Noncommutative Quantum Field Theories and UV/IR Mixing

Our present-day understanding of fundamental constituents of matter and their interactions is based on the Standard Model of particle physics, which relies on quantum gauge field theories. On the other hand, the large scale dynamical behaviour of spacetime is understood via the general theory of relativity of Einstein. The merging of these two complementary aspects of nature, quantum and gravity, is one of the greatest goals of modern fundamental physics, the achievement of which would help us understand the short-distance structure of spacetime, thus shedding light on the events in the singular states of general relativity, such as black holes and the Big Bang, where our current models of nature break down. The formulation of quantum field theories in noncommutative spacetime is an attempt to realize the idea of nonlocality at short distances, which our present understanding of these different aspects of Nature suggests, and consequently to find testable hints of the underlying quantum behaviour of spacetime. The formulation of noncommutative theories encounters various unprecedented problems, which derive from their peculiar inherent nonlocality. Arguably the most serious of these is the so-called UV/IR mixing, which makes the derivation of observable predictions especially hard by causing new tedious divergencies, to which our previous well-developed renormalization methods for quantum field theories do not apply. In the thesis I review the basic mathematical concepts of noncommutative spacetime, different formulations of quantum field theories in the context, and the theoretical understanding of UV/IR mixing. In particular, I put forward new results to be published, which show that also the theory of quantum electrodynamics in noncommutative spacetime defined via Seiberg-Witten map suffers from UV/IR mixing. Finally, I review some of the most promising ways to overcome the problem. The final solution remains a challenge for the future.

On the Geometry of Infinite-Dimensional Grassmannian Manifolds and Gauge Theory

This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.

Mathematical models on the impact of noise and dyadic molecular structures on the properties of a cardiac myocyte

In cardiac myocytes (heart muscle cells), coupling of electric signal known as the action potential to contraction of the heart depends crucially on calcium-induced calcium release (CICR) in a microdomain known as the dyad. During CICR, the peak number of free calcium ions (Ca) present in the dyad is small, typically estimated to be within range 1-100. Since the free Ca ions mediate CICR, noise in Ca signaling due to the small number of free calcium ions influences Excitation-Contraction (EC) coupling gain. Noise in Ca signaling is only one noise type influencing cardiac myocytes, e.g., ion channels playing a central role in action potential propagation are stochastic machines, each of which gates more or less randomly, which produces gating noise present in membrane currents. How various noise sources influence macroscopic properties of a myocyte, how noise is attenuated and taken advantage of are largely open questions. In this thesis, the impact of noise on CICR, EC coupling and, more generally, macroscopic properties of a cardiac myocyte is investigated at multiple levels of detail using mathematical models. Complementarily to the investigation of the impact of noise on CICR, computationally-efficient yet spatially-detailed models of CICR are developed. The results of this thesis show that (1) gating noise due to the high-activity mode of L-type calcium channels playing a major role in CICR may induce early after-depolarizations associated with polymorphic tachycardia, which is a frequent precursor to sudden cardiac death in heart failure patients; (2) an increased level of voltage noise typically increases action potential duration and it skews distribution of action potential durations toward long durations in cardiac myocytes; and that (3) while a small number of Ca ions mediate CICR, Excitation-Contraction coupling is robust against this noise source, partly due to the shape of ryanodine receptor protein structures present in the cardiac dyad.

Mathematical Aspects of Financial Markets with Frictions

Frictions are factors that hinder trading of securities in financial markets. Typical frictions include limited market depth, transaction costs, lack of infinite divisibility of securities, and taxes. Conventional models used in mathematical finance often gloss over these issues, which affect almost all financial markets, by arguing that the impact of frictions is negligible and, consequently, the frictionless models are valid approximations. This dissertation consists of three research papers, which are related to the study of the validity of such approximations in two distinct modeling problems. Models of price dynamics that are based on diffusion processes, i.e., continuous strong Markov processes, are widely used in the frictionless scenario. The first paper establishes that diffusion models can indeed be understood as approximations of price dynamics in markets with frictions. This is achieved by introducing an agent-based model of a financial market where finitely many agents trade a financial security, the price of which evolves according to price impacts generated by trades. It is shown that, if the number of agents is large, then under certain assumptions the price process of security, which is a pure-jump process, can be approximated by a one-dimensional diffusion process. In a slightly extended model, in which agents may exhibit herd behavior, the approximating diffusion model turns out to be a stochastic volatility model. Finally, it is shown that when agents' tendency to herd is strong, logarithmic returns in the approximating stochastic volatility model are heavy-tailed. The remaining papers are related to no-arbitrage criteria and superhedging in continuous-time option pricing models under small-transaction-cost asymptotics. Guasoni, Rásonyi, and Schachermayer have recently shown that, in such a setting, any financial security admits no arbitrage opportunities and there exist no feasible superhedging strategies for European call and put options written on it, as long as its price process is continuous and has the so-called conditional full support (CFS) property. Motivated by this result, CFS is established for certain stochastic integrals and a subclass of Brownian semistationary processes in the two papers. As a consequence, a wide range of possibly non-Markovian local and stochastic volatility models have the CFS property.

Using Mathematical Morphology for Geometric Music Retrieval

The usual task in music information retrieval (MIR) is to find occurrences of a monophonic query pattern within a music database, which can contain both monophonic and polyphonic content. The so-called query-by-humming systems are a famous instance of content-based MIR. In such a system, the user's hummed query is converted into symbolic form to perform search operations in a similarly encoded database. The symbolic representation (e.g., textual, MIDI or vector data) is typically a quantized and simplified version of the sampled audio data, yielding to faster search algorithms and space requirements that can be met in real-life situations. In this thesis, we investigate geometric approaches to MIR. We first study some musicological properties often needed in MIR algorithms, and then give a literature review on traditional (e.g., string-matching-based) MIR algorithms and novel techniques based on geometry. We also introduce some concepts from digital image processing, namely the mathematical morphology, which we will use to develop and implement four algorithms for geometric music retrieval. The symbolic representation in the case of our algorithms is a binary 2-D image. We use various morphological pre- and post-processing operations on the query and the database images to perform template matching / pattern recognition for the images. The algorithms are basically extensions to classic image correlation and hit-or-miss transformation techniques used widely in template matching applications. They aim to be a future extension to the retrieval engine of C-BRAHMS, which is a research project of the Department of Computer Science at University of Helsinki.

A Historical Approach to Children s Physics Education : Modelling of DC-circuit Phenomena in a Small Group

This three-phase design research describes the modelling processes for DC-circuit phenomena. The first phase presents an analysis of the development of the DC-circuit historical models in the context of constructing Volta s pile at the turn of the 18th century. The second phase involves the designing of a teaching experiment for comprehensive school third graders. Among other considerations, the design work utilises the results of the first phase and research literature of pupils mental models for DC-circuit phenomena. The third phase of the research was concerned with the realisation of the planned teaching experiment. The aim of this phase was to study the development of the external representations of DC-circuit phenomena in a small group of third graders. The aim of the study has been to search for new ways to guide pupils to learn DC-circuit phenomena while emphasing understanding at the qualitative level. Thus, electricity, which has been perceived as a difficult and abstract subject, could be learnt more comprehensively. Especially, the research of younger pupils learning of electricity concepts has not been of great interest at the international level, although DC-circuit phenomena are also taught in the lower classes of comprehensive schools. The results of this study are important, because there has tended to be more teaching of natural sciences in the lower classes of comprehensive schools, and attempts are being made to develop this trend in Finland. In the theoretical part of the research an Experimental-centred representation approach, which emphasises the role of experimentalism in the development of pupil s representations, is created. According to this approach learning at the qualitative level consists of empirical operations like experimenting, observations, perception, and prequantification of nature phenomena, and modelling operations like explaining and reasoning. Besides planning teaching, the new approach can be used as an analysis tool in describing both historical modelling and the development of pupils representations. In the first phase of the study, the research question was: How did the historical models of DC-circuit phenomena develop in Volta s time? The analysis uncovered three qualitative historical models associated with the historical concept formation process. The models include conceptions of the electric circuit as a scene in the DC-circuit phenomena, the comparative electric-current phenomenon as a cause of different observable effect phenomena, and the strength of the battery as a cause of the electric-current phenomenon. These models describe the concept formation process and its phases in Volta s time. The models are portrayed in the analysis using fragments of the models, where observation-based fragments and theoretical fragements are distinguished from each other. The results emphasise the significance of the qualitative concept formation and the meaning of language in the historical modelling of DC-circuit phenomena. For this reason these viewpoints are stressed in planning the teaching experiment in the second phase of the research. In addition, the design process utilised the experimentation behind the historical models of DC-circuit phenomena In the third phase of the study the research question is as follows: How will the small group s external representations of DC-circuit phenomena develop during the teaching experiment? The main question is divided into the following two sub questions: What kind of talk exists in the small group s learning? What kinds of external representations for DC-circuit phenomena exist in the small group discourse during the teaching experiment? The analysis revealed that the teaching experiment of the small group succeeded in its aim to activate talk in the small group. The designed connection cards proved especially successful in activating talk. The connection cards are cards that represent the components of the electric circuit. In the teaching experiment the pupils constructed different connections with the connection cards and discussed, what kinds of DC-circuit phenomena would take place in the corresponding real connections. The talk of the small group was analysed by comparing two situations, firstly, when the small group discussed using connections made with the connection cards and secondly with the same connections using real components. According to the results the talk of the small group included more higher-order thinking when using the connection cards than with similar real components. In order to answer the second sub question concerning the small group s external representations that appeared in the talk during the teaching experiment; student talk was visualised by the fragment maps which incorporate the electric circuit, the electric current and the source voltage. The fragment maps represent the gradual development of the external representations of DC-circuit phenomena in the small group during the teaching experiment. The results of the study challenge the results of previous research into the abstractness and difficulty of electricity concepts. According to this research, the external representations of DC-circuit phenomena clearly developed in the small group of third graders. Furthermore, the fragment maps uncover that although the theoretical explanations of DC-circuit phenomena, which have been obtained as results of typical mental model studies, remain undeveloped, learning at the qualitative level of understanding does take place.

Studies of electronic structure by x-ray Raman and emission spectroscopy: applications to MgB2 superconductors and high pressure physics

X-ray Raman scattering and x-ray emission spectroscopies were used to study the electronic properties and phase transitions in several condensed matter systems. The experimental work, carried out at the European Synchrotron Radiation Facility, was complemented by theoretical calculations of the x-ray spectra and of the electronic structure. The electronic structure of MgB2 at the Fermi level is dominated by the boron σ and π bands. The high density of states provided by these bands is the key feature of the electronic structure contributing to the high critical temperature of superconductivity in MgB2. The electronic structure of MgB2 can be modified by atomic substitutions, which introduce extra electrons or holes into the bands. X ray Raman scattering was used to probe the interesting σ and π band hole states in pure and aluminum substituted MgB2. A method for determining the final state density of electron states from experimental x-ray Raman scattering spectra was examined and applied to the experimental data on both pure MgB2 and on Mg(0.83)Al(0.17)B2. The extracted final state density of electron states for the pure and aluminum substituted samples revealed clear substitution induced changes in the σ and π bands. The experimental work was supported by theoretical calculations of the electronic structure and x-ray Raman spectra. X-ray emission at the metal Kβ line was applied to the studies of pressure and temperature induced spin state transitions in transition metal oxides. The experimental studies were complemented by cluster multiplet calculations of the electronic structure and emission spectra. In LaCoO3 evidence for the appearance of an intermediate spin state was found and the presence of a pressure induced spin transition was confirmed. Pressure induced changes in the electronic structure of transition metal monoxides were studied experimentally and were analyzed using the cluster multiplet approach. The effects of hybridization, bandwidth and crystal field splitting in stabilizing the high pressure spin state were discussed. Emission spectroscopy at the Kβ line was also applied to FeCO3 and a pressure induced iron spin state transition was discovered.