Caos e termalização na teoria de Yang-Mills com quebra espontânea de simetria
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Data
2011-09-27
Autores
Woitek Junior, Marcio [UNESP]
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Editor
Universidade Estadual Paulista (Unesp)
Resumo
Uma das características mais importantes das teorias de gauge não-Abelianas é a não-linearidade das equações de campo clássicas. Mostra-se no contexto da teoria de Yang-Mills que essa característica pode fazer com que o campo de gauge apresente comportamento caótico. Isso pode acontecer mesmo quando estivermos considerando a dinâmica do campo na ausência de fontes, isto é, o vácuo da teoria de Yang-Mills. Discutimos a relação entre os comportamentos caótico e ergódico. Em seguida, introduzimos a formulação de Berdichevsky da Mecânica Estatística Clássica para sistemas dinâmicos Hamiltonianos que são ergódicos e possuem poucos graus de liberdade. A Mecânica Estatística de Berdichevsky é usada para estudar a situação mais simples numa teoria de gauge não-Abeliana onde as variáveis de campo são caóticas e o espaço de fase correspondente tem a propriedade geométrica necessária. Mostramos que, para os propósitos desse estudo, um par de campos escalares complexos deve ser incluído no problema. Mais precisamente, analisamos o modelo de Higgs não-Abeliano; a Lagrangiana da teoria considerada possui uma simetria SU(2). A transição de uma descrição dinâmica do sistema de YangMills-Higgs (fora do equilíbrio termodinâmico) para uma descrição termodinâmica (quando ele atingiu o equilíbrio) é investigada numericamente. Mostra-se que depois de um tempo suficientemente longo as soluções numéricas se comportam de tal maneira que o sistema pode ser descrito de um jeito mais simples através de grandezas como a temperatura, calculadas de acordo com as prescriçõees da Mecânica Estatística de equilíbrio. Estas são previstas analiticamente para comparção com os resultados numéricos...
One of the most important features of non-Abelian gauge theories is the non-linearity of the classical field equations. In the context of Yang-Mills theory it is shown that this feature can cause the gauge field to show chaotic behavior. That can happen even when we are considering the field dynamics in the absence of sources, i.e., the vacuum of the Yang-Mills theory. We discuss the connection between chaotic and ergodic behaviors. Then we introduce Berdichevsky’s formulation of Classical Statistical Mechanics for Hamiltonian dynamical systems that are both ergodic and low-dimensional. Berdichevsky’s theory of Statistical Mechanics is used to study the simplest situation in a non-Abelian gauge theory where the field variables are chaotic and the corresponding phase space has the necessary geometric property. We show that, for the purposes of this study, a pair of complex scalar fields must be introduced in the problem. More precisely, we analyse the so-called non-Abelian Higgs model; the Lagrangian of the theory we are considering has a SU(2) symmetry. The transition from a non-equilibrium dynamical description of the Yang-Mills-Higgs system to a thermodynamical description when it reaches equilibrium is numerically investigated. It is shown that after a sufficiently long time the numerical solutions behave in such a manner that the system can be described by quantities like the temperature, determined in accordance with the prescriptions of equilibrium Statistical Mechanics. These are predicted analytically for comparison with the numerical results. It is verified that there is agreement between analytical and numerical predictions so that the thermalization of the Yang-Mills-Higgs system can be explained with the aid of Berdichevsky’s Statistical Mechanics. A dynamical approach to the study... (Complete abstract click electronic access below)
One of the most important features of non-Abelian gauge theories is the non-linearity of the classical field equations. In the context of Yang-Mills theory it is shown that this feature can cause the gauge field to show chaotic behavior. That can happen even when we are considering the field dynamics in the absence of sources, i.e., the vacuum of the Yang-Mills theory. We discuss the connection between chaotic and ergodic behaviors. Then we introduce Berdichevsky’s formulation of Classical Statistical Mechanics for Hamiltonian dynamical systems that are both ergodic and low-dimensional. Berdichevsky’s theory of Statistical Mechanics is used to study the simplest situation in a non-Abelian gauge theory where the field variables are chaotic and the corresponding phase space has the necessary geometric property. We show that, for the purposes of this study, a pair of complex scalar fields must be introduced in the problem. More precisely, we analyse the so-called non-Abelian Higgs model; the Lagrangian of the theory we are considering has a SU(2) symmetry. The transition from a non-equilibrium dynamical description of the Yang-Mills-Higgs system to a thermodynamical description when it reaches equilibrium is numerically investigated. It is shown that after a sufficiently long time the numerical solutions behave in such a manner that the system can be described by quantities like the temperature, determined in accordance with the prescriptions of equilibrium Statistical Mechanics. These are predicted analytically for comparison with the numerical results. It is verified that there is agreement between analytical and numerical predictions so that the thermalization of the Yang-Mills-Higgs system can be explained with the aid of Berdichevsky’s Statistical Mechanics. A dynamical approach to the study... (Complete abstract click electronic access below)
Descrição
Palavras-chave
Caos quântico, Simetria quebrada, Quantum chaos, Broken symmetry
Como citar
WOITEK JUNIOR, Marcio. Caos e termalização na teoria de Yang-Mills com quebra espontânea de simetria. 2011. 104 f. Dissertação (mestrado) - Universidade Estadual Paulista, Instituto de Física Teórica, 2011.