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In this thesis, conformal transformations in d and two dimensions and the results of conformal symmetry in classical and quantum field theories are reviewed. After investigating the conformal group and its algebra, various aspects of conformal invariance in field theories, like conserved charges, correlation functions and the Ward identities are discussed. The central charge and the Virasoro algebra are briefly touched upon.

The dynamic structure factor of liquid para-hydrogen and ortho-deuterium in corresponding thermodynamic states, (T = 20.0 K, n = 21.24 nm⁻³) and (T = 23.0 K, n = 24.61 nm⁻³) respectively, has been computed by both the Feynman-Kleinert linearized path-integral (FK-LPI) and Ring-Polymer Molecular Dynamics (RPMD) methods and compared with Inelastic X-ray Scattering spectra. The combined use of computational and experimental methods enables a reduction in experimental uncertainties for the determination of the true sample spectrum. Furthermore, the refined experimental spectrum of para-hydrogen and ortho-deuterium is consistently reproduced by both FK-LPI and RPMD at momentum transfers lower than 12.8nm⁻¹. At larger momentum transfers the F K - LP I results agree with experiment much better for ortho-deuterium than for para-hydrogen. More specifically we found that for k ~ 20.0 nm⁻¹ para-hydrogen provides a test case for improved approximations to quantum dynamics. We meet this demand for an improved approximate quantum dynamics method by developing two classes of quasi-classical dynamics that are shown to conserve the initial quantum ensemble when used in conjunction with the Feynman-Kleinert approximation of the density operator. As shown, both classes of dynamics are able to recover the exact classical and high temperature limits of the quantum time correlation function, while a subset is able to recover the exact harmonic limit. A comparison of the approximate quantum time correlation functions obtained from both classes of dynamics are made with the exact results for the challenging model problems of the quartic and double-well potentials. It is found that this new Feynman-Kleinert Quasi-Classical Wigner (FK-QCW) method provides a great improvement over the Feynman-Kleinert implementation of the classical Wigner approximation, also known as FK-LPI, in which purely classical dynamics are used. Furthermore, it is shown that the first class of dynamics reduces to Centroid Molecular Dynamics (CMD) when used within the framework of the classical Wigner approximation for the Kubo transformed time correlation function. Finally, we apply the Feynman-Kleinert Quasi-Classical Wigner (FK- QCW) method to the same liquid para-hydrogen and ortho-deuterium system, previously studied using FK-LPI and RPMD. When applied to this challenging system, it is shown that this new FK-QCW method consistently reproduces the experimental dynamic structure factor for all momentum transfers considered. This shows that FK-QCW provides a great improvement over FK-LPI for not only model problems, but also realistic systems. Furthermore, for small momentum transfers, where RPMD is applicable, it is shown that FK-QCW provides nearly the same results as RPMD, thus suggesting that FK-QCW provides a potentially more appealing algorithm than RPMD since one is not limited to correlation functions involving linear operators. This then suggests that the FK-QCW method is a top contender in the realm of approximate quantum dynamics methods which allow for the practical evaluation of time correlation functions. / text

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In this thesis, computer simulations are used to study the properties of new colloidal systems with structured interactions. These are pair interactions that include both attraction and repulsion. Structured colloids differ from conventional colloids in which the interactions between particles are either strictly attractive or strictly repulsive. It is anticipated that these novel interactions will give rise to new microscopic structure and dynamics and therefore new material properties. Three classes of structured interactions are considered: radially structured interactions with an energetic barrier to pair association, Janus surface patterns with two hemispheres of different surface charge, and striped surface patterns. New models are developed to capture the structured interactions of these novel colloid systems. Dynamical computer simulations of these models are performed to quantify the effects of structured interactions on colloid properties. The results show that structured interactions can lead to unexpected particle ordering and novel dynamics. For Janus and striped particles, the particle order can be captured with simpler isotropic coarse-grained models. This relates the static properties of these new colloids to conventional isotropically attractive colloids (e.g. depletion attracting colloids). In contrast, Janus and striped particles are found to have substantially slower dynamics than isotropically attractive colloids. This is explained by the observation of longer-duration reversible bonds between pairs of structured particles. Dynamical mapping methods are explored to relates the dynamics of these structured colloids to isotropically attractive colloids. These methods could also facilitate future nonequilibrium simulation of structured colloids with computationally efficient coarse-grained models.

Thesis (M.S.)--Miami University, Dept. of Physics, 2005. / Title from first page of PDF document. Document formatted into pages; contains [1], v, 296 p. : ill. Includes bibliographical references (p. 393-396).

Dans cette thèse on étudie les (super)fonctions de corrélation à plusieurs points et à plusieursboucle du multiplets demi-BPS en théorie N = 4 super-Yang-Mills. Les fonctions de corrélationsont des objets dynamiques naturels à considérer dans toutes les théories conformes des champs.Elles sont des quantités finies et leur symétrie (super)conforme n’est pas brisée par des divergences.Elles contiennent des informations sur de nombreuses autres intéressantes quantités dynamiques dela théorie. Le produit opératoire engendre les règles de somme pour les fonctions à trois points et lesdimensions anormales. Dans la limite du cône de lumière, elles coïncident avec les boucles de Wilsonde lumière et avec des superamplitudes de diffusion. Cette dualité tient tant au niveau des intégralesdivergentes régularisés que au niveau de leurs intégrandes rationnels finis.La partie principale de la thèse est consacrée aux super-corrélateurs à plusieurs points au niveau Born du supermultiplet du tenseur de stress. Pour les étudier on utilise les règles de Feynman qui préservent une quantité de la supersymétrie. Donc, on reformule la théorie N = 4 SYM dans le superespace harmonique de Lorentz. On s’occupe de l’espace euclidien et on harmonise la moitié du groupe de Lorentz SU(2) × SU(2). La théorie est formulée en termes de deux demi-superchamps chiraux-analytique. L’action de la théorie est une somme de deux termes : l’action de Chern-Simons et une action non-polynomiale qui prend en compte les interactions. Puisque la formulation de l’action est chiral, la Ǭ-supersymétrie est réalisée d’un façon non-linéaire sur la paire de champs. L’action se simplifie considérablement dans la jauge axiale. On obtient les propagateurs correspondants et on formule les règles de Feynman en superspace harmonique de Lorentz. Afin d’étudier super-corrélateurs non-chiraux du supermultiplet de tenseur de stress on formule l’opérateur composite pertinent en termes de demi-superchamps chiraux-analytique ainsi. Au niveau chiral, on propose la construction par R-vertex du super-corrélateur chiral. Afin d’élucider la structure du super-corrélateur on réorganise les règles de Feynman harmoniques qui introduisent une nouvelle classe des invariants hors-shell nilpotent analytique qui sont des blocs de construction élémentaires de la super-corrélateur. Ensuite, on procède au secteur non-chiral et on constate que la dépendance de Ɵ̅ est pris en compte par une légère modification du R-vertex qui consiste à une modification des variables spatio-temporelles de la base chirale à la base analytique. Ainsi, le corrélateur non-chiral est exprimée en termes d’une classe assez particulière des invariants nilpotents non-chiraux. Dans la dernière partie de la thèse, on étudie les fonctions de corrélation à quatre points des opérateurs demi-BPS dans l’approximation de trois boucle dans la limite planaire. Cette étude est motivée par une conjecture basée sur intégrabilité pour les constantes de structure. A l’ordre de trois boucles toutes les approches de graphes de Feynman connus sont extrêmement inefficaces. Le principal obstacle est un grand nombre de diagrammes de Feynman pertinents. Cependant, le corrélateur est presque complètement fixé par ses propriétés élémentaires comme symétries, singularités et planairité. La structure de pôle et la symétrie super-conforme spécifient les intégrandes rationnelles des corrélateurs à un nombre de coefficients numériques. Les coefficients sont fixés par la planairité, la symétrie de croisement et le produit opératoire en cône de lumière des intégrandes avec diverses configurations de poids dans la limite par rapport à une paire de points. / In the present thesis we study the multi-point multi-loop (super)correlation functions of half-BPSmultiplets in N = 4 super-Yang-Mills theory. Correlation functions are natural dynamical objectsto consider in any Conformal Field Theory. They are finite quantities and their (super)conformalsymmetry is not broken by divergences. They contain information about many others interestingdynamical quantities of the theory. The Operator Product Expansion being applied to them producessum rules for three-point functions and anomalous dimensions. In the light-cone limit they coincidewith the light-like Wilson loops and scattering superamplitudes. This duality holds both at the levelof the regularized divergent integrals and at the level of their finite rational integrands.The main part of the thesis is devoted to multi-point Born level super-correlators of the stress-tensor supermultiplet. There exists a number of hints that such super-correlators are remarkable dynamicalquantities in N = 4 SYM. Studying the supercorrelators it is convenient to use the Feynman rulespreserving an amount of the supersymmetry. So, we reformulate the N = 4 SYM in the Lorentzharmonic superspace. We deal with Euclidean space and harmonize one half of the Lorentz groupSU(2) x SU(2). The theory is formulated in terms of two chiral-analytic semi-superfields one ofwhich is scalar and the other one is spinor. The action of the theory is a sum of two terms: theChern-Simons action describing the self-dual N = 4 SYM theory and a non-polynomial action whichtakes into account interactions. Since the formulation of the action is chiral the Ǭ-supersymmetry isnon-linearly realized on the pair of fields. The action considerably simplifies in the axial gauge. Wework out corresponding propagators and formulate Lorentz harmonic superspace Feynman rules. Inorder to study nonchiral supercorrelators of the stress-tensor supermultiplet we formulate the relevant composite operator in terms of the chiral-analytic semi-superfields as well.At the chiral level we propose the R-vertex construction of the chiral supercorrelator which turnsout to be rational at the Born level by construction. In order to elucidate the structure of thesupercorrelator we rearrange harmonic Feynman rules introducing a new class of off-shell analyticnilpotent (Grassmann degree two). They are simple building blocks of the super-correlator. Thenwe proceeded to the nonchiral sector and and that the dependence on Ɵ̅ is taken into account by aslight modification of the R-vertices. This modification of the R-vertices is equivalent to a change of the space-time variables from the chiral to analytic bases. So the non-chiral correlator is expressed in terms of a rather special class of non-chiral nilpotent invariants.In the last part of the thesis we study four-point correlation functions of half-BPS operators inthe three-loop approximation in the planar limit. This study is motivated by an integrability basedconjecture for the structure constants. At the three-loop order all known Feynman graph approachesare extremely inefficient. The main obstacle is a huge number of relevant Feynman diagrams andthe complexity of the corresponding loop integrals. However the correlator is almost completely fixedby its elementary properties like symmetries, singularities and planarity. The pole structure andthe super-conformal symmetry specify the rational integrands of the correlators up to a number ofnumerical coefficients. We fix these coefficients using planarity, the crossing symmetry and comparingthe light-cone OPE of the correlator integrands with various weight configurations in the light-likelimit with respect to a pair of points.