Quantum field theories are very good at describing the world around us but use complicated computations that cannot always be solved exactly. Introducing conformal symmetry to quantum field theory can reduce this complexity and allow for quite simple calculation in the best case. This report aims to describe the critical part of the Ising model in 2 dimensions using conformal field theory while assuming only some knowledge of quantum mechanics and complex analysis from the reader. This is done by using the book Conformal Field Theory as the source for information about conformal field theory. / Kvantfältteorier är mycket bra på att beskriva verkligheten runt om oss men de använder sig av avancerade beräkningar som inte alltid kan lösas exakt. Genom att ge systemet konform symmetri så kan dessa avancerade beräkningar förenklas och bli ganska enkla i de bästa fallen. Målet med denna rapport är att beskriva hur en modell som kallas för "Ising model" kan beskrivas i sitt kritiska tillstånd med hjälp utav konform fältteori. Läsaren antas kunna kvantmekanik samt komplex analys. Informationen om konform fältteori hämtas ifrån boken Conformal Field Theory

The aim of this thesis is to examine the hard $S$-matrix for QED and QCD, due to Hannesdottir and Schwartz [1], both recreating known results, and finding new results in lower dimensions. Electrons by themselves are not gauge invariant, and in the massless limit do not give finite $S$-matrix elements for individual processes; one needs to add together seemingly arbitrary combinations of diagrams in order to be able to calculate the scattering cross-section. One remedy is to exchange the bare electron for a dressed state, or equivalently to include the dressing in the $S$-matrix itself. This gives rise to a formalism in which one needs only consider time cuts of an individual diagram in order to find finite $S$-matrix elements. The main result of this thesis is the examination of the results which are found by Hannesdottir and Schwartz in a model system of fewer dimensions. This may be of use for toy model calculations in QCD, where 2+1-dimensional QED better models certain phenomena than does 3+1-dimensional QED.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24410号 / 理博第4909号 / 新制||理||1701(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)准教授 福間 將文, 教授 橋本 幸士, 准教授 吉岡 興一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Charged or rotating black holes possess an inner horizon beyond which determinism is lost. However, the strong cosmic censorship conjecture claims that even small perturbations will turn the horizon into a singularity beyond which the spacetime is inextendible, preventing the loss of determinism. Motivated by this conjecture, this dissertation studies free scalar quantum fields on various black-hole spacetimes to test whether quantum effects can lead to the formation of a singularity at the inner horizon in cases where classical perturbations cannot. The starting point is the investigation of the behaviour of real-scalar-field observables near the inner horizon of Reissner-Nordström-de Sitter spacetimes. Using semi-analytical methods, we find that quantum effects can indeed uphold the censorship conjecture. Subsequently, we consider charged scalar fields on the same spacetime and observe that a first-principle calculation is essential to accurately describe the quantum effects at the inner horizon. As a first step towards an extension of these results to rotating black holes, we rigorously construct the Unruh state for the real scalar field on slowly rotating Kerr-de Sitter spacetimes. We show that it is a well-defined Hadamard state and can therefore be used to compute expectation values of the stressenergy tensor and other non-linear observables.:1 Introduction 7 2 An introduction to quantum fields and black holes 13 2.1 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 A brief introduction to AQFT . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 An introduction to microlocal analysis . . . . . . . . . . . . . . . . . . . 24 2.4 An introduction to black-hole spacetimes . . . . . . . . . . . . . . . . . 28 2.4.1 The Reissner-Nordström-de Sitter spacetime . . . . . . . . . . . 28 2.4.2 The Kerr-de Sitter spacetime . . . . . . . . . . . . . . . . . . . . 32 2.5 Free scalar fields in black-hole spacetimes . . . . . . . . . . . . . . . . . 37 3 Computing the energy flux of the real scalar field 43 3.1 Strong cosmic censorship on RNdS . . . . . . . . . . . . . . . . . . . . 43 3.2 The Klein-Gordon equation on RNdS . . . . . . . . . . . . . . . . . . . 45 3.3 Extension to the charged scalar field on RNdS . . . . . . . . . . . . . . . 52 3.4 The energy flux at the Cauchy horizon . . . . . . . . . . . . . . . . . . . 53 4 The charged scalar field in Reissner-Nordström-de Sitter 63 4.1 The Unruh state for the charged scalar field . . . . . . . . . . . . . . . . 65 4.2 The renormalized current . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3 The current in the Unruh state - numerical results . . . . . . . . . . . . . 80 4.4 The charged scalar field at the inner horizon . . . . . . . . . . . . . . . . 86 5 The Unruh state on Kerr-de Sitter 97 5.1 Null geodesics in the Kerr-de Sitter spacetime . . . . . . . . . . . . . . . 98 5.2 The Unruh state on Kerr-de Sitter . . . . . . . . . . . . . . . . . . . . . . 107 5.3 The Hadamard property of the Unruh state . . . . . . . . . . . . . . . . . 120 5.3.1 The Hadamard condition in O . . . . . . . . . . . . . . . . . . . 123 5.3.2 The Hadamard condition on M\O . . . . . . . . . . . . . . . . . 128 6 Summary and discussion 139 A Bibliography 143

We consider the Casimir effect for a scalar field interacting with another scalar field that is confined to two half spaces. This model is aimed to mimic the interaction of the photon field with matter in two slabs. We use Dirichlet boundary conditions on the interfaces for the fields in the half spaces and calculate their one-loop contribution to the wave equation for the other field. We perform the ultraviolet renormalization and develop a convenient formalism for the calculation of the vacuum energy in this configuration.

BSCCO-2212 is a high-temperature cuprate superconductor whose microscopic behavior is currently poorly understood. In particular, it is unclear whether its order parameter is consistent with s-wave or d-wave symmetry. It has been suggested that its order parameter might take one of several forms that are consistent with d-wave behavior. We present some calculations using the many-body theory approach to superconductivity that suggest that such order parameters would lead to a suppression of the critical temperature in the presence of impurities. Because some experiments have suggested the critical temperature of BSCCO-2212 is relatively independent of the concentration of impurities, this lends support to the hypothesis that its order parameter has s-wave symmetry.

In the first part of this thesis we study the generalization of the recent algebraic approach to classical field theory by proposing a more general setting based on the manifold of smooth sections of a non-trivial fiber bundle. Central is the notion of observables/functionals over such sections, i.e. appropriate smooth functions on them. The kinematic will be further specified by means of Peierls brackets, which in turn are defined via the causal propagators of linearized field equations. In the second part we implement deformation quantization of the algebras obtained previously in the simpler setting of scalar field theory. Wick powers and time ordered products for quantum field theories in curved spacetimes are defined by giving a set of axioms which, when implemented, defines uniquely, up to some classifiable ambiguities, the aforementioned quantities. Those ambiguities are known to be tightly restrained by locality, covariance and other regularity conditions. One of the additional constraints used was to require continuous and analytic dependence on the metric and coupling parameters. It was recently shown that this rather strong requirement could be weakened, in the case of Wick powers, to the so-called parametrized microlocal spectrum condition. We therefore show the existence of Wick powers satisfying the above condition and extend this axiom to time ordered products, while reestablishing the usual uniqueness and existence results in light of the new constraint.

No / We analyze the possibility and efficiency of nonholonomic control over quantum devices with exponentially large number of Hilbert space dimensions. We show that completely controllable devices of this type can be assembled from elementary units of arbitrary physical nature, and can be employed efficiently for universal quantum computations and simulation of quantum-field dynamics. As an example we describe a toy device that can perform Toffoli-gate transformations and discrete Fourier transform on 9 qubits.

Fedosov has described a geometro-algebraic method to construct in a canonical way a deformation of the Poisson algebra associated with a finite-dimensional symplectic manifold (\\\"phase space\\\"). His algorithm gives a non-commutative, but associative, product (a so-called \\\"star-product\\\") between smooth phase space functions parameterized by Planck\\\'s constant ℏ, which is treated as a deformation parameter. In the limit as ℏ goes to zero, the star product commutator goes to ℏ times the Poisson bracket, so in this sense his method provides a quantization of the algebra of classical observables. In this work, we develop a generalization of Fedosov\\\'s method which applies to the infinite-dimensional symplectic \\\"manifolds\\\" that occur in Lagrangian field theories. We show that the procedure remains mathematically well-defined, and we explain the relationship of this method to more standard perturbative quantization schemes in quantum field theory.

This thesis considers the problem of determining Casimir-Lifshitz forces in inhomogeneous media. The ground-state energy of the electromagnetic field in a piston-geometry is discussed. When the cavity is empty, the Casimir pressure on the piston is finite and independent of the small-scale physics of the media that compose the mirrors. However, it is demonstrated that, when the cavity is filled with an inhomogeneous dielectric medium, the Casimir energy is cut-off dependent. The local behavior of the stress tensor commonly used in calculations of Casimir forces is also determined. It is shown that the usual expression for the stress tensor is not finite anywhere within such a medium, whatever the temporal dispersion or index profile, and that this divergence is unlikely to be removed by modifying the regularisation. These findings suggest that the value of the Casimir pressure may be inextricably dependent on the detailed behavior of the mirror and the medium at large wave vectors. This thesis also examines two exceptions to this rule: first, the case of an idealised metamaterial is considered which, when introduced into a cavity, reduces the magnitude of the Casimir force. It is shown that, although the medium is inhomogeneous, it does not contribute additional scattering events but simply modifies the effective length of the cavity, so the predicted force is finite and can be stated exactly. Secondly, a geometric argument is presented for determining a Casimir stress in a spherical mirror filled with the inhomogeneous medium of Maxwell's fish-eye. This solution questions the idea that the Casimir force of a spherical mirror is repulsive, but prompts additional questions concerning regularisation and the role of non-local effects in determining Casimir forces.