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In this thesis, I study entanglement in quantum field theory, using methods from operator algebra theory. More precisely, the thesis covers original research on the entanglement properties of the free fermionic field. After giving a pedagogical introduction to algebraic methods in quantum field theory, as well as the modular theory of Tomita-Takesaki and its relation to entanglement, I present a coherent framework that allows to solve Tomita-Takesaki theory for free fermionic fields in any number of dimensions. Subsequently, I use the derived machinery on the free massless fermion in two dimensions, where the formulae can be evaluated analytically. In particular, this entails the derivation of the resolvent of restrictions of the propagator, by means of solving singular integral equations. In this way, I derive the modular flow, modular Hamiltonian, modular correlation function, R\'enyi entanglement entropy, von-Neumann entanglement entropy, relative entanglement entropy, and mutual information for multi-component regions. All of this is done for the vacuum and thermal states, both on the infinite line and the circle with (anti-)periodic boundary conditions. Some of these results confirm previous results from the literature, such as the modular Hamiltonian and entanglement entropy in the vacuum state. The non-universal solutions for modular flow, modular correlation function, and R\'enyi entropy, however are new, in particular at finite temperature on the circle. Additionally, I show how boundaries of spacetime affect entanglement, as well as how one can define relative (entanglement) entropy and mutual information in theories with superselection rules. The findings regarding modular flow in multi-component regions can be summarised as follows: In the non-degenerate vacuum state, modular flow is multi-local, in the sense that it mixes the field operators along multiple trajectories, with one trajectory per component. This was already known from previous literature but is presented here in a more explicit form. In particular, I present the exact solution for the dynamics of the mixing process. What was not previously known at all, is that the modular flow of the thermal state on the circle is infinitely multi-local even for a connected region, in the sense that it mixes the field along an infinite, discretely distributed set, of trajectories. In the limit of high temperatures, all trajectories but the local one are pushed towards the boundary of the region, where their amplitude is damped exponentially, leaving only the local result. At low temperatures, on the other hand, these trajectories distribute densely in the region to either---for anti-periodic boundary conditions---cancel, or---for periodic boundary conditions---recover the non-local contribution due to the degenerate vacuum state. Proceeding to spacetimes with boundaries, I show explicitly how the presence of a boundary implies entanglement between the two components of the Dirac spinor. By computing the mutual information between the components inside a connected region, I show quantitatively that this entanglement decreases as an inverse square law at large distances from the boundary. In addition, full conformal symmetry (which is explicitly broken due to the presence of a boundary) is recovered from the exact solution for modular flow, far away from the boundary. As far as I know, all of these results are new, although related results were published by another group during the final stage of this thesis. Finally, regarding relative entanglement entropy in theories with superselection sectors, I introduce charge and flux resolved relative entropies, which are novel measures for the distinguishability of states, incorporating a charge operator, central to the algebra of observables. While charge resolved relative entropy has the interpretation of being a ``distinguishability per charge sector'', I argue that it is physically meaningless without placing a cutoff, due to infinite short-distance entanglement. Flux resolved relative entropy, on the other hand, overcomes this problem by inserting an Aharonov-Bohm flux and thus passing to a variant of the grand canonical ensemble. It takes a well defined value, even without putting a cutoff, and I compute its value between various states of the free massless fermion on the line, the charge operator being the total fermion number. / In dieser Dissertation untersuche ich quantenmechanische Verschränkung mittels Methoden aus Theorie der Operatoralgebren. Genauer gesagt stelle ich eigene Forschung über die Verschränkungseigenschaften des freien Fermions vor. Die Arbeit beginnt mit einer pädagogischen Einführung in algebraische Quantenfeldtheorie und stellt die modulare Theorie nach Tomita und Takesaki, sowie ihre Verbindung zu Verschränkung vor. Darauffolgend stelle ich einen vollständigen Satz an Werkzeugen vor, mit dem Tomita-Takesaki-Theorie für freie fermionische Felder in beliebiger Anzahl von Dimenionen gelöst werden kann. Diese Werkzeuge wende ich dann auf das freie, masselose Dirac-Fermion in zwei Dimensionen an, wo die hergeleiteten Formeln exakt gelöst werden können. Dies beinhaltet insbesondere die Herleitung der Resolvente von Einschränkungen des Propagators mittels der analytischen Lösung singulärer Integralgleichungen. Daraus ergeben sich schließlich der modulare Fluss, der modulare Hamiltonian, der modulare Korrelator, Rényi Verschränkungsentropien, von-Neumann Verschränkungsentropien, relative Verschränkungsentropie und Transinformation für nicht-zusammenhängende Verschränkungsgebiete. Dies alles wird im Vakuum und bei endlicher Temperatur ausgearbeitet, für ein Fermion sowohl auf der Geraden, als auch auf dem Kreis mit (anti-)periodischen Randbedingungen. Einige der Ergebnisse, besipielsweise der modulare Hamiltonian und von-Neumann Verschränkungsentropie, bestätigen Resultate aus bereits existierender Literatur. Die nicht-universellen Lösungen für den modularen Fluss, den modularen Korrelator und die Rényi Verschränkungsentropie dagegen sind neu, insbesondere für den Fall des thermischen Zustandes auf dem Kreis. Zusätzlich demonstriere ich den Einfluss von Rändern der Raumzeit auf Verschränkung und zeige, wie man relative Entropie und Transinformation in Theorien mit Superselektionsregeln definieren kann. Die Ergebnisse bezüglich modularen Flusses in nicht-zusammenhängenden Gebieten lassen sich wie folgt zusammenfassen: Im nicht-entarteten Vakuum ist der modulare Fluss multi-lokal, was bedeutet, dass er Feldoperatoren entlang mehrerer Trajektorien -- eine pro Zusammenhangskomponente der Region -- untereinander vermischt. Dies war bereits vorher bekannt, allerdings folgt es sich hier in expliziter Form aus exakten Lösungen. Ein vollkommen neues Ergebnis ist, dass der modulare Fluss des thermischen Zustandes auf dem Kreis sogar für zusammenhängende Regionen multi-lokal ist: Er mischt Feldoperatoren entlang unendlich vieler, diskret verteilter Trajektorien in der Verschränkungsregion. Im Hochtemperaturgrenzwert befinden sich alle diese Trajektorien, bis auf die lokale, nahe am Rand der Region, wo ihre Amplitude exponentiell gedämpft wird -- es bleibt nur die lokale Lösung. Bei tiefen Temperaturen dagegen sind die Trajektorien dicht in der Region verteilt, sodass sie entweder (bei antiperiodischen Randbedingungen) sich durch destruktive Interferenz gegenseitig aufheben oder (bei periodischen Randbedingungen) durch konstruktive Interferenz einen nicht-lokalen Term erzeugen, der auf das entartete Vakuum zurückgeführt werden kann. Im Falle von Raumzeiten mit Rand zeige ich explizit, wie der Rand Verschränkung zwischen beiden Komponenten des Dirac-Spinors impliziert. Mit zunehmdendem Abstand vom Rand nimmt diese Verschränkung invers quadratisch ab, wie ich quantitativ durch Berechnung der Transinformation zwischen den Komponenten in einem zusammenhängenden Gebiet zeige. Zusätzlich lässt sich die volle konforme Symmetrie der Theorie (die durch den Rand explizit gebrochen wird) aus der exakten Lösung für den modularen Fluss wiederherstellen, indem man den Grenzwert eines weit entfernten Randes betrachtet. Meines Wissens sind alle diese Resultate neu, allerdings wurden während der Fertigstellung dieser Dissertation verwandte Ergebnisse von einer anderen Arbeitsgruppe veröffentlicht. Die letzten Resultate in dieser Arbeit beziehen sich auf die Untersuchung relativer Entropie in Systemen mit Superselektionsregeln. Hier führe ich neue informationstheoretische Maße für die Unterscheidbarkeit von Zuständen ein: Die ladungs- und flussbezogenen relativen Entropien. Beide werden mittels eines Ladungsoperators aus dem Zentrum der Observablenalgebra definiert. Während die ladungsbezogene relative Entropie sich physikalisch als "Unterscheidbarkeit pro Ladungssektor" interpretieren lässt, argumentiere ich, dass sie nur innerhalb eines Regularisierungsschemas physikalisch bedeutsam ist, da die universell unendliche Verschränkung auf kurzen Längenskalen sonst zu Widersrpüchen führt. Flussbezogene relative Entropie dagegen hat dieses Problem nicht: Durch das Hinzufügen eines Aharonov-Bohm-Flusses betrachtet man hier eine lokale Variante des großkanonischen Ensembles, wodurch sie sich auch ohne Regularisierung definieren und berechnen lässt. Ich berechne ihren Wert zwischen verschiedenen Zuständen des freien masselosen Fermions auf der Geraden. Die erhaltene Ladung ist hierbei die Gesamtzahl der Fermionen im System.

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[pt] Neste trabalho apresentamos a teoria de Tomita-Takesaki para uma álgebra de Von Neumann M com vetor cíclico separante u. Usamos o caso finito dimensional para motivar a teoria, depois prosseguimos para os argumentos analíticos geralmente empregados para provar o caso infinito dimensional. Também calculamos os operadores modulares da teoria para três exemplos padrão. Na mecânica estatística quântica, os estados de equilíbrio termodinâmico de um sistema físico com um número de partículas e volume finito são modelados pelos estados de Gibbs, enquanto no caso infinito eles são modelados pelos chamados estados KMS através da abordagem de álgebra de operadores. Mostramos como a teoria de Tomita-Takesaki fornece estados KMS naturais e a unicidade da evolução temporal do sistema físico para esses estados. / [en] In this work we present the Tomita-Takesaki theory for a Von Neumann algebra M with cyclic separating vector u. We use the finite-dimensional case to motivate the theory, and then proceed to the analytical arguments usually employed to prove the infinite dimensional case. Also, we calculate the modular operators from the theory for three standard examples. In quantum statistical mechanics, the thermodynamic equilibrium states of a physical system with finitely many particles and finite volume are modeled by Gibbs states, while in the infinite case they are modeled by the so called KMS states through the operator-algebraic approach.We show how Tomita-Takesaki theory provides natural KMS states and the uniqueness of the time evolution of the physical system for those states.

Software engineers need to understand the structure of the programs they construct. This task is made difficult by the intangible nature of software, and its complexity, size and changeability. Static analysis tools can help by extracting information from source code and conveying it to software engineers. However, the information provided by typical tools is limited, and some potentially rich veins of information - particularly metrics and visualisations - are under-utilised because developers cannot easily acquire or make use of the data. This thesis documents new tools and techniques for static analysis of software. It addresses the problem of generating parsers directly from standard grammars, thus avoiding the com-mon practice of customising grammars to comply with the limitations of a given parsing al-gorithm, typically LALR(1). This is achieved by a new parser generator that applies a range of bottom-up parsing algorithms to produce a hybrid parsing automaton. Consequently, we can generate more powerful deterministic parsers - up to and including LR(k) - without incurring the combinatorial explosion that makes canonical LR(k) parsers impractical. The range of practical parsers is further extended to include GLR, which was originally developed for natural language parsing but is shown here to also have advantages for static analysis of programming languages. This emphasis on conformance to standard grammars im-proves the rigour of static analysis tools and allows clearer definition and communication of derived information, such as metrics. Beneath the syntactic structure of software (exposed by parsing) lies the deeper semantic structure of declarations, scopes, classes, methods, inheritance, invocations, and so on. In this work, we present a new tool that performs semantic analysis on parse trees to produce a comprehensive semantic model suitable for processing by other static analysis tools. An XML pipeline approach is used to expose the syntactic and semantic models of the software and to derive metrics and visualisations. The approach is demonstrated producing several types of metrics and visualisations for real software, and the value of static analysis for informing software engineering decisions is shown.

Made available in DSpace on 2016-04-27T14:32:20Z (GMT). No. of bitstreams: 1 Helena Maria dos Santos Felicio.pdf: 3499112 bytes, checksum: c42ad53fe4a8e674128a02923fa4f38c (MD5) Previous issue date: 2008-10-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This study analyzes the construction of a curriculum with the potential for emancipation in educational quotidian experience. It is based on the analysis of educative practices developed by educators in the initial years of primary education who work in an educational context that attends students in a community that is in the process of being taken out of a shantytown in the municipality of São José dos Campos (SP). The educational context investigated comprises a partnership between a regular public school and a formal educational institution in the community, in order to develop the educational process amongst the children and teenagers in the integral education provided. The investigation was developed from a qualitative standpoint, including bibliographical research, participatory observation, documental analysis and interviews with the educators from the two institutions comprising the context analyzed. We observed that the need to consider the social context the institutions are installed in led to the proposal for curricular redesign in order to construct and conceive of an extended curriculum that integrates in itself with the formal and informal dimension of education, as articulating dimensions. Based on this analysis it was possible to check the possibility of building pedagogical practices with the potential for emancipation to the extent that the institutions and educators are capable of deconstructing their conceptions of what its is to teach and learn, generally marked by traces of transmission, to develop practices based on the principle of construction, interaction, exchange, respect, and creativity with the students themselves. We conclude in this study that a curriculum with the potential for emancipation could be built when the educational institutions are encouraged to take a critical look at their quotidian experience, being able to propose action that is the fruit of interaction with the community and culture they are in / O presente trabalho objetiva analisar a construção de um currículo com perspectivas emancipatórias, no cotidiano educacional, a partir da análise das práticas educativas, desenvolvidas pelos educadores dos anos iniciais do Ensino Fundamental, que trabalham em um contexto educacional que atende educandos de uma comunidade instituída a partir de um processo de desfavelização, no município de São José dos Campos (SP). O contexto educacional investigado é constituído por uma parceria entre uma escola de ensino regular pública e uma instituição de educação não formal, instalados na comunidade com a intenção de desenvolver o processo educacional das crianças e dos adolescentes, na dimensão de uma educação integral. A investigação desenvolveu-se em uma perspectiva qualitativa, compreendendo pesquisa bibliográfica, observação participante, análise documental e entrevistas com os educadores das duas instituições que compreendem o contexto analisado. Observamos que a partir da constatação da necessidade de considerar o contexto social em que as instituições estão instaladas, foi proposto um redimensionamento curricular, na direção de uma construção / concepção ampliada de currículo, que integrasse em si a dimensão formal e não formal da educação, como dimensões articuladoras. A partir da analise foi possível verificar a possibilidade da construção de práticas pedagógicas com perspectivas emancipatórias na medida em que instituições e educadores foram capazes de desconstruir suas concepções do que é ensinar e aprender, em geral marcadas pelos traços da transmissão, para desenvolver práticas fundamentadas no princípio da construção, da interação, da troca, do respeito, da criatividade com os próprios educandos. Concluímos, com este estudo, que um currículo com perspectivas emancipatórias pode ser construído quando as instituições educacionais se encorajarem a olhar, criticamente, para o seu cotidiano, sendo capazes de propor ações que sejam frutos de uma interação com a comunidade e a cultura onde as mesmas estão instituídas

This PhD thesis deals with the use of structural methods of objects identification for industrial robots operation. First, the present state of knowledge in the field is described, i.e. the whole process of objects recognition with the aid of common methods of the syntactic analysis. The main disadvantage of these methods is that is impossible to recognize objects whose digitalized image is corrupted in some ways (due to excessive noise or image disturbances), objects are therefore deformed. Further, other methods for the recognition of deformed objects are described. These methods use structural description of objects for object recognition, i.e. methods which determine the distance between attribute descriptions of images. The core part of this PhD thesis begins in Chapter 5, where deformation grammars, capable of description of all possible object deformations, are described. The only complication in the analysis is the ambiguity of the deformation grammar, which lowers the effectiveness of the analysis. Further, PhD thesis deals with the selection and modification of a proper parser, which is able to analyze a deformation grammar effectively. Three parsers are described: the modified Earley parser, the modified Tomita parser and the modified hybrid LRE(k) parser. As for the modified Earley’s parser, ways of its effective implementation are described. One of the necessary parts of the object recognition is providing the invariances, which this PhD thesis covers in detail, too. Finally, the results of described algorithms are mentioned (successfulness and speed of deformed objects recognition) and suggested testing environment and implemented algorithms are described. In conclusion, all determined possibilities of deformation grammars and their results are summarized.