Tesis sobre el tema "Spinory"

This thesis reviews the method of Mafra, Schlotterer, and Stieberger (2011) for computing the full colour ordered N-point open superstring amplitude using the Pure Spinor formalism. We introduce relevant elements of super Yang-Mills theory and examine the basics of the Pure Spinor formalism, with a focus on tools for amplitude computation. We then define a series of objects with increasingly useful BRST transformation properties, which greatly simplify the calculations, and show how these properties can be determined using a diagrammatic method. Finally, we use the explicit four- and five-point amplitude computations as stepping stones to compute the general N-point amplitude, which factors into a set of kinematic integrals multiplying SYM subamplitudes.

Orientador: Julio Marny Hoff da Silva
Resumo: Espinores exóticos surgem quando a topologia da variedade $M$ tomada como sendo o espaço-tempo é suposta ser não-trivial, no sentindo que seu grupo fundamental é não-trivial: $\pi_1(M) \neq 0$. Assim, um novo termo exótico $\partial_\mu \theta$ surge na equação dinâmica destes espinores, e novas propriedades se apresentam. A não-trivialidade de $\pi_1(M)$ pode ser diretamente ligada a própria existência de buracos negros. Assim, estudamos, nesta tese, relações entre estruturas espinoriais exóticas e a taxa de emissão de radiação Hawking por buracos negros assintoticamente \textit{flat} em Relatividade Geral, encontrando equações diferenciais para o termo exótico, o que dá a possibilidade de inferir uma forma explícita para $\theta$. Também, tratamos aqui dos chamados espinores RIM, que são espinores que respeitam uma equação dinâmica não-linear chamada de equação não-linear de Heisenberg. Apresentamos dois lemas relativos a estes espinores: um deles encontrando restrições para ocorrer a decomposição de espinores de Dirac em termos de espinores RIM, e outro que nega a existência de espinores RIM exóticos, ou seja, relaciona a existência de espinores RIM a própria estrutura topológica do espaço-tempo. Ainda, encontramos um método de classificarmos os espinores RIM nas classes de Lounesto. Por fim, apresentamos, na forma de dois teoremas, maneiras de deformar homotopicamente tais espinores no que chamamos de \textit{spinor-plane}.
Abstract: Exotic spinors emerge when the topology associatd to the manifold $M$, which is token as being the spacetime, is suppose to be non-trivial, in the sense that its fundamental group is non-trivial: $\pi_1(M) \neq 0$. Thus, a new exotic term $\partial_\mu \theta$ rises from the dynamical equation related to these spinors, and new properties are in order. The non-triviality of $\pi_1(M)$ may be directly linked to the very existence of black holes. In this vein, we study some relations between exotic spinorial structures and the Hawking radiation emission rate by asymptotically flat black holes solutions of General Relativity, finding an equation from which an explicity form for the exotic term could be inferred. Moreover, we work on the so-called RIM spinors, which are spinor fields satisfying a non-linear dynamical equation known as Heiseing non-linear equation. We present two \textit{lemmata} related to these spinors: one of them gives us restrictions for the decompostion of Dirac fields in terms of RIM spinors to occur, while the other deny the existence of exotic RIM spinors, i.e., it relates the very existence of RIM spinors to the spacetime topological structure. Besides, we develop a classifying method for RIM spinors into the Lounesto classes. Finally, we present, in the form of two theorems, ways to homotopically deform such spinors in what we call the spinor-plane.
Doutor

This thesis is based on five articles, four of which have been published in the Journal of Mathematical Physics, Physical Review D, Modern Physics Letters A and Journal of High Energy Physics. The fifth has been submitted to Mathematika. In these works we study several distinct problems within the broad subject area of Mathematical Physics. The common feature is that all these works deal with rotations of one form or another. In particular, we show an equivalence between the massless and massive Dirac equations and models based on the concept of rotating material points. We also solve an open problem in Einstein-Cartan theory, namely, we find a natural matter source for a non-trivial spin angular momentum tensor. Finally, we construct a complete class of non-standard (non-local) spinor field theories and examine their possible applications in Cosmology.

This thesis is concerned with the theory of spinors, embeddings and everywhere invariance with applications to general relativity. The approach is entirely geometric with particular emphasis on the use of natural structures. A clear indication of the interaction between the above topics is given; this Interaction then sheds light on various aspects of general relativity theory. The main ideas discussed are:- (i) Spinors, conformal structure and the spacetime projective null bundle framework. (ii) Spaces of embeddings. (ill) Embeddings and spin structure. (iv) Null embeddings and the null limit (a technique for obtaining differential equations on null hypersurfaces). (v) Quasi-local momentum. (vi) The space of metrics, natural group actions and generalized conformal structure. (vii) Everywhere invariance and the invariance equation as a method for obtaining spacetime symmetries. Three appendices are also provided:- These give comprehensive summaries of the theories of principal bundles, conformal structure and asymptotic simplicity.

Ultracold atomic gases have established themselves as quantum systems, which are clean and offer a high degree of control over crucial parameters. They are well isolated from their environment and thus offer the possibility to study coherent many-body dynamics. In this thesis, we address the dynamics of ultracold Fermions with large spin. Fermionic spinor gases differ from the typical situation in condensed matter physics, due to both the presence of the trap and the possibility of having fermions with large (>1/2) spin. Compared to the spin-1/2 case, large spin fermions must have one of two possible new properties. Either they obey an enhanced SU(N) symmetry, or they feature spin-changing collisions and a quadratic Zeeman shift. Here, we address the latter case. In the weakly interacting scenario, there are three different regimes. For very weak interactions, the system is in the collisionless regime and interactions can be taken into account on a mean-field level. For stronger interactions, collisions ensure local equilibrium and the system is described by hydrodynamic equations. For the intermediate regime however, there is no simple description. Moreover, the scattering cross-section for spin-changing and spin-conserving collisions can be different for large-spin fermions and we find a situation, where the system is hydrodynamic with respect to one process but not the other. In this thesis, a semi-classical Boltzmann equation with full spin coherence is developed, which allows to interpolate between the collisionless and hydrodynamic regime in the presence of the trap and for large spins. This approach goes beyond mean-field theory and treats the single-particle dynamics as an open system coupled to an environment given by all other particles. We find good agreement with experiments performed in the group of Klaus Sengstock at Hamburg University, using ultracold Potassium-40. We begin by investigating the effect of the harmonic trap on a collisionless system. We find a dynamical mechanism for spin-segregation, the mean-field driven creation of two domains of opposite magnetization in phase-space. The effect finds a transparent explanation when introducing the concept of dynamically induced long-range interactions, occurring when the fast phase-space rotation induced by a strong parabolic trap effectively smears out the contact interactions. Further results in this thesis have been achieved in collaboration with the experimental group in Hamburg. In the first project, we study the collective excitations of a trapped four-component Fermi gas. Long wavelength spin waves are excited by using a magnetic field gradient to wind up a spin spiral. During the subsequent dynamics, the spin components oscillate in the trap, while the total density remains constant. The dynamics can be understood quantitatively by disentangling it into dipolar, nematic and octupolar configurations. In a further experiment with spin-9/2 fermions, it was found that spin-changing interactions can lead to collective and coherent oscillations of the spin state of the whole Fermi sea with long lifetimes. It is found theoretically, that these giant oscillations are protected from spatial dephasing by dynamically induced long-range interactions. We identify the suppression of such oscillations in the high-density regime as the consequence of incoherent non-forward scattering. In the last project, we study collision processes in ultracold Potassium in greater detail. We find that they can be arranged in 3 categories: Spin-changing vs. spin-conserving collisions, processes depending on density vs. processes depending on density gradients and forward vs. lateral scattering. With this categorization, as well as the exact dependence of each process on scattering lengths and momenta, we can explain and simulate not only the coherent mean-field driven oscillations, but also relaxation effects that appear to be incoherent on the single-particle level
Gases atómicos ultrafríos han establecido como sistemas cuánticos limpias que ofrecen un alto grado de control sobre parámetros cruciales. Están bien aisladas de su entorno y por eso ofrecen la posibilidad de estudiar la dinámica coherente de muchos cuerpos. En esta tesis, estudiamos la dinámica de fermiones ultrafríos con spin largo. Gases espinoriales fermiónicos difieren de la situación típica en la física de materia condensada por la presencia de la trampa y la posibilidad de tener un spin largo (> 1/2). En comparación con el caso de spin 1/2, fermiones de espín largo deben tener una de dos posibles propiedades nuevas. Obedecen a una simetría ampliada SU(N), o muestran colisiones spin-cambiante y un efecto Zeeman cuadrático. Aqui tratamos el segundo caso. En el escenario de interacciónes débiles, hay tres regímenes diferentes. Para interacciones muy débiles, el sistema está en el régimen sin colisiones e interacciones se puede describir en un nivel de campo medio. Para interacciones fuertes, las colisiones garantizan el equilibrio local y el sistema es descrito por ecuaciones hidrodinámicas. Para el régimen intermedio, no hay una descripción sencilla. Ademas, la sección transversa de dispersión para colisiones spin-cambiantes y de spin-conservación puede ser diferente para fermiones de espín largo. Encontramos una situación, donde el sistema es hidrodinámico con respecto a un proceso, pero no a la otra. En esta tesis desarrollamos una ecuación de Boltzmann semi-clásica, que permite interpolar el régimen intermedio, en presencia de la trampa y para espín largo. Este enfoque trata la dinámica de un cuerpo como un sistema abierto, acoplado a un entorno determinado por todas las atomos demás. Encontramos un buen acuerdo con experimentos realizados en el grupo de Klaus Sengstock en la Universidad de Hamburgo, hechos con potasio-40 ultrafrío. Comenzamos investigando el efecto de la trampa armónica en un sistema sin colisiones. Encontramos un mecanismo dinámico par la segregación de spin, la creación de dos dominios de magnetización opuesta en el espacio fásico, impulsada por el campo medio. Encontramos una explicación transparente de este efecto con la introducción del concepto de interacciones de largo alcance inducidos dinámicamente, que se forma cuando una fuerte trampa parabólica desenfoque eficazmente las interacciones de contacto. Otros resultados de esta tesis han sido realizados en colaboración con el grupo experimental en Hamburgo. En el primer proyecto, estudiamos las excitaciones colectivas de un gas de Fermi atrapada, con cuatro componentes de spin. Ondas de spin con larga longitud de onda se excitan mediante un gradiente de campo magnético. Durante la dinámica siguiente, los componentes de spin oscilan en la trampa, mientras que la densidad total permanece constante. Podemos entender esta dinámica cuantitativamente desligandola en configuraciones dipolares, nemáticos y octupolares de espín. En un experimento siguiente con fermiones de spin 9/2, se encontró que las interacciones spin-cambiando pueden activar oscilaciones colectivas y coherentes del estado de spin de todo el mar de Fermi con duración larga. Descubrimos teóricamente, que estas oscilaciones gigantes están protegidos de desfase espacial por las interacciones de largo alcance inducidos dinámicamente. Identificamos la supresión de tales oscilaciones en el régimen de alta densidad como la consecuencia de la dispersión incoherente lateral. En el último proyecto, estudiamos los procesos de colisión en potasio ultrafrío en mas detalle. Podemos organizarlos en tres categorías: Colisiones spin-cambiante vs. spin-conservación, procesos dependiente de la densidad vs. gradientes de densidad y colisiones hacia adelante vs. laterales. Con esta clasificación y la dependencia en la longitud de dispersión y momentos, podemos explicar y simular no sólo las oscilaciones coherentes impulsados por el campo medio, sino también efectos de relajación

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Nesta tese, mostramos o cálculo da função de partição dos espinores puros. O cálculo será executado de dois modos diferentes usando o método de fantasma-para-fantasma (até o décimo segundo nível massivo) e usando o método do ponto fixo (até o quinto nível massivo). Após incluir a contribuição das variáveis do setor da matéria ('qui POT. m, 'teta POT. alfa', 'rô IND. alfa'), nós derivamos o espectro massivo da supercorda aberta. Embora os espinores puros sejam variáveis bosônicas, a função de partição dos espinores puros contém estados fermiônicos os quais começam aparecer a partir do segundo nível massivo. Estes estados fermiônicos vêm de funções que não são bem definidas globalmente no espaço dos espinores puros, e estão relacionados ao fantasma b no formalismo de spinores puros para a supercorda
In this thesis, we have calculated the partition function of pure spinors. The computation is performed by using two different methods, namely ghosts-for-ghosts (up to the twelth mass-level) and fixed point (up to the fifth mass-level) techniques. After adding the contribution from the ('qui POT. m, 'teta POT. alfa', 'rô IND. alfa') matter variables, we reproduce the massive open superstring spectrum. Even though pure spinor variables are bosonic, the pure spinor partition function contains fermionic states which first appear at the second mass-level. These fermionic states come from functions which are not globally defined in pure spinor space, and are related to the b ghost in the pure spinor formalism for the superstring

Orientador: Nathan Jacob Berkovits
Banca: Victor de Oliveira Rivelles
Banca: Nelson Ricardo de Freitas Braga
Banca: Yuri Aisaka
Banca: Dáfni Fernanda Zenedin Marchioro
Resumo: Nesta tese, mostramos o cálculo da função de partição dos espinores puros. O cálculo será executado de dois modos diferentes usando o método de fantasma-para-fantasma (até o décimo segundo nível massivo) e usando o método do ponto fixo (até o quinto nível massivo). Após incluir a contribuição das variáveis do setor da matéria ('qui POT. m, 'teta POT. alfa', 'rô IND. alfa'), nós derivamos o espectro massivo da supercorda aberta. Embora os espinores puros sejam variáveis bosônicas, a função de partição dos espinores puros contém estados fermiônicos os quais começam aparecer a partir do segundo nível massivo. Estes estados fermiônicos vêm de funções que não são bem definidas globalmente no espaço dos espinores puros, e estão relacionados ao fantasma b no formalismo de spinores puros para a supercorda
Abstract: In this thesis, we have calculated the partition function of pure spinors. The computation is performed by using two different methods, namely ghosts-for-ghosts (up to the twelth mass-level) and fixed point (up to the fifth mass-level) techniques. After adding the contribution from the ('qui POT. m, 'teta POT. alfa', 'rô IND. alfa') matter variables, we reproduce the massive open superstring spectrum. Even though pure spinor variables are bosonic, the pure spinor partition function contains fermionic states which first appear at the second mass-level. These fermionic states come from functions which are not globally defined in pure spinor space, and are related to the b ghost in the pure spinor formalism for the superstring
Doutor

The purpose of this thesis is to give a fully gauge-natural formulation of gravitation theory, which turns out to be essential for a correct geometrical formulation of the coupling between gravity and spinor fields. In Chapter 1 we recall the necessary background material from differential geometry and introduce the fundamental notion of a gauge-natural bundle. Chapter 2 is devoted to expounding the general theory of Lie derivatives, its specialization to the gauge-natural context and, in particular, to spinor structures. In Chapter 3 we describe the geometric approach to the calculus of variations and the theory of conserved quantities. Then, in Chapter 4 we give our gauge-natural formulation of the Einstein (-Cartan) -Dirac theory and, on applying the formalism developed in the previous chapter, derive a new gravitational superpotential, which exhibits an unexpected freedom of a functorial origin. Finally, in Chapter 5 we complete the picture by presenting the Hamiltonian counterpart of the Lagrangian formalism developed in Chapter 3, and proposing a multisymplectic derivation of bi-instantaneous dynamics. Appendices supplement the core of the thesis by providing the reader with useful background information, which would nevertheless disrupt the main development of the work. Appendix A is devoted to a concise account of categories and functors. In Appendix B we review some fundamental notions on vector fields and flows, and prove a simple, but useful, proposition. In Appendix C we collect the basic results that we need on Lie groups, Lie algebras and Lie group actions on manifolds. Finally, Appendix D consists of a short introduction to Clifford algebras and spinors.

This work reports on the construction of a new-generation system capable to create single-mode spinor Bose-Einstein condensates of 87Rb, and non-destructively probe them using optical Faraday rotation. This system brings together many of the stateof-the-art technologies in ultra-cold physics in a minimalist design which was possible due to the prolific advances in the field respect to the pioneering experiments (Cornell's, Ketterle's, and Chapman's groups). There is rich phenomena that can be potentially studied in this system from the study of predicted novel quantum phases and topologies to entanglement and spin squeezing which are useful for quantum information and interferometry. The potential of this system make it suitable to answer fundamental questions on the phase transition to a condensed and ferromagnetic state. In particular, this work describes theoretically and experimentally, the atomic spin coherence, which is relevant for applications like coherent sensing of magnetic fields. In this direction, our findings demonstrate the characteristics of our system make it a sensor with the best predicted energy resolution per unit bandwidth (~10^-2 h) among all the different technologies applied to magnetometry. The thesis is structured as follows: Part I is dedicated to the mathematical description of the relevant interactions. First, the interaction of optical polarization and atomic spin polarization is reviewed, with special attention to ac-Stark shifts, which are used to generate a conservative trapping potential and Faraday rotation effects that are used for non-destructive spin detection. Second, the interaction of the atoms with a magnetic field is presented. And finally, the mean-filed theory of spinor Bose-Einstein condensation is summarized. The dynamics of a spin-1system in this picture is described by a three-component Gross-Pitaevskii equation. Part II contains three chapters describing the implemented technologies and techniques used in the experiment to create and characterize a spinor condensate. The first chapter describes the ultrahigh vacuum, magnetic fields, lasers, spectroscopy and imaging needed to create a magneto optical trap (MOT) and transfer those atoms into an optical dipole trap (ODT). We implemented a non-standard loading technique based on the semicompensation of the strong differential lightshift induced by the ODT which profits from the effective dark-MOT created at the trap position. In the second chapter we detail, theoretically and experimentally, the all-optical evaporation process employed to achieve condensation in less than five second after the loading. In the final chapter the spin manipulation and read-out techniques are presented. Because there is no observable associated to the spin angle, we exploit the Faraday rotation effect and Stern-Gerlach imaging in order to retrieve information about the spin dynamics. Finally in Part III, we consider the potential of a spinor BEC as a magnetic sensor. The measurement of fundamental properties defining the sensitivity of the sensor are detailed. Those properties are the volume, the temporal coherence and the readout noise. We present a model of the magnetic field environment and its repercussion on the noise of the magnetometer. In the last chapter we present our perspectives to the possible applications of our system.
Este trabajo compila los detalles experimentales de un aparato de "nueva generación" capaz de crear condensados Espinoriales de 87Rb en un único dominio magnético, y de obtener información del estado de espín en una forma no destructiva explotando el efecto Faraday. Este aparato conjunta algunas de las tecnologías de punta aplicadas a física de gases ultrafrios en un diseño minimalista. Estas tecnologías se han podido desarrollar debido a los prolíficos avances en el campo, respecto a los experimentos pioneros en los grupos de Cornell, Ketterle y Chapman. Una rica cantidad de fenómenos pueden ser estudiados en este sistema, desde el estudio de novedosas fases y topologías cuánticas hasta la aplicación de entrelazamiento y estados comprimidos relevantes en información cuántica e interferometría. Su potencial lo hace un buen candidato para responder preguntas acerca de la naturaleza de las transiciones ferromagnética y de condensación. En particular, este trabajo describe teorética y experimentalmente la coherencia del estado de espín, el cual, es relevante en aplicaciones como la medición coherente de campos magnéticos. En este sentido, nuestros resultados demuestran que las características de nuestro condensado espinorial lo hacen el sensor con la mejor resolución en energía por unidad de ancho de banda (~10^-2 h ), de entre todas las tecnologías aplicadas a magnetometría. Esta tesis se estructura de la siguiente manera: Part I está dedicada a la descripción matemática de las interacciones relevantes. Primero la interacción entre la luz y el espín atómico es revisada, con especial énfasis en el desplazamiento ac-Stark, que es explotado para generar un potencial conservador, así como en las medidas no destructivas del espín via efecto Faraday. En segundo lugar, estudiamos la dinámica de espín bajo la interacción Zeeman entre los átomos y un campo magnético que varía en el tiempo. Finalmente es brevemente tratada la teoría de campo medio (mean-field theory) que describe los condensados espinoriales en la forma de una ecuación de Gross-Pitaevskii multicomponente. Part II contiene tres capítulos que detallan la tecnologías y técnicas usadas en el experimento para crear y caracterizar el condensado. El primer capítulo describe el ultra-alto vacío, los campos magnéticos, láseres, espectroscopía e imaging usados para crear una trampa magneto-óptica (MOT), y para transferir esos átomos en una trampa dipolar óptica (ODT). Nosotros implementamos una técnica poco estandard para cargar la ODT, la cual se basa en compensar medianamente el excesivo lightshift diferencial inducido por nuestra ODT. Esta técnica nos ayuda a crear una dark-MOT efectiva con la que podemos conseguir altas densidades de átoms en la ODT. En el segundo capítulo detallamos la evaporación que es "all-optical", con la que podemos conseguir un condensado en menos de 5 s de evaporación. En el capítulo final describimos las técnicas para crear arbitrarios estados de espín y cómo detectarlos. Para esto último explotamos el efecto Faraday y capturamos imágenes Stern-Gerlach. Finalmente en Part III, estudiamos las propiedades de coherencia, tiempo de vida y extensión espacial del condensado. Detallamos el sistema especialmente en el contexto de sensores magnéticos. Además, presentamos un modelo del campo magnético ambiental y sus repercusiones en el ruido del magnetómetro. En el último capítulo hablamos de algunas de las alternativas aplicaciones de nuestro sistema.

Questo elaborato ha lo scopo di ricavare formalmente gli spinori di Dirac e di Majorana. Per far ciò vengono analizzati i gruppi di Lorentz e di Poincaré. A partire dalle rappresentazioni irriducibili del gruppo ristretto ortocrono di Lorentz troviamo le rappresentazioni spinoriali sinistrorse e destrorse, da cui quella di Dirac. Viene descritto allora il gruppo SL(2;C) ed il suo omomorfismo con L↑+. Attraverso lo studio delle sue rappresentazioni equivalenti, infatti, sono esplicitate le leggi di trasformazione degli spinori. Si costruisce quindi lo spinore di Dirac e, richiedendone l’invarianza sotto l’azione della matrice di coniugazione di carica, si ottiene lo spinore di Majorana.

In this thesis, we have investigated several fluctuation-driven phenomena in ultracold spinor Bose gases. In Bose-Einstein condensates of hyperfine spin-two (F=2) atoms, it is shown that zero-point quantum fluctuations completely lift the accidental continuous degeneracy in quantum spin nematic phases predicted by mean field analysis, and these fluctuations select out two distinct spin nematic states with higher symmetries. It is further shown that fluctuations can drive a novel type of coherent spin dynamics which is very sensitive to the variation of quantum fluctuations controlled by magnetic fields or potential depths in optical lattices. These results have indicated fundamental limitations of precision measurements based on mean field theories. In addition, fluctuation-driven coherent spin dynamics studied here is a promising tool to probe correlated fluctuations in many body systems. In another system -- a two-dimension superfluid of spin-one (F=1) Na²³ atoms -- we have investigated spin correlations associated with half quantum vortices. It is shown that when cold atoms become superfluid below a critical temperature a unique nonlocal topological order emerges simultaneously due to fluctuations in low dimensional systems. Our simulation have indicated that there exists a nonlocal softened pi-spin disclination structure associated with a half-quantum vortex although spin correlations are short ranged. We have also estimated fluctuation-dependent critical frequencies for half-quantum vortex nucleation in rotating optical traps. These results indicate that the strongly fluctuating ultracold spinor system is a promising candidate for studying topological orders that are the focus of many other fields.

This work is concerned with two examples of the interactions between differential geometry and analysis, both related to spinors. The first example is the Dirac operator on conformal spin manifolds with boundary. I aim to demonstrate that the analysis of the Dirac operator is a natural generalisation of complex analysis to manifolds of arbitrary dimension, by providing, as far as possible, elementary proofs of the main analytical results about the boundary behaviour of solutions to the Dirac equation. I emphasise throughout the conformal invariance of the theory, and also the usefulness of the Clifford algebra formalism. The main result is that there is a conformally invariant Hilbert space of boundary values of harmonic spinors, and that the pointwise evaluation map defines a conformally invariant metric on the interior. Along the way, many results from complex analysis are generalised to arbitrary (Riemannian or conformal spin) manifolds, such as the Cauchy integral formula, the Plemelj formula, and the L2-boundedness of the Hilbert transform. The second example concerns the geometry of the twistor operator and the analysis of differential operators arising in twistor theory. I study the differential equations on a complex quadric induced by holomorphic vector bundles on its twistor space. In 4 dimensions, there is already a beautiful example of such a relationship, the Ward correspondence between holomorphic vector bundles trivial on twistor lines, and self-dual connections. There are many generalisations of twistor theory to higher dimensions, but it is not clear how best to generalise the Ward correspondence. Consequently, I focus on 6 dimensional geometry, and one possible generalisation proposed by Atiyah and Hitchin, and investigated by Manin and Minh. I study a number of differential equations produced by this 6 dimensional twistor construction, with a view to reconstructing the holomorphic vector bundle on the twistor space from these equations. While this aim has not been realised, some progress has been made.

Qu’est-ce que la liberté chez Spinoza ? Comment la démontre-t-il dans l’Ethique ? Le point de départ de la réflexion est d’examiner le rapport entre l’esprit et le corps chez Spinoza et de démontrer comment cette forme de relation rend possible le projet de libération de l’homme établi par Spinoza dans l’Ethique. Or, le projet de libération de l’homme chez Spinoza commence par une libération de l’homme de sa vie affective. Ainsi, il faut examiner le fondement ontologique des affects pour les expliquer dans le même ordre causal qui détermine le corps et l’esprit. Mais, Spinoza n’envisage pas seulement de répondre à la question du mystère existentiel qui concerne la réalisation de notre liberté dans l’existence, mais aussi de fonder une liberté d’ordre métaphysique, pour établir son fondement ontologique de la vie affective humaine. L’analyse consistera à démontrer que la conception du conatus est essentielle pour la conception nouvelle de la liberté que propose Spinoza. Car, le conatus est la manifestation dans l’existence de la détermination de chaque être de la nature par une cause nécessaire et immanente qui est Dieu. C’est donc à titre d’hypothèse qu’est travaillé le lien entre la conception spinoziste de la liberté et la réalisation de notre conatus dans l’existence
What is liberation according to Spinoza? How does he demonstrate it in his work Ethics? A jumping off point for answering that question is to examine the relationship between mind and body in Spinoza, and demonstrate how the nature of that relationship makes the liberation of man possible as established by Spinoza in Ethics. Incidentally, the project of a man's liberation in Spinoza begins with the liberation of man from his affective life. Thus, we must examine the ontological basis of man's emotions in order to explain them in the same causal order that determines body and mind. But Spinoza does not only seek to answer the question of the existential mystery behind the realisation of our existential liberation, he also seeks to pave the way towards a metaphysical liberation, to lay the ontological foundation for the emotional life of man. This analysis will consist of demonstrating that the concept of conatus is essential to the new concept of liberation as espoused by Spinoza. Here, conatus is the manifestation of a necessary and immanent cause, which is God, in the existence of the determination of every being in nature. Thus, the hypothesis at work here is the link between Spinoza's concept of liberation and the realisation of our existential conatus

S'interroger sur le rapport du rationnel a la pensee chez spinoza, c'est d'abord questionner la formalisation mathematique qui caracterise sa pensee philosophique. Nous voulons montrer qu'une telle formalisation n'exclut aucunement l'univers tel que nous le percevons puisqu'au contraire, elle permet d'elaborer des modeles geometriques pour eclairer des points essentiels, tels dieu, les attributs et les modes. Mais surtout, si elle se trouve placee au cur de la reflexion spinoziste, c'est qu'elle n'est rien d'autre que la pensee meme, qui doit, pour etre pensee, suivre les criteres rigoureux de la formalisation mathematique. Il est des lors possible de suivre le double chemin de la genese de la pensee et de ce qui est pense. Pour fonder son ontologie, spinoza reprend les bases de la physique de descartes, ce qui de prime abord apparait paradoxal, en raison de leurs positions philosophiques conflictuelles. Toutefois, une analyse plus precise permet d'etablir que le rapport a descartes est le meme en physique et en philosophie. La reprise des lois cartesiennes n'est aucunement pure passivite, elle implique au contraire une profonde transformation de celles-ci, l'objet fondamental etant non plus la chose en tant que telle, mais ses relations aux autres. Il en resulte une physique de la dynamique et non de la statique, d'un monde qui, comme celui de la pensee, est homogene, symetrique, infini. Cependant, l'unite entre la pensee rationnelle et la pensee mathematique pose a toutes deux des difficultes redoutables. C'est pourtant au travers de cette tension meme, qu'est mise en lumiere l'absolue unite de la pensee spinoziste.

On doit nécessairement aborder le problème des rapports entre les sciences de la nature et la philosophie quand il s'agit d'étudier la manière dont la pensée philosophique du dix-septième siècle a conçu le monde matériel. Plusieurs philosophes-savants ont essayé de fonder la nouvelle physique de ce siècle sur des raisons métaphysiques et d'élaborer une physique cohérente en ayant recours seulement à la disposition, la configuration et le mouvement des parties de la matière. Une analyse des textes de Spinoza qui portent sur la philosophie naturelle permet d'inscrire sa physique dans le registre de cette physique déterministe et mécaniste ; cette étude systématique de la manière dont ce philosophe rend raison de la nature de la matière et de ses changements doit suivre de près le développement de sa pensée afin de mieux dégager le lien entre les thèses proprement philosophiques de Spinoza et sa conception de la nature. En commençant par un commentaire détaillé de la seconde partie des principes de la philosophie de Descartes, ce travail se propose de situer la pensée spinoziste par rapport aux thèses essentielles de la physique cartésienne, et de montrer que le déterminisme qui caractérise cette physique est associé, aux yeux de Spinoza, à une philosophie de l'immanence. Une étude de l'échange épistolaire entre Spinoza et Boyle (par l'intermédiaire d’Oldenburg) qui traite des questions physiques, aussi bien qu'une analyse des axiomes, des définitions et des lemmes qui constituent l'abrégé de physique de la seconde partie de l'éthique, permettent de préciser le caractère de la physique de Spinoza, aussi bien que les fondements ontologiques que son système fournit au mécanisme. On peut ainsi montrer que Spinoza élabore une conception originale de la nature, tout en fondant la rationalité de sa physique sur des concepts intelligibles qu'elle partage en commun avec la science de son temps, dont elle exploite les acquis.

Les conditions pour atteindre la liberté et la béatitude restent ancrées, d’après Spinoza, dans la durée, c'est-à-dire pendant que l'union de l'esprit et le corps dure, étant donné qu'il n'y a pas un au-delà de la mort. Tout effort pour connaître à partir de la raison et de l'intuition doit s'inscrire dans le régime de l'affectivité, ce qui implique étudier le régime par rapport à les « coordonnées temporelles de constitution de l'affectivité » : (1) le fait que le mode doive arriver à l'adéquation pour résoudre la différence entre le nombre infini d'idées et sa capacité limitée pour les former claire et distinctement de façon simultanée (simultanéité et différence attributives) ; (2) le corps est constitué dans un conflit entre l'activité et la passivité, il est un « mixte » puisqu'il est affectant tant qu'affecté (simultanéité modale) (3) cette constitution fait de l'individu un individu désirant et « sujet » aux passions, quand sa puissance diminue
According to Spinoza, conditions to attain freedom and happiness depend on duration; in other words, since there is nothing beyond death, they are attainable while the unity of body and mind endures. Any effort to acquire knowledge through reason and intuition must be effected within the system of affects. This entails studying said system in connection with the three temporal coordinates constituting affectivity: 1. The fact that mode (the individual) must achieve adequacy to solve the difference between the infinite number of ideas and his or her limited capacity to simultaneously form ideas clearly and distinctly (attributive simultaneity and difference); 2. The conflict itself between activity and passivity constitutes the body; and, 3. This constitution turns the individual into a desiring individual (one who desires), “subject” to passions when his or her power diminishes

La rencontre Spinoza-Machiavel est analysée à trois niveaux différents après avoir reconstruit le status quaestionis au XXe siècle dans le premier chapitre. Dans le deuxième chapitre, on analyse la présence des textes et de l'image de Machiavel dans la bibliothèque de Spinoza ainsi que la présence et la fonction stratégique du nom de Machiavel dans le TP. Dans le troisième chapitre on analyse la présence des argumentations de Machiavel dans les deux grandes oeuvres politiques de Spinoza, (TTP et TP), ainsi que la continuité et la discontinuité de l'usage de ces argumentations à l'intérieur des deux oeuvres. Dans le quatrième chapitre on analyse la modification des structures métaphysiques profondes du TdIE à l'Ethica produit par la rencontre avec Machiavel et la problématique historico-politique dont il est porteur : cette modification conduit Spinoza à articuler le temps et l'eternité d'une façon tout-à-fait nouvelle et non pas réductible aux deux grandes lignes du spinozisme qui traversent la modernité : la philosophie de l'histoire et le nihilisme.

Spinoza distingue les concepts de durée, d'éternité et de temps à travers toute son œuvre. Il part du principe que toute essence (ce qui est intelligible) est une vérité éternelle, c'est-à-dire nécessairement vraie, indépendamment de toute relation au temps. L'éternité, qui n'est pas une durée infinie, est définie par l'existence qui suit nécessairement d'une essence, et la durée par l'existence qui ne remplit pas cette condition. Or, selon son étude sur Descartes et la lettre 12, seul l'être unique absolument infini est éternel, l'existence d'aucun de ses effets n'étant nécessairement liée à une essence. En revanche, dans l'éthique, en dehors de dieu, l'éternité est attribuée à tous les modes infinis, mais aussi à des modes finis, tels que les âmes de toutes les choses singulières, dans la mesure où leur existence est conçue comme une conséquence nécessaire de la seule définition d'une chose éternelle. La distinction entre la durée et le temps est, pour l'essentiel, conçue toujours de la même façon. Si la durée est quelque chose de réel et de positif, en dehors de la pensée, le temps présuppose les êtres pensants. Il n'est qu'un auxiliaire de l'imagination, qui nous sert a délimiter la durée
Spinoza makes a distinction between the concepts of duration, eternity and time throughout his entire work. He starts off by assuming that any essence (that is the intelligible) is an eternal truth, necessarily true, which bears no relation to time. Eternity, which is not infinite, is defined by the existence that necessarily follows from an essence and duration by the existence that does not meet this condition. Now, according to his study on Descartes and to letter 12, only the unique absolutely infinite being is eternal, the existence of none of his effects being necessarily connected to an essence. On the other hand, in the ethics, apart from god, eternity is attributed to all infinite modes but also to finite modes such as the souls of all singular things insofar as their existence is conceived to follow solely from the definition of an eternal thing. Most importantly, the distinction between duration and time is always conceived in the same manner. If duration is something real and positive, outside of thought, time presupposes thinking beings. It is a mere appendage to imagination which helps us determine duration

Semiconductor microcavities offer the possibility to strongly confine light in a small cavity volume. Here the light interacts strongly with the electronic excitations of the quantum wells, which are embedded in the cavity, giving rise to a new kind of quasiparticle called exciton-polaritons or polaritons. These polaritons are the superposition of a photon and an exciton and inherit a light effective mass from the photon part and strong inter-particle interactions from the exciton part. Polaritons have extremely rich physics such as Bose-Einstein condensation (BEC) and super fluidity, to name a few. Thanks to their spin properties and fast dynamics polaritons could have potential applications in ultrafast optoelectronics such as optical switches. Under certain conditions the strong coupling does not sustain and the Eigenstates of the system change to the uncoupled cavity and exciton mode, which is called the weak coupling regime. In this thesis non-linear effects in the strong and in the weak coupling regime are investigated. In particular a crossover between a photon and a polariton laser is observed. Distribution functions and the dynamic behaviour of the long-range coherence confirms great similarities with BEC and exhibit the transition between two coherent states. Following these observations we study the spinor nature of polaritons and photons. In single shot experiments the spontaneous symmetry breaking at the phase transition to a coherent state was shown. In a nearly isotropic system the phase of the order parameter was chosen spontaneously and showed strong variations from shot to shot. This phenomenon which was once identified as the smoking gun for BEC was observed in a polariton and a photon laser. The spinor nature of polariton condensates was further exploited by studying the transport of spin by a propagating polariton condensate. Whilst travelling through the sample the spin experiences the optical spin-Hall effect and coherently precesses around an effective magnetic field. We observe up to four complete revolutions of the pseodospin around the effective magnetic field and the formation of a spin pattern that extends to 300 microns.

A survey of the spectral properties of the classical Dirac operator on a Riemannian spin manifold is made. Killing spinors, which are special eigenfunctions of the Dirac operator, are studied and necessary conditions for their existence are given. Killing spinors on $ IR sp{n}$, $S sp{n}$ and $H sp{n}$ are also computed explicitly. Finally the transformation law for Dirac operator under conformal change of the metric is computed and a lower bound for the eigenvalues is given.

We use a map introduced by Berkovits to determine the first levels of the theta-expansion of vertex operators for leading Regge trajectory bosons in the pure spinor formalism. This expansion is then used to confirm that in the pure spinor formalism, the 3-point disc scattering of any leading Regge trajectory bosons reproduces the corresponding Ramond-Neveu-Schwarz result found by Schlotterer.

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Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
Esta tese é baseada em parte do meu trabalho de doutoramento e tem como objetivo apresentar uma análise detalhada de algumas propriedades recém abordadas do fantasma b composto no formalismo de espinores puros. Primeiramente será feita uma revisão dos formalismos mínimo e não-mínimo. Em seguida, será apresentada a construção do fantasma b passo a passo, inlcuindo correções quânticas. Por fim, serão estudadas em detalhes suas propriedades fundamentais, ue vão desde a nilpotência até a definição de um possível conjugado, o fantasma c
This thesis is based in part of my work during the Ph.D. and aims to present a detailed analysis of some newly studied properties of the composite non-minimal pure spinor b ghost. First, a review of the minimal and non-minimal pure spinor formalisms will be presented. Then, the construction of the non-minimal b ghost will be done step-by-step, including quantum corrections. Finally, some of its fundamental properties will be studied i detail, ranging from nilpotency until the definition of a possible canonical conjugate, the c ghost
FAPESP: 2009/17516-4

Orientador: Nathan Jacob Berkovits
Banca: Horatiu Nastase
Banca: Andrei Mikhailov
Banca: Victor de Oliveira Rivelles
Banca: Vladimir Perchin
Resumo: Esta tese é baseada em parte do meu trabalho de doutoramento e tem como objetivo apresentar uma análise detalhada de algumas propriedades recém abordadas do fantasma b composto no formalismo de espinores puros. Primeiramente será feita uma revisão dos formalismos mínimo e não-mínimo. Em seguida, será apresentada a construção do fantasma b passo a passo, inlcuindo correções quânticas. Por fim, serão estudadas em detalhes suas propriedades fundamentais, ue vão desde a nilpotência até a definição de um possível conjugado, o fantasma c
Abstract: This thesis is based in part of my work during the Ph.D. and aims to present a detailed analysis of some newly studied properties of the composite non-minimal pure spinor b ghost. First, a review of the minimal and non-minimal pure spinor formalisms will be presented. Then, the construction of the non-minimal b ghost will be done step-by-step, including quantum corrections. Finally, some of its fundamental properties will be studied i detail, ranging from nilpotency until the definition of a possible canonical conjugate, the c ghost
Doutor

Dans ce manuscrit, nous présentons une étude expérimentale d'un gaz de Bose de spin 1 avec des interactions antiferromagnétiques avec des atomes de sodium ultra-froids dans l'état hyperfin F=1. Les trois composantes Zeeman sont piégées simultanément dans des pièges dipolaires optiques. Nous obtenons un condensat de Bose-Einstein spineur par refroidissement évaporatif et nous étudions ses propriétés magnétiques. Il y a deux types d’interactions dans le système: des interactions de contact qui ne changent pas les populations des composantes Zeeman et des interactions d'échange de spin qui les modifient. Une compétition entre l'énergie Zeeman et l'énergie d'échange impose l'ordre magnétique dans le système.Nous étudions dans un premier temps les phases magnétiques de condensats de Bose-Einstein spineurs a température quasi nulle. L'état fondamental comporte deux phases qui sont observées en variant le champ magnétique (donc l'énergie Zeeman quadratique) et la magnétisation de l'échantillon. Dans la phase antiferromagnétique, le spin de l'échantillon est simplement selon l'axe du champ magnétique. Dans la phase polaire, une composante transverse apparait pour minimiser l'énergie Zeeman. Pour une magnétisation nulle, le condensat spineur forme un nématique de spin. Cet état, nommé par analogie avec la phase nématique dans les cristaux liquides, est caractérisée par des fluctuations de spin orthogonales à un axe particulier, mais sans préférer une des deux direction sur cet axe. Dans chacune des deux phases, l'ordre nématique se manifeste par un minimisation de la longueur du spin transverse en imposant une valeur particulière ($pi$) de la phase relative des composantes Zeeman ${theta = phi_{+1} + phi_{-1} - 2 phi_{0}}$. Nous mesurons la longueur du spin transverse en analysant le bruit de spin après une rotation.Dans un second temps, nous étudions la thermodynamique d'un gaz de Bose de spin 1 près de la température critique pour la condensation de Bose-Einstein. Nous mesurons plusieurs scénarios de condensation séquentiels en fonction de la magnétisation et du champ magnétique. La température critique mesurée révèle que les interactions ont un effet important quand la condensation d'une composante se fait en présence d'un condensat dans une autre composante. Nous utilisons une théorie d'Hartree-Fock simplifiée, en négligeant les interactions d’échange de spin. Nous constatons que les résultats expérimentaux sont en bon accord. Cependant, pour de bas champs magnétiques, le diagramme de phase thermodynamique est largement modifié par les interactions d'échange de spin, ce qui pose de nouvelles questions sur leur rôle a température finie
In this manuscript, we present an experimental study of a Spin 1 Bose gas with antiferromagnetic interactions with ultracold sodium atoms in the F=1 manifold. The three Zeeman components are trapped simultaneously in optical dipole traps. By performing evaporative cooling, we obtain quasi-pure spinor Bose-Einstein condensates of which we study the magnetic properties. There are two types of interactions between the constituents of the system: Contact interactions that do not change the Zeeman populations and spin-exchange contact interactions that do. A competition between Zeeman energy and the spin-exchange energy sets the magnetic ordering in the system.We first study the magnetic phases of spinor Bose-Einstein condensates near zero temperature. The ground state present two phases that are observed by varying the magnetic field (hence the quadratic Zeeman energy) and the magnetization of the sample. In the antiferromagnetic phase, the spin of the sample is purely along the direction of the magnetic field. In the broken-axisymmetry phase, a transverse component appears in order to minimize the Zeeman energy. For zero magnetization, the spinor condensate forms a spin nematic. This state, named in analogy with the liquid crystal nematic phase, is characterized by spin fluctuations orthogonal to a particular axis, with no preferred direction along that axis. In both phases, spin nematic order manifests as a minimization of the transverse spin length that is realized by enforcing a particular value ($pi$) of the relative phase of the Zeeman components $theta = phi_{+1} + phi_{-1} - 2 phi_0$. We measure the transverse spin length by analyzing spin noise after a spin rotation.Second, we study the thermodynamics of an antiferromagnetic spin 1 Bose gas next to the critical temperature for Bose-Einstein condensation. We measure several sequential condensation scenarii depending on the magnetization and the magnetic field. The measured critical temperatures reveal a large effect of interactions when one of the Zeeman component condenses in presence of a condensate in another component. We use a simplified Hartree-Fock theory, neglecting the spin exchange interactions and note a good agreement with our data. However, for low magnetic fields, the thermodynamic phase diagram is strongly modified which raises new open questions about the role of spin exchange interactions at finite temperatures

Was im späten 18. Jahrhundert mit der Spinoza-Begeisterung von Johann Wolfgang Goethe, Johann Gottfried Herder und Friedrich Heinrich Jacobi begann, setzt sich im 19. Jahrhundert fort: Es ist das Jahrhundert einer intensiven Beschäftigung mit Spinoza und seiner Philosophie. Die spezifische Intensität dieser Beschäftigung wirkt bis heute nach, wie im folgenden gezeigt werden soll. Was im 19. Jahrhundert stattfindet, ist keine Spinoza-Rezeption, keine bloße Interpretation von Leben und Werk, sondern der Anfang eines historischen Begreifens, das Rezeption wie Interpretation bis heute bestimmt. In der Beschäftigung mjt Spinoza hat das 19. Jahrhundert Formen des philosophiehistorischen Denkens ausgebildet, die immer noch prägend sind.

Cette thèse porte sur l’unité du concept de liberté chez Spinoza. Aussi passe-t-elle par les trois objets privilégiés du philosophe, Dieu, l’homme et la multitude. A ces trois objets correspond une certaine forme de la liberté : liberté métaphysique, liberté éthique et liberté politique. Etant données, d’une part la définition de la liberté (en EID7), d’autre part l’affirmation faite par Spinoza selon laquelle seul Dieu est cause libre, le problème se pose en effet de savoir dans quel sens ou dans quelle mesure l’homme peut, du point de vue éthique et du point de vue politique, être dit libre. Nous avons tenté de résoudre le problème par l’étude d’un aspect peu ou pas étudié de la théorie spinoziste des affects, la notion de jouissance. Nous avons commencé par une étude du champ lexical motivé par l’absence de définition du terme fruitio, pour s’étendre ensuite vers d’autres termes proches, tels le gaudium (qui a deux sens différents chez Spinoza) et la delectatio. Ensuite, nous avons situé ces termes du point de vue de l’histoire de la philosophie au sein du débat auquel ils appartenaient : la question de la jouissance dans la philosophie médiévale chrétienne. Ainsi nous avons été amenés à nous interroger sur le concept central d’Amor intellectualis Dei, qui est ce en quoi consiste la liberté, à partir de l’idée de jouissance, et ce dans les deux domaines de Dieu et de l’homme. Enfin, nous avons montré qu’en politique l’unité du concept de liberté était maintenue aussi grâce à l’idée de jouissance (obtinentia). Bref, c’est toujours l’être qui jouit de sa puissance constitutive en tant que telle, parce qu’il existe et qu’il agit nécessairement, qui est dit libre
This thesis refers to the unity of the concept of freedom in Spinoza, focusing in the three privileged objects of the philosopher: God, man and multitude. To each one belongs a certain form of freedom: metaphysical, ethical and political. Given that, in one hand, the definition of freedom (E1D7) and in the other hand the affirmation made by Spinoza, that God is the only free cause, the problem is indeed, knowing in which sense or measure man can, from the ethical and political point of view, called free. We have tried to solve the problem studying a barely or not studied at all issue of Spinoza’s theory of the affectus: the notion of fruitio (enjoyment). We started by studying the lexical field motivated by the absence of definition of the term fruitio, in order to go further with nearby terms, such as gaudium (which has two different senses in Spinoza) and the delectatio. Furthermore, we have placed these points of view of the history of philosophy in the core of the discussion where they belong: the question of fruitio in Christian medieval philosophy. Thus, we question ourselves about the main concept of Amor intellectualis Dei, which is freedom itself, starting from the idea of fruitio, and the same in both God’s and man’s domains. Finally, we showed that in politics, the unity of the concept of freedom was sustained also due to the idea of obtinentia (enjoyment). In brief, we call free the being that enjoys its constitutive power as such, because it exists and acts out of necessity

Les avancées de la biologie contemporaine, posent de façon nouvelle des problèmes philosophiques anciens. Ceux des rapports entre le vivant et l'inanimé, entre le corps et l'esprit, l'erreur et la vérité, sont les plus évidents. La philosophie de Spinoza, bien que datant du 17e siècle, apporte à ces problèmes des solutions plus pertinentes que la plupart des philosophies plus récentes, développées dans les siècles qui l'ont suivie. En retour, les acquis actuels des sciences physiques et biologiques, notamment des neurosciences cognitives, permettent de porter un nouveau regard sur certaines notions propres à la philosophie de Spinoza, telles que sa «petite physique», la nature cause de soi, la notion de matière, l'essence des choses, les genres de connaissance, qui acquièrent de ce fait un surcroît d'actualité
Old philosophical problems are raised in renewed ways by advances in biology of today. Most obvious are the problems of relationship between living and non-living, mind and body, error en truth. Spinoza's philosophy, although from 17th century, offers solutions to these problems more relevant than most more recent philosophies. In return, present knowledge from physical and biological sciences, especially cognitive neurosciences, can provide a new look at some specifically Spinozist notions such as his "little physics", Nature as cause of itself, the notion of matter, the essence of a thing, kinds of knowledge, which gain all the more interest from a present day point of view

To think a Deleuzian politics requires to understand the philosophical markleft in his work by Spinoza. Both philosophers’ paths crisscross in their problems, questions and consequences. In this paper three concepts will be put in relation: critique, right and crowd. The aim is to elucidate how man can open new ways of political experience from the conjunction of two philosophies articulated upon a common ground: immanence and life.
Pensar una política en Gilles Deleuze exige comprender la huella filosófica que ha dejado en su obra Baruch Spinoza. Las sendas de ambos filósofos se entrecruzan en problemas, preguntas y consecuencias. Hay tres conceptos que serán puestos en relación en el presente trabajo: crítica, derecho  y  multitud.  El  objetivo  es elucidar cómo se pueden abrir nuevos modos de experiencia política a partir de la conjunción de dos filosofías articuladas por un suelo común: la inmanencia y la vida.