A eletrodinâmica escalar generalizada de Duffin-Kemmer-Petiau, uma análise funcional de sua dinâmica quântica covariante e o equilíbrio termodinâmico
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Data
2016-02-26
Autores
Nogueira, Anderson Antunes [UNESP]
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Universidade Estadual Paulista (Unesp)
Resumo
Este trabalho tem como objetivo explorar a dinâmica quântica de interação entre partículas escalares e vetoriais e estudar o equilíbrio termodinâmico dessas partículas no ensemble gran-canônico. A dinâmica de interação, escrita em uma linguagem covariante entre o campo de matéria (escalar) e o campo intermediador de interação (vetorial), apresenta uma simetria de calibre local, U(1) no caso quântico e SO(4) no equilíbrio termodinâmico. Sendo assim dividimos o trabalho em dois setores. No primeiro setor analisamos sistematicamente a interação quântica entre partículas escalares (mésons) e partículas vetoriais (fótons) no contexto da eletrodinâmica quântica escalar generalizada de Duffin-Kemmer-Petiau (GSDKP). Para isso quantizamos a teoria, utilizando uma abordagem funcional. Construímos a estrutura Hamiltoniana do sistema seguindo a metodologia de Dirac, o procedimento de Faddeev-Senjanovic para obter a amplitude de transição no calibre de Coulomb generalizado e o método de Faddeev-Popov-DeWitt para escrever a amplitude de transição anterior de maneira covariante na condição de calibre no-mixing. Daí, escrevendo o funcional gerador via Schwinger, as equações de Schwinger-Dyson (SD) e as identidades de Ward-Takahashi (WT) são obtidas. Como introdução à análise das correções radiativas, fizemos um cálculo quantitativo para ver os tipos de divergências superficiais (ultravioleta) que poderiam aparecer na teoria. Depois apresentamos um cálculo explícito das primeiras correções radiativas (1-laço) associadas ao propagador do fóton, propagador do méson, vértice e, estudamos a função de 4 pontos (fóton-fóton) utilizando o método de regularização dimensional, em que a simetria de calibre é manifesta. Como veremos, uma consequência do estudo é que a álgebra de DKP assegura o funcionamento das identidades de WT nas primeiras correções radiativas proibindo certas divergências no ultravioleta. Com o conhecimento das divergências no ultravioleta (UV) e no infravermelho (IV) abordadas nas correções radiativas, estabelecemos o Programa de Renormalização multiplicativo para esta teoria na camada de massa. O fato do propagador do campo escalar possuir uma nova estrutura divergente na massa de Podolsky nos levou a analisar as correções radiativas a 2-laços. Do propagador do fóton definimos o tensor de polarização e com este, de maneira fenomenológica, analisando o comportamento assintótico das funções de Green para altos momentos, abordamos a dependência da constante de estrutura com a escala de energia. No segundo setor estudamos o Formalismo de Matsubara-Fradkin (MF) para descrever campos em equilíbrio termodinâmico. Para isso foi necessário construir as equações em equilíbrio termodinâmico que descrevessem o setor escalar e vetorial e a posteriori extrair a função de partição. Ao construir o setor vetorial, percebemos o surgimento e a importância dos campos fantasmas e sua conexão com a simetria de Bechi-Rouet-Stora-Tyutin (BRST). No caso da escolha de calibre covariante no-mixing, foi necessário contornar o surgimento de uma estrutura pseudo-diferencial. Analisando a função de partição associada aos fótons livres de Podolsky via método dos parâmetros fictícios, percebemos o fato da simetria BRST assegurar que a função de partição não depende das escolhas covariantes ao fixarmos o calibre. As condições de Lorenz, no-mixing e Lorenz generalizado são amarradas pela simetria BRST e esse fato está contido em uma afirmação geral em teorias de calibre a temperatura finita, atribuída ao trabalho de Tyutin, de que a física não depende das escolhas de calibre, covariantes ou não, devido a simetria BRST. Por fim, com a funções de partição em mãos, construímos as equações de Schwinger-Dyson-Fradkin (SDF) e as identidades de Ward-Takahashi-Fradkin (WTF) em equilíbrio termodinâmico.
This work has as aim to explore the quantum dynamics of interaction between scalar and vectorial particles and to study the thermodynamic equilibrium of these particles in the gran-canonical ensemble. The dynamics of interaction, written in a covariance language, between the matter field (scalar) and the field that intermediate the interaction (vectorial) exhibit a local gauge symmetry, U(1) in a quantum case and SO(4) in a thermodynamic equilibrium. Therefore we divided the work into two sections. In the first section we analyze systematically the quantum interaction between the scalar particles (mesons) and vectorial particles (photons) in the context of the generalized scalar Duffin-Kemmer-Petiau quantum electrodynamics (GSDKP). For this we use the functional approach to quantize the theory. We built the hamiltonian structure by the Dirac methodology, utilize the Faddeev-Senjanovic procedure to obtain the transition amplitude in the generalized Coulomb gauge and the Faadeev-Popov-DeWitt method to write the covariant form of the previously amplitude in the no-mixing gauge condition. Then writing the functional generator by Schwinger, the Schwinger-Dyson (SD) equations and the Ward-Takahashi (WT) identities are obtained. As an introductory analysis to the first radiative corrections we make a quantitative calculus to see the types of ultraviolet (UV) superficial divergences that appear in the theory. After this we show an explicit calculation of the first radiative corrections (1-loop) associated with the photon propagator, meson propagator, vertex and the 4 point function (photon-photon) utilizing the dimensional regularization method, where the gauge symmetry is manifest. As we will see one of the consequences of the study is that the DKP algebra ensures the functioning of the WT identities in the first radiative corrections prohibiting certain UV divergences. With the knowledge of the UV divergences and de infrared (IR) addressed in the radiative corrections we established the multiplicative renormalization procedure to this theory in the mass shell. The fact that the meson propagator has a new divergence structure in terms of the Podolsky mass took us to analyze the radiative correction at 2-loops. With the photon propagator we define the polarization tensor and in a phenomenological manner, analyzing the asymptotic behavior of Green's functions for higher momentum, we derive the dependence of the structure constant by the scale of energy. In the second section we study the Matsubara-Fradkin (MF) formalism to describe fields in thermodynamical equilibrium. For this it was necessary to construct the equations in thermodynamic equilibrium that describe the scalar sector and vectorial sector and then extract the partition function. When we construct the vectorial sector we realize the emergence and the importance of the ghost fields and their connection to the Bechi-Rouet-Stora-Tyutin (BRST) symmetry. In the case of the no-mixing gauge condition was necessary to contour a pseudo-differential structure. Analyzing the free partition function associated with the free Podolsky photons by the method of fictitious parameters we realize that the BRST symmetry ensures that it does not depend of the covariant choices when we fix de gauge. The Lorenz condition, no-mixing and generalized Lorenz are tied by the BRST symmetry and this fact is contained in a general statement in gauge theories at finite temperature, assigned by Tyutin work, that the physics doesn't depend of the gauge choices, covariant or not, due to BRST symmetry. Lastly, with the partition function in hands, we construct the Schwinger-Dyson-Fradkin (SDF) and the Ward-Takahashi-Fradkin (WTF) in thermodynamic equilibrium.
This work has as aim to explore the quantum dynamics of interaction between scalar and vectorial particles and to study the thermodynamic equilibrium of these particles in the gran-canonical ensemble. The dynamics of interaction, written in a covariance language, between the matter field (scalar) and the field that intermediate the interaction (vectorial) exhibit a local gauge symmetry, U(1) in a quantum case and SO(4) in a thermodynamic equilibrium. Therefore we divided the work into two sections. In the first section we analyze systematically the quantum interaction between the scalar particles (mesons) and vectorial particles (photons) in the context of the generalized scalar Duffin-Kemmer-Petiau quantum electrodynamics (GSDKP). For this we use the functional approach to quantize the theory. We built the hamiltonian structure by the Dirac methodology, utilize the Faddeev-Senjanovic procedure to obtain the transition amplitude in the generalized Coulomb gauge and the Faadeev-Popov-DeWitt method to write the covariant form of the previously amplitude in the no-mixing gauge condition. Then writing the functional generator by Schwinger, the Schwinger-Dyson (SD) equations and the Ward-Takahashi (WT) identities are obtained. As an introductory analysis to the first radiative corrections we make a quantitative calculus to see the types of ultraviolet (UV) superficial divergences that appear in the theory. After this we show an explicit calculation of the first radiative corrections (1-loop) associated with the photon propagator, meson propagator, vertex and the 4 point function (photon-photon) utilizing the dimensional regularization method, where the gauge symmetry is manifest. As we will see one of the consequences of the study is that the DKP algebra ensures the functioning of the WT identities in the first radiative corrections prohibiting certain UV divergences. With the knowledge of the UV divergences and de infrared (IR) addressed in the radiative corrections we established the multiplicative renormalization procedure to this theory in the mass shell. The fact that the meson propagator has a new divergence structure in terms of the Podolsky mass took us to analyze the radiative correction at 2-loops. With the photon propagator we define the polarization tensor and in a phenomenological manner, analyzing the asymptotic behavior of Green's functions for higher momentum, we derive the dependence of the structure constant by the scale of energy. In the second section we study the Matsubara-Fradkin (MF) formalism to describe fields in thermodynamical equilibrium. For this it was necessary to construct the equations in thermodynamic equilibrium that describe the scalar sector and vectorial sector and then extract the partition function. When we construct the vectorial sector we realize the emergence and the importance of the ghost fields and their connection to the Bechi-Rouet-Stora-Tyutin (BRST) symmetry. In the case of the no-mixing gauge condition was necessary to contour a pseudo-differential structure. Analyzing the free partition function associated with the free Podolsky photons by the method of fictitious parameters we realize that the BRST symmetry ensures that it does not depend of the covariant choices when we fix de gauge. The Lorenz condition, no-mixing and generalized Lorenz are tied by the BRST symmetry and this fact is contained in a general statement in gauge theories at finite temperature, assigned by Tyutin work, that the physics doesn't depend of the gauge choices, covariant or not, due to BRST symmetry. Lastly, with the partition function in hands, we construct the Schwinger-Dyson-Fradkin (SDF) and the Ward-Takahashi-Fradkin (WTF) in thermodynamic equilibrium.
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Palavras-chave
Quantização funcional, Sistemas com vínculos, Formalismo de Matsubara-Fradkin, Functional quantization, Constrained systems, Matsubara-Fradkin formalism