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Is studied in this work two classes of solutions of scalar field theories. The first chapter deals with one-dimensional static fields models that give rise to topological defects, having its stability assured by arguments topological. This type of solution has a simple mathematical approach, however have greatly interest physical. The second chapter discusses a field theory with stationary solutions which is associated the a Noethers charge, coming from an internal symmetry of the field. It is called Q-balls the objects described in the second chapter. The main objective will be to use the method of deformation to find new solutions type Q-ball.

Estuda-se neste trabalho duas classes de soluções de teorias de campos escalares. O primeiro capítulo aborda modelos de campos estáticos unidimensionais que dão origem a defeitos topológicos, tendo sua estabilidade assegurada por argumentos topológicos. Esse tipo de solução tem uma abordagem simples matematicamente, no entanto apresenta muito interesse físico. O segundo capítulo aborda uma teoria de campo com soluções estacionárias a qual está associada uma carga de Noether, provinda de uma simetria interna do campo. Chama-se de Q-balls os objetos descritos no segundo capítulo. O objetivo principal será utilizar o método de deformação para encontrar novas soluções tipo Q-ball.

Advisors/Committee Members: q=%2Bpublisher%3A%22Universidade%20Federal%20da%20Para%C3%83%C2%ADba%22%20%2Bcontributor%3A%28%22Dionisio%20Bazeia%20Filho%22%29&pagesize-30">Dionisio Bazeia Filho.

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Dans cette thèse, nous nous intéressons à plusieurs aspects des systèmes de racines des algèbres de Lie simples. Dans un premier temps, nous étudions les coordonnées des vecteurs propres des matrices de Cartan. Nous commençons par généraliser les travaux de physiciens qui ont montré que les masses des particules dans la théorie des champs de Toda affine sont égales aux coordonnées du vecteur propre de Perron  – Frobenius de la matrice de Cartan. Puis nous adoptons une approche différente, puisque nous utilisons des résultats de la théorie des singularités pour calculer les coordonnées des vecteurs propres de certains systèmes de racines. Dans un deuxième temps, en s'inspirant des idées de Givental, nous introduisons les matrices de Cartan q-déformées et étudions leur spectre et leurs vecteurs propres. Puis, nous proposons une q-déformation des équations de Toda et construisons des 1-solitons solutions en adaptant la méthode de Hirota, d'après les travaux de Hollowood. Enfin, notre intérêt se porte sur un ensemble de transformations agissant sur l'ensemble des bases ordonnées de racines comme le groupe de tresses. En particulier, nous étudions les bases distinguées, qui forment l'une des orbites de cette action, et des matrices que nous leur associons.

In this thesis, our goal is to study various aspects of root systems of simple Lie algebras. In the first part, we study the coordinates of the eigenvectors of the Cartan matrices. We start by generalizing the work of physicists who showed that the particle masses of the affine Toda field theory are equal to the coordinates of the Perron  – Frobenius eigenvector of the Cartan matrix. Then, we adopt another approach. Namely, using the ideas coming from the singularity theory, we compute the coordinates of the eigenvectors of some root systems. In the second part, inspired by Givental's ideas, we introduce q-deformations of Cartan matrices and we study their spectrum and their eigenvectors. Then, we propose a q-deformation of Toda's equations et compute 1-solitons solutions, using the Hirota's method and Hollowood's work. Finally, our interest is focused on a set of transformations which induce an action of the braid group on the set of ordered root basis. In particular, we study an orbit for this action, the set of distinguished basis and some associated matrices.

Advisors/Committee Members: q=%2Bpublisher%3A%22Universit%26%23233%3B%20Toulouse%20III%26%23160%3B%26%238211%3B%20Paul%20Sabatier%22%20%2Bcontributor%3A%28%22Schechtman%2C%20Vadim%22%29&pagesize-30">Schechtman, Vadim (thesis director).

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES

Is studied in this work two classes of solutions of scalar field theories. The first chapter deals with one-dimensional static fields models that give rise to topological defects, having its stability assured by arguments topological. This type of solution has a simple mathematical approach, however have greatly interest physical. The second chapter discusses a field theory with stationary solutions which is associated the a Noether s charge, coming from an internal symmetry of the field. It is called Q-balls the objects described in the second chapter. The main objective will be to use the method of deformation to find new solutions type Q-ball.

Estuda-se neste trabalho duas classes de soluções de teorias de campos escalares. O primeiro capítulo aborda modelos de campos estáticos unidimensionais que dão origem a defeitos topológicos, tendo sua estabilidade assegurada por argumentos topológicos. Esse tipo de solução tem uma abordagem simples matematicamente, no entanto apresenta muito interesse físico. O segundo capítulo aborda uma teoria de campo com soluções estacionárias a qual está associada uma carga de Noether, provinda de uma simetria interna do campo. Chama-se de Q-balls os objetos descritos no segundo capítulo. O objetivo principal será utilizar o método de deformação para encontrar novas soluções tipo Q-ball.

Advisors/Committee Members: q=%2Bpublisher%3A%22Universidade%20Federal%20da%20Para%C3%83%C2%AD%C3%82%C2%ADba%3B%20Programa%20de%20P%C3%83%C2%B3s-Gradua%C3%83%C2%A7%C3%83%C2%A3o%20em%20F%C3%83%C2%ADsica%3B%20UFPB%3B%20BR%3B%20F%C3%83%C2%ADsica%22%20%2Bcontributor%3A%28%22Bazeia%20Filho%2C%20Dionisio%22%29&pagesize-30">Bazeia Filho, Dionisio.

▼ The symplectic sp(4) algebra provides a natural framework for studying proton-neutron (pn) and like-nucleon pairing correlations as well as higher-J pn interactions in nuclei when protons and neutrons occupy the same shell. While these correlations manifest themselves most clearly in the binding energies of 0+ ground states, they also have a large effect on the spectra of excited isobaric analog 0+ states. With a view towards nuclear structure applications, a fermion realization of sp(4) is explored and its q-deformed extension, sp(4)q, is constructed for single and multiple shells. The su(2)(q) substructures that enter are associated with isospin symmetry and with identical-particle and pn pairing. We suggest a non-deformed as well as a q-deformed algebraic descriptions of pairing for even-A nuclei of the mass 32 < A < 164 region. A Hamiltonian with a symplectic dynamical symmetry is constructed and its eigenvalues are fit to the relevant Coulomb corrected experimental 0+ state energies in both the “classical” and “deformed” cases. While the non-deformed microscopic theory yields results that are comparable to other models for light nuclei, the present approach succeeds in providing a reasonable estimate for interaction strength parameters as well as a detailed investigation of isovector pairing, symmetry energy and symmetry breaking effects. It also reproduces the relevant ground and excited 0+ state energies and predicts some that are not yet measured. The model successfully interprets fine features driven by pairing correlations and higher-J nuclear interactions. In a classification scheme that is inherent to the sp(4) algebraic approach, a finite energy difference technique is used to investigate two-particle separation energies, irregularities found around the N = Z region, and like-particle and pn isovector pairing gaps. The analysis identifies a prominent staggering behavior between groups of even-even and odd-odd nuclides that is due to discontinuities in the pairing and symmetry terms. While the “classical” limit of the theory provides good overall results, the analysis also shows that q-deformation can be used to gain a better understanding of higher-order effects in the interaction within each individual nucleus.