Topological Quantum Computation - An Analysis of an Anyon Model Based on Quantum Double Symmetries

There exists various suggestions for building a functional and a fault-tolerant large-scale quantum computer. Topological quantum computation is a more exotic suggestion, which makes use of the properties of quasiparticles manifest only in certain two-dimensional systems. These so called anyons exhibit topological degrees of freedom, which, in principle, can be used to execute quantum computation with intrinsic fault-tolerance. This feature is the main incentive to study topological quantum computation. The objective of this thesis is to provide an accessible introduction to the theory. In this thesis one has considered the theory of anyons arising in two-dimensional quantum mechanical systems, which are described by gauge theories based on so called quantum double symmetries. The quasiparticles are shown to exhibit interactions and carry quantum numbers, which are both of topological nature. Particularly, it is found that the addition of the quantum numbers is not unique, but that the fusion of the quasiparticles is described by a non-trivial fusion algebra. It is discussed how this property can be used to encode quantum information in a manner which is intrinsically protected from decoherence and how one could, in principle, perform quantum computation by braiding the quasiparticles. As an example of the presented general discussion, the particle spectrum and the fusion algebra of an anyon model based on the gauge group S_3 are explicitly derived. The fusion algebra is found to branch into multiple proper subalgebras and the simplest one of them is chosen as a model for an illustrative demonstration. The different steps of a topological quantum computation are outlined and the computational power of the model is assessed. It turns out that the chosen model is not universal for quantum computation. However, because the objective was a demonstration of the theory with explicit calculations, none of the other more complicated fusion subalgebras were considered. Studying their applicability for quantum computation could be a topic of further research.

Topological stability through tame retractions

A smooth map is said to be stable if small perturbations of the map only differ from the original one by a smooth change of coordinates. Smoothly stable maps are generic among the proper maps between given source and target manifolds when the source and target dimensions belong to the so-called nice dimensions, but outside this range of dimensions, smooth maps cannot generally be approximated by stable maps. This leads to the definition of topologically stable maps, where the smooth coordinate changes are replaced with homeomorphisms. The topologically stable maps are generic among proper maps for any dimensions of source and target. The purpose of this thesis is to investigate methods for proving topological stability by constructing extremely tame (E-tame) retractions onto the map in question from one of its smoothly stable unfoldings. In particular, we investigate how to use E-tame retractions from stable unfoldings to find topologically ministable unfoldings for certain weighted homogeneous maps or germs. Our first results are concerned with the construction of E-tame retractions and their relation to topological stability. We study how to construct the E-tame retractions from partial or local information, and these results form our toolbox for the main constructions. In the next chapter we study the group of right-left equivalences leaving a given multigerm f invariant, and show that when the multigerm is finitely determined, the group has a maximal compact subgroup and that the corresponding quotient is contractible. This means, essentially, that the group can be replaced with a compact Lie group of symmetries without much loss of information. We also show how to split the group into a product whose components only depend on the monogerm components of f. In the final chapter we investigate representatives of the E- and Z-series of singularities, discuss their instability and use our tools to construct E-tame retractions for some of them. The construction is based on describing the geometry of the set of points where the map is not smoothly stable, discovering that by using induction and our constructional tools, we already know how to construct local E-tame retractions along the set. The local solutions can then be glued together using our knowledge about the symmetry group of the local germs. We also discuss how to generalize our method to the whole E- and Z- series.

Non-Gaussianities and Preheating from N-flation and Magnetogenesis via rotating cosmic string loops

In this thesis we examine multi-field inflationary models of the early Universe. Since non-Gaussianities may allow for the possibility to discriminate between models of inflation, we compute deviations from a Gaussian spectrum of primordial perturbations by extending the delta-N formalism. We use N-flation as a concrete model; our findings show that these models are generically indistinguishable as long as the slow roll approximation is still valid. Besides computing non-Guassinities, we also investigate Preheating after multi-field inflation. Within the framework of N-flation, we find that preheating via parametric resonance is suppressed, an indication that it is the old theory of preheating that is applicable. In addition to studying non-Gaussianities and preheatng in multi-field inflationary models, we study magnetogenesis in the early universe. To this aim, we propose a mechanism to generate primordial magnetic fields via rotating cosmic string loops. Magnetic fields in the micro-Gauss range have been observed in galaxies and clusters, but their origin has remained elusive. We consider a network of strings and find that rotating cosmic string loops, which are continuously produced in such networks, are viable candidates for magnetogenesis with relevant strength and length scales, provided we use a high string tension and an efficient dynamo.

Konstruktioner i interaktion. : de e som resurs i samtal

This study concerns the most common word pair in spoken Swedish, de e (it is, third person pronoun + copula-verb in present tense). The aim of the study is twofold, with an empirical aim and a theoretical aim. The empirical aim is to investigate if and how the string de e can be understood and described as a construction in its own right with characteristics that distinguishes it from other structures and resources in spoken Swedish. The theoretical aim is to test how two different linguistic theories and methods, interactional linguistics and construction grammar, can be combined and used to describe and explain patterns in languaging that traditional grammar does take into account. The empirical analysis is done within the interactional linguistic framework with sequence analyses of excerpts from authentic conversation data. The data consists of approximately ten hours of recorded conversation from Finland and Sweden. The sequence analysis suggests that the string de e really is used as a resource in its own right. In most cases, the string is also used in ways consistent with abstract grammatical patterns described by traditional grammar. Nevertheless, there are instances where de e is used in ways not described before: with numerals and infinitive phrases as complements, without any complements at all and together with certain complements (bra, de) in idiomatic ways. Furthermore, in the instances where de e is used according to known grammatical patterns the function of the particular string de e is clearly contextually specific and in various ways linked to the micro-context in which it is used. A new model is suggested for understanding and concluding the results from the sequence analyses. It consists of two different types of constructions grammatical and interactional. The grammatical constructions show how the string is used in eleven structurally different ways. The interactional constructions show seven different sequential positions and functions in which the string occurs. The two types of constructions are also linked to each other as potentials. This is a new way to describe how interactants use and responds to a concrete string like de e in conversation.

Koko talo soi : Klemetti-Opisto suomalaisen musiikkikulttuurin kehittäjänä 1953-1968

This study aims to examine the operations and significance of the Klemetti Institute (Klemetti-Opisto) as a developer of Finnish music culture from 1953 to 1968 during the term of office of the Institute s founder and first director, Arvo Vainio. The Klemetti Institute was originally established as a choir institute, but soon expanded to offer a wide range of music courses. In addition to providing courses for choir leaders and singers, the Institute began its orchestral activities as early as the mid-1950s. Other courses included ear training seminars as well as courses for young people s music instructors and in playing the kantele (a Finnish string instrument) and solo singing. More than 20 types of courses were offered over the 16-year period. The Klemetti Institute s courses were incorporated into the folk high school courses offered by the Orivesi Institute (Oriveden Opisto) and were organised during the summer months of June and July. In addition to funding based on the Folk High School Act, financial assistance was obtained from various foundations and funds, such as the Wihuri Foundation. This study is linked to the context of historical research. I examine the Klemetti Institute s operations chronologically, classifying instruction into different course types, and analyse concert activities primarily in the section on the Institute s student union. The source material includes the Klemetti Institute archives, which consist of Arvo Vainio s correspondence, student applications, register books and cards, journals and student lists, course albums and nearly all issues of the Klemettiläinen bulletin. In addition, I have used focused interviews and essays to obtain extensive data from students and teachers. I concentrate on primary school teachers, who accounted for the majority of course participants. A total of more than 2,300 people participated in the courses, nearly half of whom took courses during at least two summers. Primary school teachers accounted for 50% to 70% of the participants in most courses and constituted an even larger share of participants in some courses, such as the music instructor course. The Klemetti Institute contributed to the expansion throughout Finland of a new ideal for choral tone. This involved delicate singing which strives for tonal purity and expressiveness. Chamber choirs had been virtually unheard of in Finland, but the Klemetti Institute Chamber Choir popularised them. Chamber choirs are characterised by an extensive singing repertoire ranging from the Middle Ages to the present. As the name suggests, chamber choirs were originally rather small mixed choirs. Delicate singing meant the avoidance of extensive vibrato techniques and strong, heavy forte sounds, which had previously been typical of Finnish choirs. Those opposing and shunning this new manner of singing called it ghost singing . The Klemetti Institute s teachers included Finland s most prominent pedagogues and artists. As the focused essays, or reminiscences as I call them, show, their significance for the students was central. I examine extensively the Klemetti Institute s enthusiastic atmosphere, which during the early years was characterised by what some writers described as a hunger for music . In addition to distributing a new tonal ideal and choir repertoire, the Klemetti Institute also distributed new methods of music education, thus affecting the music teaching of Finnish primary schools, in particular. The Orff approach, which included various instruments, became well known, although some of Orff s ideas, such as improvisation and physical exercise, were initially unfamiliar. More important than the Orff approach was the in-depth teaching at the Klemetti Institute of the Hungarian ear training method known as the Kodály method. Many course participants were among those launching specialist music classes in schools, and the method became the foundation for music teaching in many such schools. The Klemetti Institute was also a pioneer in organising orchestra camps for young people. The Klemetti Institute promoted Finnish music culture and played an important role in the continuing music education of primary school teachers. Keywords: adult education, Grundtvigian philosophy, popular enlightenment, Klemetti Institute, Kodály method, choir singing, choir conducting, music history, music education, music culture, music camp, Orff approach, Orff-Schulwerk, Orivesi Institute, instrument teaching, free popular education, communality, solo singing, voice production

Movement-associated proteins of Potato virus A: attachment to virus particles and phosphorylation

The particles of Potato virus A (PVA; genus Potyvirus) are helically constructed filaments that contain multiple copies of a single type of coat-protein (CP) subunit and a single copy of genome-linked protein (VPg), attached to one end of the virion. Examination of negatively-stained virions by electron microscopy revealed flexuous, rod-shaped particles with no obvious terminal structures. It is known that particles of several filamentous plant viruses incorporate additional minor protein components, forming stable complexes that mediate particle disassembly, movement or transmission by insect vectors. The first objective of this work was to study the interaction of PVA movement-associated proteins with virus particles and how these interactions contribute to the morphology and function of the virus particles. Purified particles of PVA were examined by atomic force microscopy (AFM) and immuno-gold electron microscopy. A protrusion was found at one end of some of the potyvirus particles, associated with the 5' end of the viral RNA. The tip contained two virus-encoded proteins, the genome-linked protein (VPg) and the helper-component proteinase (HC-Pro). Both are required for cell-to-cell movement of the virus. Biochemical and electron microscopy studies of purified PVA samples also revealed the presence of another protein required for cell-to-cell movement the cylindrical inclusion protein (CI), which is also an RNA helicase/ATPase. Centrifugation through a 5-40% sucrose gradient separated virus particles with no detectable CI to a fraction that remained in the gradient, from the CI-associated particles that went to the pellet. Both types of particles were infectious. AFM and translation experiments demonstrated that when the viral CI was not present in the sample, PVA virions had a beads-on-a-string phenotype, and RNA within the virus particles was more accessible to translation. The second objective of this work was to study phosphorylation of PVA movement-associated and structural proteins (CP and VPg) in vitro and, if possible, in vivo. PVA virion structural protein CP is necessary for virus cell-to-cell movement. The tobacco protein kinase CK2 was identified as a kinase phosphorylating PVA CP. A major site of CK2 phosphorylation in PVA CP was identified as a single threonine within a CK2 consensus sequence. Amino acid substitutions affecting the CK2 consensus sequence in CP resulted in viruses that were defective in cell-to-cell and long-distance movement. The CK2 regulation of virion assembly and cell-to-cell movement by phosphorylation of CP was possibly due to the inhibition of CP binding to viral RNA. Four putative phosphorylation sites were identified from an in vitro phosphorylated recombinant VPg. All four were mutated and the spread of mutant viruses in two different host plants was studied. Two putative phosphorylation site mutants (Thr45 and Thr49) had phenotypes identical to that of a wild type (WT) virus infection in both Nicotiana benthamiana and N. tabacum plants. The other two mutant viruses (Thr132/Ser133 and Thr168) showed different phenotypes with increased or decreased accumulation rates, respectively, in inoculated and the first two systemically infected leaves of N. benthamiana. The same mutants were occasionally restricted to single cells in N. tabacum plants, suggesting the importance of these amino acids in the PVA infection cycle in N. tabacum.

Noncommutative Gravitation as a Gauge Theory of Twisted Poincaré Symmetry

Gravitaation kvanttiteorian muotoilu on ollut teoreettisten fyysikkojen tavoitteena kvanttimekaniikan synnystä lähtien. Kvanttimekaniikan soveltaminen korkean energian ilmiöihin yleisen suhteellisuusteorian viitekehyksessä johtaa aika-avaruuden koordinaattien operatiiviseen ei-kommutoivuuteen. Ei-kommutoivia aika-avaruuden geometrioita tavataan myös avointen säikeiden säieteorioiden tietyillä matalan energian rajoilla. Ei-kommutoivan aika-avaruuden gravitaatioteoria voisi olla yhteensopiva kvanttimekaniikan kanssa ja se voisi mahdollistaa erittäin lyhyiden etäisyyksien ja korkeiden energioiden prosessien ei-lokaaliksi uskotun fysiikan kuvauksen, sekä tuottaa yleisen suhteellisuusteorian kanssa yhtenevän teorian pitkillä etäisyyksillä. Tässä työssä tarkastelen gravitaatiota Poincarén symmetrian mittakenttäteoriana ja pyrin yleistämään tämän näkemyksen ei-kommutoiviin aika-avaruuksiin. Ensin esittelen Poincarén symmetrian keskeisen roolin relativistisessa fysiikassa ja sen kuinka klassinen gravitaatioteoria johdetaan Poincarén symmetrian mittakenttäteoriana kommutoivassa aika-avaruudessa. Jatkan esittelemällä ei-kommutoivan aika-avaruuden ja kvanttikenttäteorian muotoilun ei-kommutoivassa aika-avaruudessa. Mittasymmetrioiden lokaalin luonteen vuoksi tarkastelen huolellisesti mittakenttäteorioiden muotoilua ei-kommutoivassa aika-avaruudessa. Erityistä huomiota kiinnitetään näiden teorioiden vääristyneeseen Poincarén symmetriaan, joka on ei-kommutoivan aika-avaruuden omaama uudentyyppinen kvanttisymmetria. Seuraavaksi tarkastelen ei-kommutoivan gravitaatioteorian muotoilun ongelmia ja niihin kirjallisuudessa esitettyjä ratkaisuehdotuksia. Selitän kuinka kaikissa tähänastisissa lähestymistavoissa epäonnistutaan muotoilla kovarianssi yleisten koordinaattimunnosten suhteen, joka on yleisen suhteellisuusteorian kulmakivi. Lopuksi tutkin mahdollisuutta yleistää vääristynyt Poincarén symmetria lokaaliksi mittasymmetriaksi --- gravitaation ei-kommutoivan mittakenttäteorian saavuttamisen toivossa. Osoitan, että tällaista yleistystä ei voida saavuttaa vääristämällä Poincarén symmetriaa kovariantilla twist-elementillä. Näin ollen ei-kommutoivan gravitaation ja vääristyneen Poincarén symmetrian tutkimuksessa tulee jatkossa keskittyä muihin lähestymistapoihin.

Noncommutativity and Some of Its Present-Day Developments

This masters thesis explores some of the most recent developments in noncommutative quantum field theory. This old theme, first suggested by Heisenberg in the late 1940s, has had a renaissance during the last decade due to the firmly held belief that space-time becomes noncommutative at small distances and also due to the discovery that string theory in a background field gives rise to noncommutative field theory as an effective low energy limit. This has led to interesting attempts to create a noncommutative standard model, a noncommutative minimal supersymmetric standard model, noncommutative gravity theories etc. This thesis reviews themes and problems like those of UV/IR mixing, charge quantization, how to deal with the non-commutative symmetries, how to solve the Seiberg-Witten map, its connection to fluid mechanics and the problem of constructing general coordinate transformations to obtain a theory of noncommutative gravity. An emphasis has been put on presenting both the group theoretical results and the string theoretical ones, so that a comparison of the two can be made.

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.

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.

Luonnollisten lukujen joukon määriteltävyys funktioalgebroissa

Let X be a topological space and K the real algebra of the reals, the complex numbers, the quaternions, or the octonions. The functions form X to K form an algebra T(X,K) with pointwise addition and multiplication. We study first-order definability of the constant function set N' corresponding to the set of the naturals in certain subalgebras of T(X,K). In the vocabulary the symbols Constant, +, *, 0', and 1' are used, where Constant denotes the predicate defining the constants, and 0' and 1' denote the constant functions with values 0 and 1 respectively. The most important result is the following. Let X be a topological space, K the real algebra of the reals, the compelex numbers, the quaternions, or the octonions, and R a subalgebra of the algebra of all functions from X to K containing all constants. Then N' is definable in , if at least one of the following conditions is true. (1) The algebra R is a subalgebra of the algebra of all continuous functions containing a piecewise open mapping from X to K. (2) The space X is sigma-compact, and R is a subalgebra of the algebra of all continuous functions containing a function whose range contains a nonempty open set of K. (3) The algebra K is the set of reals or the complex numbers, and R contains a piecewise open mapping from X to K and does not contain an everywhere unbounded function. (4) The algebra R contains a piecewise open mapping from X to the set of the reals and function whose range contains a nonempty open subset of K. Furthermore R does not contain an everywhere unbounded function.

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.

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.