Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

The Spin-Statistics Relation and Noncommutative Quantum Field Theory

The efforts of combining quantum theory with general relativity have been great and marked by several successes. One field where progress has lately been made is the study of noncommutative quantum field theories that arise as a low energy limit in certain string theories. The idea of noncommutativity comes naturally when combining these two extremes and has profound implications on results widely accepted in traditional, commutative, theories. In this work I review the status of one of the most important connections in physics, the spin-statistics relation. The relation is deeply ingrained in our reality in that it gives us the structure for the periodic table and is of crucial importance for the stability of all matter. The dramatic effects of noncommutativity of space-time coordinates, mainly the loss of Lorentz invariance, call the spin-statistics relation into question. The spin-statistics theorem is first presented in its traditional setting, giving a clarifying proof starting from minimal requirements. Next the notion of noncommutativity is introduced and its implications studied. The discussion is essentially based on twisted Poincaré symmetry, the space-time symmetry of noncommutative quantum field theory. The controversial issue of microcausality in noncommutative quantum field theory is settled by showing for the first time that the light wedge microcausality condition is compatible with the twisted Poincaré symmetry. The spin-statistics relation is considered both from the point of view of braided statistics, and in the traditional Lagrangian formulation of Pauli, with the conclusion that Pauli's age-old theorem stands even this test so dramatic for the whole structure of space-time.

Patient exposure monitoring and radiation qualities in two-dimensional digital x-ray imaging

The methods for estimating patient exposure in x-ray imaging are based on the measurement of radiation incident on the patient. In digital imaging, the useful dose range of the detector is large and excessive doses may remain undetected. Therefore, real-time monitoring of radiation exposure is important. According to international recommendations, the measurement uncertainty should be lower than 7% (confidence level 95%). The kerma-area product (KAP) is a measurement quantity used for monitoring patient exposure to radiation. A field KAP meter is typically attached to an x-ray device, and it is important to recognize the effect of this measurement geometry on the response of the meter. In a tandem calibration method, introduced in this study, a field KAP meter is used in its clinical position and calibration is performed with a reference KAP meter. This method provides a practical way to calibrate field KAP meters. However, the reference KAP meters require comprehensive calibration. In the calibration laboratory it is recommended to use standard radiation qualities. These qualities do not entirely correspond to the large range of clinical radiation qualities. In this work, the energy dependence of the response of different KAP meter types was examined. According to our findings, the recommended accuracy in KAP measurements is difficult to achieve with conventional KAP meters because of their strong energy dependence. The energy dependence of the response of a novel large KAP meter was found out to be much lower than with a conventional KAP meter. The accuracy of the tandem method can be improved by using this meter type as a reference meter. A KAP meter cannot be used to determine the radiation exposure of patients in mammography, in which part of the radiation beam is always aimed directly at the detector without attenuation produced by the tissue. This work assessed whether pixel values from this detector area could be used to monitor the radiation beam incident on the patient. The results were congruent with the tube output calculation, which is the method generally used for this purpose. The recommended accuracy can be achieved with the studied method. New optimization of radiation qualities and dose level is needed when other detector types are introduced. In this work, the optimal selections were examined with one direct digital detector type. For this device, the use of radiation qualities with higher energies was recommended and appropriate image quality was achieved by increasing the low dose level of the system.

Comprehensive Two-Dimensional Gas Chromatography: Instrumental and Methodological Development

Comprehensive two-dimensional gas chromatography (GC×GC) offers enhanced separation efficiency, reliability in qualitative and quantitative analysis, capability to detect low quantities, and information on the whole sample and its components. These features are essential in the analysis of complex samples, in which the number of compounds may be large or the analytes of interest are present at trace level. This study involved the development of instrumentation, data analysis programs and methodologies for GC×GC and their application in studies on qualitative and quantitative aspects of GC×GC analysis. Environmental samples were used as model samples. Instrumental development comprised the construction of three versions of a semi-rotating cryogenic modulator in which modulation was based on two-step cryogenic trapping with continuously flowing carbon dioxide as coolant. Two-step trapping was achieved by rotating the nozzle spraying the carbon dioxide with a motor. The fastest rotation and highest modulation frequency were achieved with a permanent magnetic motor, and modulation was most accurate when the motor was controlled with a microcontroller containing a quartz crystal. Heated wire resistors were unnecessary for the desorption step when liquid carbon dioxide was used as coolant. With use of the modulators developed in this study, the narrowest peaks were 75 ms at base. Three data analysis programs were developed allowing basic, comparison and identification operations. Basic operations enabled the visualisation of two-dimensional plots and the determination of retention times, peak heights and volumes. The overlaying feature in the comparison program allowed easy comparison of 2D plots. An automated identification procedure based on mass spectra and retention parameters allowed the qualitative analysis of data obtained by GC×GC and time-of-flight mass spectrometry. In the methodological development, sample preparation (extraction and clean-up) and GC×GC methods were developed for the analysis of atmospheric aerosol and sediment samples. Dynamic sonication assisted extraction was well suited for atmospheric aerosols collected on a filter. A clean-up procedure utilising normal phase liquid chromatography with ultra violet detection worked well in the removal of aliphatic hydrocarbons from a sediment extract. GC×GC with flame ionisation detection or quadrupole mass spectrometry provided good reliability in the qualitative analysis of target analytes. However, GC×GC with time-of-flight mass spectrometry was needed in the analysis of unknowns. The automated identification procedure that was developed was efficient in the analysis of large data files, but manual search and analyst knowledge are invaluable as well. Quantitative analysis was examined in terms of calibration procedures and the effect of matrix compounds on GC×GC separation. In addition to calibration in GC×GC with summed peak areas or peak volumes, simplified area calibration based on normal GC signal can be used to quantify compounds in samples analysed by GC×GC so long as certain qualitative and quantitative prerequisites are met. In a study of the effect of matrix compounds on GC×GC separation, it was shown that quality of the separation of PAHs is not significantly disturbed by the amount of matrix and quantitativeness suffers only slightly in the presence of matrix and when the amount of target compounds is low. The benefits of GC×GC in the analysis of complex samples easily overcome some minor drawbacks of the technique. The developed instrumentation and methodologies performed well for environmental samples, but they could also be applied for other complex samples.

Jumping numbers of a simple complete ideal in a two-dimensional regular local ring

The multiplier ideals of an ideal in a regular local ring form a family of ideals parametrized by non-negative rational numbers. As the rational number increases the corresponding multiplier ideal remains unchanged until at some point it gets strictly smaller. A rational number where this kind of diminishing occurs is called a jumping number of the ideal. In this manuscript we shall give an explicit formula for the jumping numbers of a simple complete ideal in a two dimensional regular local ring. In particular, we obtain a formula for the jumping numbers of an analytically irreducible plane curve. We then show that the jumping numbers determine the equisingularity class of the curve.

Noncommutative Quantum Field Theory : Problem of Time and Some Applications

In this thesis the current status and some open problems of noncommutative quantum field theory are reviewed. The introduction aims to put these theories in their proper context as a part of the larger program to model the properties of quantized space-time. Throughout the thesis, special focus is put on the role of noncommutative time and how its nonlocal nature presents us with problems. Applications in scalar field theories as well as in gauge field theories are presented. The infinite nonlocality of space-time introduced by the noncommutative coordinate operators leads to interesting structure and new physics. High energy and low energy scales are mixed, causality and unitarity are threatened and in gauge theory the tools for model building are drastically reduced. As a case study in noncommutative gauge theory, the Dirac quantization condition of magnetic monopoles is examined with the conclusion that, at least in perturbation theory, it cannot be fulfilled in noncommutative space.