three cases, the axis of the infinitely long tube of radius a coincides with the cosmic string. Our main objectives in this paper are to analyze vacuum expected values (VEVs) of the current density, jP, fermionic condensate (FC) e and the VEV of the fermionic energy-momentum tensor, Ti". In order to do that, we explicitly construct the complete set of normalized wave-functions for each configuration of magnetic field. We show that in the region outside the tube, the current density, the FC and the VEV of the energy-momentum tensor are decomposed into two parts: the first ones correspond to the zero-thickness magnetic flux contributions, and the seconds are induced by the non-trivial structure of the magnetic field, named core-induced contributions. The latter present specific forms depending on the magnetic field configuration considered. We also show that the VEV of the energy-momentum tensor is diagonal, obeys the conservation condition and its trace is expressed in terms of the fermionic condensate.
Nesta Tese, consideramos urn campo fermiônico massivo e carregado no espaço-tempo de uma corda cósmica ideal na presença de um campo magnético confinado em um tubo cilindrico de raio finito a. Levamos em conta três configurações para o campo magnético: (i) uma casca cilindrica de raio a, (ii) um campo magnetico proporcional a 1/r e (iii) urn campo magnetico constante. Nos três casos, o eixo de simetria da corda cósmica coincide corn o eixo de simetria do tubo cilindrico de campo magnetico, dispostos ao longo do eixo z. Nossos principais objetivos nesta Tese sao analisar os valores esperados no yam° (VEV) da densidade de corrente, ji", do condensado fermionico (FC) e o VEV do tensor energia-momento (TEM), Ti". Para isto, construfmos urn conjunto completo de fungoes de onda de Dirac normalizadas para cada configuragao de campo magnetic° e mostramos que na regiao fora do tubo, a densidade de corrente, o CF e o VEV do TEM sao decompostos como a soma de duas partes. A primeira corresponde a contribuigao da linha de fluxo magnetic° que corre ao longo da corda cOsmica ideal, e a segunda contribuigao é induzida devido a estrutura nao trivial de campo magnetic°. Mostramos tambem que o VEV do tensor energia-momento é diagonal, obedece a condigao de conservagao e que seu trago é expresso ern termos do condensado fermionico.

Cette thèse concerne l'étude des propriétés de champs scalaires classiques et quantiques en présence d'un environnement inhomogène et/ou dépendant du temps. Nous nous concentrerons sur des modèles pouvant être décrits, fondamentalement ou de manière effective, par un espace-temps courbe contenant un horizon des événements. Nous verrons en particulier comment une correspondance mathématique, provenant d'une symétrie de Lorentz effective à basse énergie, permet de relier les comportements des ondes dans un cadre non relativiste à la physique des trous noirs, quelles en sont les limites et dans quelle mesure les résultats ainsi obtenus sont og analogues fg à leurs pendants gravitationnels. Après un premier chapitre d'introduction rappelant quelques bases de relativité générale puis une dérivation de la radiation de Hawking et de la correspondance avec des systèmes non relativistes, je présenterai le détail de quatre travaux effectués durant ma thèse. Les autres articles écrits dans ce cadre sont résumés dans le dernier chapitre, précédant une conclusion générale. Mes collaborateurs et moi nous sommes concentrés sur trois aspects du comportement des champs près de l'analogue d'un horizon des événements dans des modèles avec une symétrie de Lorentz effective à basse énergie. Le premier concerne les effets non linéaires, cruciaux pour comprendre l'évolution de la radiation de Hawking ainsi que pour les réalisations expérimentales mais auparavant peu étudiés. Nous montrerons comment ceux-ci déterminent les possibles comportements aux temps longs pour des systèmes stables ou instables. Le second aspect a trait aux effets linéaires et quantiques, en particulier la radiation de Hawking elle-même, son devenir lorsque l'horizon est continûment effacé, ainsi que les diverses instabilités à même de survenir dans différents modèles. Enfin, nous avons participé à l'élaboration, à l'analyse et à l'étude d’expériences dites de og gravité analogue fg dans des condensats de Bose-Einstein et des systèmes hydrodynamiques ou acoustiques, dont je rapporte les principaux résultats
The present thesis deals with some properties of classical and quantum scalar fields in an inhomogeneous and/or time-dependent background, focusing on models where the latter can be described as a curved space-time with an event horizon. While naturally formulated in a gravitational context, such models extend to many physical systems with an effective Lorentz invariance at low energy. We shall see how this effective symmetry allows one to relate the behavior of perturbations in these systems to black-hole physics, what are its limitations, and in which sense results thus obtained are “analogous” to their general relativistic counterparts. The first chapter serves as a general introduction. A few notions from Einstein's theory of gravity are introduced and a derivation of Hawking radiation is sketched. The correspondence with low-energy systems is then explained through three important examples. The next four chapters each details one of the works completed during this thesis, updated and slightly reorganized to account for new developments which occurred after their publication. The other articles I contributed to are summarized in the last chapter, before the general conclusion. My collaborators and I focused on three aspects of the behavior of fields close to the (analogue) event horizon in models with an effective low-energy Lorentz symmetry. The first one concerns nonlinear effects, which had been given little attention in view of their crucial importance for understanding the evolution in time of Hawking radiation as well as for experimental realizations. We showed in particular how they determine the late-time behavior in stable and unstable configurations. The second aspect concerns linear and quantum effects. We studied the Hawking radiation itself in several models and what replaces it when continuously erasing the horizon. We also characterized and classified the different types of linear instabilities which can occur. Finally, we contributed to the design and analysis of “analogue gravity” experiments in Bose-Einstein condensates, hydrodynamic flows, and acoustic setups, of which I report the main results

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In this thesis we study the dynamics of charge carriers, and the electronic properties, of the C60 fullerene molecule. Characterizing it by a geometric bias. In inertial reference systems and when we have your material under rotation content. Initially we discussed the scientific advent of carbon allotropes, and the importance of modelling its derivates at low energies. We show that at low energies, the graphene - the two-dimensional carbon allotrope form - can be described for a non-massive theory of free fermions. At a second moment, we extended the nonmassive free fermions theory for the C60 molecule. Assuming the hexagonal graphene network can be entered in fullerene when we introduce topological defects. A brief study of topological defects in condensed matter was done. And soon after, we made a description these defects via a non-Euclidean geometry. Showing how the charge carriers in the network see the defects like gauge fields. Then we began to expose the results of this thesis. First we assume the fullerene by a two-dimensional spherical metric with defects, containing a fictitious t’Hooft-Polyakov monopole in its center. TheC60 is still subjected to the action of an Aharonov-Bohm flux arising of a magnetic wire running through its poles. So we get the spectrum, and the prediction of a persistent current in the molecule. Finally we return to the analysis of the molecule, now with your content of matter under rotation. For this, we studied a metric Gödel-type with spherical symmetry. We discussed the problem of causality and obtain the spectrum and the persistent current in terms of the vorticity (W) of spacetime.
Nesta tese estudamos a dinâmica de portadores de carga, e as propriedades eletrônicas, na molécula de fulerenoC60. Caracterizando-a por um viés geométrico. Tanto em sistemas de referência inercial, como quando temos seu conteúdo de matéria sob rotação. Inicialmente abordamos o advento científico das formas alotrópicas do carbono e a importância da modelagem a baixas energias dos seus derivados. Onde mostramos que no limite de baixas energias, o grafeno - que trata-se da forma alótropica bidimensional do carbono - pode ser descrito por uma teoria de férmions livres sem massa. Num segundo momento estendemos a teoria de férmions não massivos para a molécula de C60. Assumindo que a rede hexagonal do grafeno pode inscrever o C60 ao introduzirmos alguns defeitos topológicos. Um breve estudo sobre os defeitos topológicos na matéria condensada foi feito. Onde, logo em seguida, partimos para uma descrição de tais defeitos via uma geometria não-euclidiana. Mostrando como os portadores de carga no meio enxergam os defeitos como campos de gauge. Em seguida começamos a expor os resultados desta tese. Primeiramente assumimos tratar o fulereno por uma métrica de uma esfera bidimensional com defeitos, e contendo um monopolo de t’Hooft-Polyakov fictício em seu centro. O C60 é ainda submetido a ação de um fluxo de Aharonov-Bohm advindo de uma corda magnética quiral transpassando seus polos. Obtemos assim o espectro e a predição de uma corrente persistente na molécula. Por fim retomamos a análise da molécula, agora com seu conteúdo de matéria sob rotação. Para isso assumimos tratar o fulereno por uma métrica do tipo Gödel com simetria esférica. Discutimos o problema da causalidade e obtemos espectro e corrente persistente em termos da vorticidade (W) do espaço-tempo.

A presente dissertação tem como objeto de estudo sistemas da física da matéria condensada que sejam capazes de simular sistemas gravitacionais, tais como buracos negros e universos em expansão, onde processos quânticos tomam parte. Neste estudo nos debruçamos principalmente sobre o modelo do fluido e condensados de Bose-Einstein. No modelo do fluido exploramos a geometria efetiva que surge e os problemas de back-reaction e dos modos trans-planckianos de campos quânticos. No modelo baseado em condensados exploramos sua faceta cosmológica e a possibilidade de campos maciços. Além destes dois modelos de grande relevância na literatura, ainda expomos os análogos em cordas elásticas e os baseados em ondas na superfícies de fluidos e uma análise geral baseada no formalismo lagrangeano para campos.
This dissertation has as object of study systems of condensate-matter physics which can simulate gravitational systems like black holes and expanding universes where quantum processes take place. In this study we lay attention mainly on the fluid model and on Bose-Einstein-condensate-based models. In the fluid model we explore the features of the emergent geometry and other problems like the back-reaction and the trans-planckian modes of quantum fields. In the condensate-based models we explore their cosmological aspects and the possibility for massive fields. Moreover, we shall present two other models, the elastic string and the surface-wave-based models in fluids, and a very general analysis based on the Lagrangean formalism for fields.

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Nesta tese, exploram-se diferentes aspectos e aplicações das teorias gravitacionais com correções quânticas. O texto é dividido em três partes principais. Na primeira parte, são consideradas as soluções linearizadas em diferentes teorias de gravitação com derivadas superiores. O potencial Newtoniano é calculado nos modelos locais, super-renormalizáveis no nível quântico, e mostra-se que a singularidade Newtoniana é cancelada devido a contribuição dos modos massivos extras. Logo depois, o colapso gravitacional de uma pequena massa é estudado na gravitação não-local livre de fantasmas, sendo o principal resultado a ausência da singularidade na solução do campo gravitacional e a possibilidade da não formação do miniburaco negro como resultado do colapso. Na segunda parte, algumas questões sobre a inflação induzida pela anomalia conforme são estudadas. É discutida a possibilidade da transição entre os períodos de inflação estável para instável. É mostrado que esta transição é automática se as correções quânticas nesse período forem desprezadas. Em seguida, considera-se o efeito de termos que violam as simetrias de CPT e Lorentz na inflação induzida pela anomalia conforme. É demonstrado que os novos termos responsáveis por violar essas simetrias não afetam a dinâmica do fator de escala da métrica. Por fim, na terceira parte as correções quânticas para o modelo dos Galileons e para as teorias dos campos massivos tensoriais antissimétricos são obtidas. É mostrado que o propagador da teoria dos Galileons recebe correções quânticas com derivadas superiores e que o teorema de não-renormalização do modelo dos Galileons permanece, de uma maneira generalizada, válido na região das baixas energias. Depois, por meio de cálculos explícitos das correções quânticas semiclássicas não-locais é confirmada a equivalência quântica entre os modelos dos campos tensoriais antissimétricos massivos com a teoria de Proca e com o modelo do campo escalar massivo mínimo.
In this thesis, different aspects and applications of gravitational theories with quan-tum corrections are explored. The text is divided into three main parts. In the first part, the linearized solutions in different gravity theories with higher derivatives are considered. The Newtonian potential is calculated in the local models, super-renormalizable at the quantum level, and it is shown that the Newtonian singularity is cancelled due to the contributions of the extra massive modes. Then the gravitational collapse of a small mass is studied in non-local ghost-free gravity, being the main result the absence of singularity in the gravitational field solution and the possibility of non-mini black hole formation as the collapse result. In the second part, some issues about anomaly-induced inflation are studied. It is discussed the possibility of the transition between stable to unstable periods of inflation. It is shown that this transition is automatic if the quantum corrections in this period are neglected. In the following, we consider the effect of CPT and Lorentz-violating terms in the conformal anomaly-induced inflation. It is shown the new terms responsible to violate these symmetries do not affect the dynamics of the metric scale factor. Finally, in the third part, the quantum corrections for the Galileon model and for the theory of the massive antisymmetric tensor fields are obtained. It is shown that the propaga-tor of Galilean theory receives quantum corrections with higher derivatives and that the non-renormalization theorem for Galileon models remains, in a generalized way, valid in the low-energy region. Then, by means of explicit calculations of non-local semiclassical quantum corrections, the quantum equivalence between the massive antisymmetric tensor field models with the Proca theory and minimal massive scalar field model is confirmed.

Les deux principales prédictions de la théorie quantique des champs en espace-temps courbe, à savoir la radiation de Hawking et la production de paires de particules ayant lieu dans un espace-temps non stationnaire, n'ont jamais été testé expérimentalement et impliquent toutes deux des processus à ultra haute énergie. En conséquence, de telles prédictions doivent être considérées prudemment. En utilisant l'analogie avec des systèmes de matière condensée mise en avant par Unruh, leur analogue pourrait être testé en laboratoire. Par ailleurs, dispersion et dissipation sont toujours présentes dans de tels systèmes, ce qui régularise la théorie à courte distances. Lors d'expériences destinées à tester les prédictions citées ci-dessus, le bruit thermique modifiera le résultat. En effet, il existe une compétition entre l'émission stimulée dudit bruit thermique et l'émission spontanée issue du vide quantique. Afin de mesurer la radiation de Hawking analogue et de l'analogue des productions de paires (souvent appelé effet Casimir dynamique), il est alors nécessaire de calculer les conséquence de la dispersion et de la dissipation, ainsi que d'identifier des observables permettant de certifier que l'amission spontanée a eu lieu. Dans cette thèse, nous analyserons d'abord les effets de la dispersion et de la dissipation à la fois sur la radiation de Hawking et sur la production de paires de particules. Afin d'obtenir des résultats explicites, nous travaillerons avec l'espace-temps de de Sitter. Les symétries de la théorie nous permettront d'obtenir des résultats exacts. Ceux-ci seront alors appliqués aux trous noirs grâce aux ressemblances entre la région proche du trou noir et l'espace de de Sitter. Afin d’introduire de la dissipation, nous considérerons un modèle exactement soluble permettant de modéliser n'importe quel taux de dissipation. Dans un tel modèle, le champ est couplé de manière linéaire à un environnement contenant un ensemble dense de degrés de liberté. Dans un tel contexte, nous étudierons l'intrication des particules produites. Ensuite, nous considérerons des systèmes de matière condensée spécifiques, à savoir les condensats de Bose et les polaritons. Nous analyserons les effets de la dissipation sur l'intrication de l’effet Casimir dynamique. Enfin, nous étudieront de manière générique l'intrication de la radiation de Hawking en présence de dispersion pour des systèmes analogues
The two main predictions of quantum field theory in curved space-time, namely Hawking radiation and cosmological pair production, have not been directly tested and involve ultra high energy configurations. As a consequence, they should be considered with caution. Using the analogy with condensed matter systems put forward by Unruh, their analogue versions could be tested in the lab. Moreover, the high energy behavior of these systems is known and involved dispersion and dissipation, which regulate the theory at short distances. When considering experiments which aim to test the above predictions, the thermal noise will contaminate the outcome. Indeed, there will be a competition between the stimulated emission from thermal noise and the spontaneous emission out of vacuum. In order to measure the quantum analogue Hawking radiation, or the analogue pair production also called dynamical Casimir effect, one should thus compute the consequences of ultraviolet dispersion and dissipation, and identify observables able to establish that the spontaneous emission took place. In this thesis, we first analyze the effects of dispersion and dissipation on both Hawking radiation and pair particle production. To get explicit results, we work in the context of de Sitter space. Using the extended symmetries of the theory in such a background, exact results are obtained. These are then transposed to the context of black holes using the correspondence between de Sitter space and the black hole near horizon region. To introduce dissipation, we consider an exactly solvable model producing any decay rate. In such a model, the field is linearly coupled to an environment containing a dense set of degrees of freedom. We also study the quantum entanglement of the particles so produced. In a second part, we consider explicit condensed matter systems, namely Bose Einstein condensates and exciton-polariton systems. We analyze the effects of dissipation on entanglement produced by the dynamical Casimir effect. As a final step, we study the entanglement of Hawking radiation in the presence of dispersion for a generic analogue system

Über die letzten Jahrzehnte hat sich die nichtkommutative Geometrie zu einem etablierten Teilgebiet der reinen Mathematik und der theoretischen Physik entwickelt. Die Entdeckung, dass gewisse Grenzfälle der Quantengravitation und Stringtheorie zu nichtkommutativer Geometrie führen, motivierte die Suche nach Physik jenseits des Standardmodells der Elementarteilchenphysik und der Einstein'schen allgemeinen Relativitätstheorie im Rahmen von nichtkommutativen Geometrien. Einen ergiebigen Ansatz zu letzteren Theorien, welcher Deformationsquantisierung (Sternprodukte) mit Methoden aus der Theorie der Quantengruppen kombiniert, wurde von der Gruppe um Julius Wess entwickelt. Die resultierende Gravitationstheorie ist nicht nur imstande nichtkommutative Effekte der Raumzeit zu beschreiben, sondern sie erfüllt ebenfalls ein generalisiertes allgemeines Kovarianzprinzip, welches durch eine deformierte Hopf Algebra von Diffeomorphismen beschrieben wird. Gegenstand des ersten Teils dieser Dissertation ist es Symmetriereduktion im Rahmen von nichtkommutativer Gravitation zu verstehen und damit exakte Lösungen der nichtkommutativen Einstein'schen Gleichungen zu konstruieren. Diese Untersuchungen sind von großer Bedeutung um den physikalischen Inhalt dieser Theorien herauszuarbeiten und den Kontakt zu Anwendungen, z.B. im Rahmen nichtkommutativer Kosmologie und Physik schwarzer Löcher, herzustellen. Wir verallgemeinern die übliche Methode der Symmetriereduktion, welche eine Standardtechnik im Auffinden von Lösungen der Einstein'schen Gleichungen ist, auf nichtkommutative Gravitation. Es wird gezeigt, dass unsere Methode zur nichtkommutativen Symmetriereduktion für ein gegebenes symmetrisches System zu bevorzugten Deformationen führt. Für Abelsche Drinfel'd Twists klassifizieren wir alle konsistenten Deformationen von räumlich flachen Friedmann-Robertson-Walker Kosmologien und des Schwarzschild'schen schwarzen Loches. Aufgrund der deformierten Symmetriestruktur dieser Modelle können wir viele Beispiele von exakten Lösungen der nichtkommutativen Einstein'schen Gleichungen finden, bei welchen das nichtkommutative Metrikfeld mit dem klassischen übereinstimmt. Im Fokus des zweiten Teils sind Quantenfeldtheorien auf nichtkommutativen gekrümmten Raumzeiten. Dazu entwickeln wir einen neuen Formalismus, welcher algebraische Methoden der Quantenfeldtheorie mit nichtkommutativer Differentialgeometrie verknüpft. Als Resultat unseres Ansatzes erhalten wir eine Observablenalgebra für skalare Quantenfeldtheorien auf einer großen Klasse von nichtkommutativen gekrümmten Raumzeiten. Es wird eine präzise Relation zwischen dieser Algebra und der Observablenalgebra der undeformierten Quantenfeldtheorie hergeleitet. Wir studieren ebenfalls explizite Beispiele von deformierten Wellenoperatoren und finden, dass im Gegensatz zu dem einfachsten Modell des Moyal-Weyl deformierten Minkowski-Raumes, im Allgemeinen schon die Propagation freier Felder durch die nichtkommutative Geometrie beeinflusst wird. Die Effekte von konvergenten Deformationen werden in einfachen Spezialfällen untersucht, und wir beobachten neue Aspekte in diesen Quantenfeldtheorien, welche sich in formalen Deformationen nicht zeigten. Zusätzlich zu der erwarteten Nichtlokalität finden wir, dass sich die Beziehung zwischen der deformierten und der undeformierten Quantenfeldtheorie nichttrivial verändert. Wir beweisen, dass dies zu einem verbesserten Verhalten der nichtkommutativen Theorie bei kurzen Abständen, d.h. im Ultravioletten, führt. Im dritten Teil dieser Arbeit entwickeln wir Elemente eines leistungsfähigeren, jedoch abstrakteren, mathematischen Ansatzes zur Beschreibung der nichtkommutativen Gravitation. Das Hauptaugenmerk liegt auf globalen Aspekten von Homomorphismen zwischen und Zusammenhängen auf nichtkommutativen Vektorbündeln, welche fundamentale Objekte in der mathematischen Beschreibung von nichtkommutativer Gravitation sind. Wir beweisen, dass sich alle Homomorphismen und Zusammenhänge der deformierten Theorie mittels eines Quantisierungsisomorphismus aus den undeformierten Homomorphismen und Zusammenhängen ableiten lassen. Es wird ebenfalls untersucht wie sich Homomorphismen und Zusammenhänge auf Tensorprodukte von Moduln induzieren lassen. Das Verständnis dieser Induktion erlaubt es uns die nichtkommutative Gravitationstheorie von Wess et al. um allgemeine Tensorfelder zu erweitern. Als eine nichttriviale Anwendung des neuen Formalismus erweitern wir unsere Studien zu exakten Lösungen der nichtkommutativen Einstein'schen Gleichungen auf allgemeinere Klassen von Deformationen
Over the past decades, noncommutative geometry has grown into an established field in pure mathematics and theoretical physics. The discovery that noncommutative geometry emerges as a limit of quantum gravity and string theory has provided strong motivations to search for physics beyond the standard model of particle physics and also beyond Einstein's theory of general relativity within the realm of noncommutative geometries. A very fruitful approach in the latter direction is due to Julius Wess and his group, which combines deformation quantization (star-products) with quantum group methods. The resulting gravity theory does not only include noncommutative effects of spacetime, but it is also invariant under a deformed Hopf algebra of diffeomorphisms, generalizing the principle of general covariance to the noncommutative setting. The purpose of the first part of this thesis is to understand symmetry reduction in noncommutative gravity, which then allows us to find exact solutions of the noncommutative Einstein equations. These are important investigations in order to capture the physical content of such theories and to make contact to applications in e.g. noncommutative cosmology and black hole physics. We propose an extension of the usual symmetry reduction procedure, which is frequently applied to the construction of exact solutions of Einstein's field equations, to noncommutative gravity and show that this leads to preferred choices of noncommutative deformations of a given symmetric system. We classify in the case of abelian Drinfel'd twists all consistent deformations of spatially flat Friedmann-Robertson-Walker cosmologies and of the Schwarzschild black hole. The deformed symmetry structure allows us to obtain exact solutions of the noncommutative Einstein equations in many of our models, for which the noncommutative metric field coincides with the classical one. In the second part we focus on quantum field theory on noncommutative curved spacetimes. We develop a new formalism by combining methods from the algebraic approach to quantum field theory with noncommutative differential geometry. The result is an algebra of observables for scalar quantum field theories on a large class of noncommutative curved spacetimes. A precise relation to the algebra of observables of the corresponding undeformed quantum field theory is established. We focus on explicit examples of deformed wave operators and find that there can be noncommutative corrections even on the level of free field theories, which is not the case in the simplest example of the Moyal-Weyl deformed Minkowski spacetime. The convergent deformation of simple toy-models is investigated and it is shown that these quantum field theories have many new features compared to formal deformation quantization. In addition to the expected nonlocality, we obtain that the relation between the deformed and the undeformed quantum field theory is affected in a nontrivial way, leading to an improved behavior of the noncommutative quantum field theory at short distances, i.e. in the ultraviolet. In the third part we develop elements of a more powerful, albeit more abstract, mathematical approach to noncommutative gravity. The goal is to better understand global aspects of homomorphisms between and connections on noncommutative vector bundles, which are fundamental objects in the mathematical description of noncommutative gravity. We prove that all homomorphisms and connections of the deformed theory can be obtained by applying a quantization isomorphism to undeformed homomorphisms and connections. The extension of homomorphisms and connections to tensor products of modules is clarified, and as a consequence we are able to add tensor fields of arbitrary type to the noncommutative gravity theory of Wess et al. As a nontrivial application of the new mathematical formalism we extend our studies of exact noncommutative gravity solutions to more general deformations

This is a diploma thesis on Buchdahl's equations for the description of massive particles of arbitrary spin s/2. On 4-dimensional, globally hyperbolic Lorentzian spacetime manifolds, existence of advanced and retarded Green's operators is proved, the Cauchy problem for Buchdahl's equations is solved globally and two possible constructions for quantizing Buchdahl fields using CAR algebras in the fashion of [Dimock 1982] are given.

In this work the extension of the criterion for local thermal equilibrium by Buchholz, Ojima and Roos to curved spacetime as introduced by Schlemmer is investigated. Several problems are identified and especially the instability under time evolution which was already observed by Schlemmer is inspected. An alternative approach to local thermal equilibrium in quantum field theories on curved spacetimes is presented and discussed. In the following the dynamic system of the linear field and matter perturbations in the generic model of inflation is studied in the view of ambiguity of quantisation. In the last part the compatibility of the temperature fluctuations of the cosmic microwave background radiation with local thermal equilibrium is investigated.:1. Introduction 5 2. Technical Background 10 2.1. The Free Scalar Field on a Globally Hyperbolic Spacetime . . . . . . 10 2.1.1. Construction of the Scalar Field . . . . . . . . . . . . . . . . . 10 2.1.2. Algebra of Wick Products . . . . . . . . . . . . . . . . . . . . 13 2.1.3. Local Covariance Principle . . . . . . . . . . . . . . . . . . . . 17 2.2. Local Thermal Equilibirum . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.1. Global Thermodynamic Equilibrium - KMS States . . . . . . 21 2.2.2. Local Thermal Observables . . . . . . . . . . . . . . . . . . . 24 2.2.3. LTE on Flat Spacetime . . . . . . . . . . . . . . . . . . . . . . 29 2.2.4. LTE in Cosmological Spacetimes . . . . . . . . . . . . . . . . 32 2.3. Linear Scalar Cosmological Perturbations . . . . . . . . . . . . . . . . 34 2.3.1. Robertson-Walker Cosmology . . . . . . . . . . . . . . . . . . 35 2.3.2. Mathematical Background . . . . . . . . . . . . . . . . . . . . 38 2.3.3. Technical Framework and Formulae . . . . . . . . . . . . . . . 40 2.3.4. The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . 46 2.3.5. The Sachs-Wolfe Effect for Adiabatic Perturbations . . . . . . 49 3. Towards a Refinement of the LTE Condition on Curved Spacetimes 54 3.1. Non-Minimal Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.1.1. Commutator Distribution . . . . . . . . . . . . . . . . . . . . 55 3.1.2. KMS Two-Point Function . . . . . . . . . . . . . . . . . . . . 57 3.1.3. Balanced Derivatives . . . . . . . . . . . . . . . . . . . . . . . 61 3.2. Conformally Static Spacetimes . . . . . . . . . . . . . . . . . . . . . . 65 3.2.1. Conformal KMS States . . . . . . . . . . . . . . . . . . . . . . 66 3.2.2. Extrinsic LTE in de Sitter Spacetime . . . . . . . . . . . . . . 71 3.3. Massive Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3.1. Properties of the Model . . . . . . . . . . . . . . . . . . . . . 78 3.3.2. Bogoliubov Transformation . . . . . . . . . . . . . . . . . . . 80 3.3.3. Thermal Observables . . . . . . . . . . . . . . . . . . . . . . . 82 3.4. Towards an Alternative Concept . . . . . . . . . . . . . . . . . . . . . 91 3.4.1. Problems and Open Questions Concerning LTE . . . . . . . . 92 3.4.2. Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . 94 3.4.3. Positivity Inequalities . . . . . . . . . . . . . . . . . . . . . . . 96 3.4.4. Macroobservable Interpretation . . . . . . . . . . . . . . . . . 100 3.5. An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4. Cosmological Perturbation Theory 105 4.1. Dynamics of Perturbations in Inflation . . . . . . . . . . . . . . . . . 106 4.1.1. CCR Quantisation is Ambiguous . . . . . . . . . . . . . . . . 106 4.1.2. Canonical Symplectic Form . . . . . . . . . . . . . . . . . . . 111 4.1.3. The Algebraic Point of View . . . . . . . . . . . . . . . . . . . 117 4.2. LTE States in Cosmology . . . . . . . . . . . . . . . . . . . . . . . . 120 4.2.1. The Link to Fluid Dynamics . . . . . . . . . . . . . . . . . . . 120 4.2.2. Incompatibility of LTE with Sachs-Wolfe Effect . . . . . . . . 125 5. Conclusion and Outlook 131 A. Technical proofs 136 A.1. Proof of Lemma 3.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 A.2. Proof of Lemma 3.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 A.3. Proof of Lemma 3.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A.4. Idea of Proof for Conjecture 3.4.3 . . . . . . . . . . . . . . . . . . . . 144 B. Introduction to Probability Theory 146 Bibliography 150 Correction of Lemma 3.1.2 155
In dieser Arbeit wird die von Schlemmer eingeführte Erweiterung des Kriteriums für lokales thermisches Gleichgewicht in Quantenfeldtheorien von Buchholz, Ojima und Roos auf gekrümmte Raumzeiten untersucht. Dabei werden verschiedene Probleme identifiziert und insbesondere die bereits von Schlemmer gezeigte Instabilität unter Zeitentwicklung untersucht. Es wird eine alternative Herangehensweise an lokales thermisches Gleichgewicht in Quantenfeldtheorien auf gekrümmten Raumzeiten vorgestellt und deren Probleme diskutiert. Es wird dann eine Untersuchung des dynamischen Systems der linearen Feld- und Metrikstörungen im üblichen Inflationsmodell mit Blick auf Uneindeutigkeit der Quantisierung durchgeführt. Zuletzt werden die Temperaturfluktuationen der kosmischen Hintergrundstrahlung auf Kompatibilität mit lokalem thermalem Gleichgewicht überprüft.:1. Introduction 5 2. Technical Background 10 2.1. The Free Scalar Field on a Globally Hyperbolic Spacetime . . . . . . 10 2.1.1. Construction of the Scalar Field . . . . . . . . . . . . . . . . . 10 2.1.2. Algebra of Wick Products . . . . . . . . . . . . . . . . . . . . 13 2.1.3. Local Covariance Principle . . . . . . . . . . . . . . . . . . . . 17 2.2. Local Thermal Equilibirum . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.1. Global Thermodynamic Equilibrium - KMS States . . . . . . 21 2.2.2. Local Thermal Observables . . . . . . . . . . . . . . . . . . . 24 2.2.3. LTE on Flat Spacetime . . . . . . . . . . . . . . . . . . . . . . 29 2.2.4. LTE in Cosmological Spacetimes . . . . . . . . . . . . . . . . 32 2.3. Linear Scalar Cosmological Perturbations . . . . . . . . . . . . . . . . 34 2.3.1. Robertson-Walker Cosmology . . . . . . . . . . . . . . . . . . 35 2.3.2. Mathematical Background . . . . . . . . . . . . . . . . . . . . 38 2.3.3. Technical Framework and Formulae . . . . . . . . . . . . . . . 40 2.3.4. The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . 46 2.3.5. The Sachs-Wolfe Effect for Adiabatic Perturbations . . . . . . 49 3. Towards a Refinement of the LTE Condition on Curved Spacetimes 54 3.1. Non-Minimal Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.1.1. Commutator Distribution . . . . . . . . . . . . . . . . . . . . 55 3.1.2. KMS Two-Point Function . . . . . . . . . . . . . . . . . . . . 57 3.1.3. Balanced Derivatives . . . . . . . . . . . . . . . . . . . . . . . 61 3.2. Conformally Static Spacetimes . . . . . . . . . . . . . . . . . . . . . . 65 3.2.1. Conformal KMS States . . . . . . . . . . . . . . . . . . . . . . 66 3.2.2. Extrinsic LTE in de Sitter Spacetime . . . . . . . . . . . . . . 71 3.3. Massive Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3.1. Properties of the Model . . . . . . . . . . . . . . . . . . . . . 78 3.3.2. Bogoliubov Transformation . . . . . . . . . . . . . . . . . . . 80 3.3.3. Thermal Observables . . . . . . . . . . . . . . . . . . . . . . . 82 3.4. Towards an Alternative Concept . . . . . . . . . . . . . . . . . . . . . 91 3.4.1. Problems and Open Questions Concerning LTE . . . . . . . . 92 3.4.2. Dynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . 94 3.4.3. Positivity Inequalities . . . . . . . . . . . . . . . . . . . . . . . 96 3.4.4. Macroobservable Interpretation . . . . . . . . . . . . . . . . . 100 3.5. An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4. Cosmological Perturbation Theory 105 4.1. Dynamics of Perturbations in Inflation . . . . . . . . . . . . . . . . . 106 4.1.1. CCR Quantisation is Ambiguous . . . . . . . . . . . . . . . . 106 4.1.2. Canonical Symplectic Form . . . . . . . . . . . . . . . . . . . 111 4.1.3. The Algebraic Point of View . . . . . . . . . . . . . . . . . . . 117 4.2. LTE States in Cosmology . . . . . . . . . . . . . . . . . . . . . . . . 120 4.2.1. The Link to Fluid Dynamics . . . . . . . . . . . . . . . . . . . 120 4.2.2. Incompatibility of LTE with Sachs-Wolfe Effect . . . . . . . . 125 5. Conclusion and Outlook 131 A. Technical proofs 136 A.1. Proof of Lemma 3.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 A.2. Proof of Lemma 3.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 A.3. Proof of Lemma 3.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 A.4. Idea of Proof for Conjecture 3.4.3 . . . . . . . . . . . . . . . . . . . . 144 B. Introduction to Probability Theory 146 Bibliography 150 Correction of Lemma 3.1.2 155