Search

In this project we studied how Category Theory can be used in the formulation of Algebraic Quantum Field Theory in curved spacetimes and how the Reeh-Schlieder property translates to general curved spacetimes. Category Theory concepts such as functors, natural transformations and natural equivalences are used in the definition of a Locally Covariant Quantum Field Theory, that arose in a context in which it was of interest to generalize Axiomatic Quantum Field Theory to curved spacetimes taking into consideration the ideas of locality and covariance. In fact, a Locally Covariant Quantum Field Theory is defined as a covariant functor, which can be related to another Locally Covariant Quantum Field Theory by a natural transformation. The equivalence between theories then becomes clear if this natural transformation is an isomorphism. Furthermore, the Reeh-Schlieder theorem is of great significance in the realm of Quantum Field Theory, since it provides a great deal of properties for the vacuum state and it has relevance in justifying applications of Tomita-Takesaki modular theory in Quantum Field Theories. It has already been proven that states with a weak form of the Reeh-Schlieder property always exist in general curved spacetimes. This was accomplished using the spacetime deformation technique and assuming the time-slice axiom in a Locally Covariant Quantum Field Theory. Neste projeto estudamos como a Teoria de Categorias pode ser usada na formulação da Teoria Quântica de Campos Algébrica em espaços-tempos curvos e como a propriedade de Reeh-Schlieder é transportada para espaços-tempos curvos gerais. Conceitos da Teoria de Categorias como funtores, transformações naturais e equivalências naturais são usados na definição de uma Teoria Quântica de Campos Localmente Covariante, que surgiu em um contexto em que se tinha interesse em generalizar a Teoria Quântica de Campos Axiomática para espaços-tempos curvos levando em consideração as ideias de localidade e covariância. De fato, uma Teoria Quântica de Campos Localmente Covariante é definida como um funtor covariante, que pode ser relacionado com outra Teoria Quântica de Campos Localmente Covariante por meio de uma transformação natural. A equivalência entre teorias se torna clara se essa transformação natural é um isomorfismo. Ademais, o teorema de Reeh-Schlieder possui grande importância no contexto da Teoria Quântica de Campos, visto que ele fornece várias propriedades do estado de vácuo e possui relevância na justificativa para aplicar a teoria modular de Tomita-Takesaki em Teorias Quânticas de Campos. Já foi provado que estados com uma forma fraca da propriedade de Reeh-Schlieder sempre existem em espaços-tempos curvos gerais. Isso foi realizado por meio da técnica de deformação do espaço-tempo e assumindo o axioma da fatiação temporal em uma Teoria Quântica de Campos Localmente Covariante.

A presente dissertação revisa um modelo, de autoria de C. Dappiaggi, K. Fredenhagen e N. Pinamonti, de um campo escalar real quântico não-interagente acoplado com a métrica de um espaço-tempo FLRW (Friedmann-Lemaítre-Robertson-Walker). Apresentamos a metodologia de quantização de campos de Klein-Gordon reais em espaçostempos globalmente hiperbólicos e discorremos sobre o procedimento de regularização do tensor de energia-momento via point-splitting. Consideramos os campos em espaços FLRW e estados adiabáticos com flutuação média de campo dado por h2i = m2 +R, com ; constantes provenientes do procedimento de regularização. A retroação do campo quântico gera a equação diferencial para o parâmetro de Hubble H(t) dada por _H (H2H2 c ) = (H2H2+ )(H2H2 ) com Hc uma constante e H pontos críticos estáveis da equação. Esse simples modelo mostra que efeitos quânticos podem, por si só, fornecer fases de de Sitter estáveis sem adição de uma constante cosmológica a priori. Mesmo que de caráter apenas qualitativo, tal resultado indica que análises cautelosas de processos de quantização são importantes para análise de efeitos cosmológicos de teorias quânticas de campos em espaços curvos. The present dissertation reviews the coupling of a scalar non-interacting quantum field with the metric of a FLRW(Friedmann-Lemaítre-Robertson-Walker) spacetime, proposed in a work by C. Dappiaggi, K. Fredenhagen and N. Pinamonti. We present methods for the quantization of a real Klein-Gordon field in globally hyperbolic spacetimes and discuss procedures for the point-splitting regularization of the stress-energy tensor. We consider those fields in FLRWspacetimes and point out adiabatic states with mean field fluctuation given by h2i = m2+R, with ; being constants that emerge from the regularization procedure. The backreaction of the quantum field provides a diferential equation for the Hubble parameter given by _H (H2H2 c ) = (H2H2+)(H2H2) with Hc a constant and H stable critical points of the equation. In this way, this simple model demonstrates that quantum efects may, by themselves, exibit stable de Sitter phases even without an introduction of a cosmological constant by hand. Althoug in a qualitative way, such result shows that, when dealing with the backreaction issue, a careful analysis of the quantization procedures is important for the analysis of cosmological efects of models of quantum field theories in curved spacetimes.

In this work, we consider the energy-momentum spectrum of lattice quantum chromodynamics (QCD) models, using the functional integral formulation, with imaginary time and in the strong coupling regime. This regime corresponds to taking the hopping parameter κ > 0 satisfying 1 >> κ >> β > 0, where β ≡ g−2 0 is the coupling parameter among the gauge fields, here defined in the SU(3) Lie algebra. We consider QCD models with fermionic quark fields with one and two flavors (isospin). For the cases 2 + 1 and 3 + 1 dimensions, one flavor, and using 4 × 4 Dirac matrices, we show the existence of mesons, as bound states of one quark and one anti-quark, which is detected by the appearance of an isolated dispersion curve in the energy-momentum spectrum in the physical Hilbert space associated with an even number of fermions. The meson masses are asymptotically given by −2 ln κ. Similar results were also obtained for the model in 2 + 1 dimensions, with one and two flavors, and using Pauli matrices 2×2. Using results previously established about the existence of baryons in the subspace with an odd number of quarks, we consider the question of the existence of bound states of two hadrons: two mesons, and one baryon and one meson, in models with dimension 2 + 1, with one and two flavors, and Pauli matrices 2 × 2. Our analysis is strongly based in analytic properties of decay for correlation functions for two and four particles, obtained using an adaptation of the decoupling of hyperplane method. The analysis of bound states of two particles is made using a Bethe-Salpeter equation on the lattice in a ladder approximation (leading order in κ). We show that bound states of two mesons or one meson and one baryon do not exist in the one flavor case. Here, Pauli repulsion overcomes any kind of attraction between the particles. For the two flavor case, restricting our analysis to the total isospin zero sector, a bound state of the type meson-meson was detected, with binding energy given approximately by 0.02359κ. The attractive potential between the two mesons comes from two kinds of contributions: exchange of a (quasi-)meson (a quark and an anti-quark) and an energy-dependent potential. We also show that, the main source of attraction is a gauge correlation effect. Using technics already developed to treat this kind of physical system, our spectral results can be extended to a rigorous analysis, i.e. a spectral analysis beyond the ladder approximation for the full model.