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.

The visual preferences for forest regeneration and field afforestation - four case studies in Finland

The overall aim of this dissertation was to study the public's preferences for forest regeneration fellings and field afforestations, as well as to find out the relations of these preferences to landscape management instructions, to ecological healthiness, and to the contemporary theories for predicting landscape preferences. This dissertation includes four case studies in Finland, each based on the visualization of management options and surveys. Guidelines for improving the visual quality of forest regeneration and field afforestation are given based on the case studies. The results show that forest regeneration can be connected to positive images and memories when the regeneration area is small and some time has passed since the felling. Preferences may not depend only on the management alternative itself but also on the viewing distance, viewing point, and the scene in which the management options are implemented. The current Finnish forest landscape management guidelines as well as the ecological healthiness of the studied options are to a large extent compatible with the public's preferences. However, there are some discrepancies. For example, the landscape management instructions as well as ecological hypotheses suggest that the retention trees need to be left in groups, whereas people usually prefer individually located retention trees to those trees in groups. Information and psycho-evolutionary theories provide some possible explanations for people's preferences for forest regeneration and field afforestation, but the results cannot be consistently explained by these theories. The preferences of the different stakeholder groups were very similar. However, the preference ratings of the groups that make their living from forest - forest owners and forest professionals - slightly differed from those of the others. These results provide support for the assumptions that preferences are largely consistent at least within one nation, but that knowledge and a reference group may also influence preferences.

Towards the Use of Radar Winds in Numerical Weather Prediction

Numerical weather prediction (NWP) models provide the basis for weather forecasting by simulating the evolution of the atmospheric state. A good forecast requires that the initial state of the atmosphere is known accurately, and that the NWP model is a realistic representation of the atmosphere. Data assimilation methods are used to produce initial conditions for NWP models. The NWP model background field, typically a short-range forecast, is updated with observations in a statistically optimal way. The objective in this thesis has been to develope methods in order to allow data assimilation of Doppler radar radial wind observations. The work has been carried out in the High Resolution Limited Area Model (HIRLAM) 3-dimensional variational data assimilation framework. Observation modelling is a key element in exploiting indirect observations of the model variables. In the radar radial wind observation modelling, the vertical model wind profile is interpolated to the observation location, and the projection of the model wind vector on the radar pulse path is calculated. The vertical broadening of the radar pulse volume, and the bending of the radar pulse path due to atmospheric conditions are taken into account. Radar radial wind observations are modelled within observation errors which consist of instrumental, modelling, and representativeness errors. Systematic and random modelling errors can be minimized by accurate observation modelling. The impact of the random part of the instrumental and representativeness errors can be decreased by calculating spatial averages from the raw observations. Model experiments indicate that the spatial averaging clearly improves the fit of the radial wind observations to the model in terms of observation minus model background (OmB) standard deviation. Monitoring the quality of the observations is an important aspect, especially when a new observation type is introduced into a data assimilation system. Calculating the bias for radial wind observations in a conventional way can result in zero even in case there are systematic differences in the wind speed and/or direction. A bias estimation method designed for this observation type is introduced in the thesis. Doppler radar radial wind observation modelling, together with the bias estimation method, enables the exploitation of the radial wind observations also for NWP model validation. The one-month model experiments performed with the HIRLAM model versions differing only in a surface stress parameterization detail indicate that the use of radar wind observations in NWP model validation is very beneficial.

Planar Edgeless Detectors for the TOTEM Experiment at the LHC

The TOTEM experiment at the LHC will measure the total proton-proton cross-section with a precision better than 1%, elastic proton scattering over a wide range in momentum transfer -t= p^2 theta^2 up to 10 GeV^2 and diffractive dissociation, including single, double and central diffraction topologies. The total cross-section will be measured with the luminosity independent method that requires the simultaneous measurements of the total inelastic rate and the elastic proton scattering down to four-momentum transfers of a few 10^-3 GeV^2, corresponding to leading protons scattered in angles of microradians from the interaction point. This will be achieved using silicon microstrip detectors, which offer attractive properties such as good spatial resolution (<20 um), fast response (O(10ns)) to particles and radiation hardness up to 10^14 "n"/cm^2. This work reports about the development of an innovative structure at the detector edge reducing the conventional dead width of 0.5-1 mm to 50-60 um, compatible with the requirements of the experiment.

Features of the geomagnetic field during the Holocene and Proterozoic

The geomagnetic field is one of the most fundamental geophysical properties of the Earth and has significantly contributed to our understanding of the internal structure of the Earth and its evolution. Paleomagnetic and paleointensity data have been crucial in shaping concepts like continental drift, magnetic reversals, as well as estimating the time when the Earth's core and associated geodynamo processes begun. The work of this dissertation is based on reliable Proterozoic and Holocene geomagnetic field intensity data obtained from rocks and archeological artifacts. New archeomagnetic field intensity results are presented for Finland, Estonia, Bulgaria, Italy and Switzerland. The data were obtained using sophisticated laboratory setups as well as various reliability checks and corrections. Inter-laboratory comparisons between three laboratories (Helsinki, Sofia and Liverpool) were performed in order to check the reliability of different paleointensity methods. The new intensity results fill up considerable gaps in the master curves for each region investigated. In order to interpret the paleointensity data of the Holocene period, a novel and user-friendly database (GEOMAGIA50) was constructed. This provided a new tool to independently test the reliability of various techniques and materials used in paleointensity determinations. The results show that archeological artifacts, if well fired, are the most suitable materials. Also lavas yield reliable paleointensity results, although they appear more scattered. This study also shows that reliable estimates are obtained using the Thellier methodology (and its modifications) with reliability checks. Global paleointensity curves during Paleozoic and Proterozoic have several time gaps with few or no intensity data. To define the global intensity behavior of the Earth's magnetic field during these times new rock types (meteorite impact rocks) were investigated. Two case histories are presented. The Ilyinets (Ukraine) impact melt rocks yielded a reliable paleointensity value at 440 Ma (Silurian), whereas the results from Jänisjärvi impact melts (Russian Karelia, ca. 700 Ma) might be biased towards high intensity values because of non-ideal magnetic mineralogy. The features of the geomagnetic field at 1.1 Ga are not well defined due to problems related to reversal asymmetries observed in Keweenawan data of the Lake Superior region. In this work new paleomagnetic, paleosecular variation and paleointensity results are reported from coeval diabases from Central Arizona and help understanding the asymmetry. The results confirm the earlier preliminary observations that the asymmetry is larger in Arizona than in Lake Superior area. Two of the mechanisms proposed to explain the asymmetry remain plausible: the plate motion and the non-dipole influence.

Quantum Space-Time and Noncommutative Gauge Field Theories

Arguments arising from quantum mechanics and gravitation theory as well as from string theory, indicate that the description of space-time as a continuous manifold is not adequate at very short distances. An important candidate for the description of space-time at such scales is provided by noncommutative space-time where the coordinates are promoted to noncommuting operators. Thus, the study of quantum field theory in noncommutative space-time provides an interesting interface where ordinary field theoretic tools can be used to study the properties of quantum spacetime. The three original publications in this thesis encompass various aspects in the still developing area of noncommutative quantum field theory, ranging from fundamental concepts to model building. One of the key features of noncommutative space-time is the apparent loss of Lorentz invariance that has been addressed in different ways in the literature. One recently developed approach is to eliminate the Lorentz violating effects by integrating over the parameter of noncommutativity. Fundamental properties of such theories are investigated in this thesis. Another issue addressed is model building, which is difficult in the noncommutative setting due to severe restrictions on the possible gauge symmetries imposed by the noncommutativity of the space-time. Possible ways to relieve these restrictions are investigated and applied and a noncommutative version of the Minimal Supersymmetric Standard Model is presented. While putting the results obtained in the three original publications into their proper context, the introductory part of this thesis aims to provide an overview of the present situation in the field.

Local numerical modelling of magnetoconvection and turbulence - implications for mean-field theories

During the last decades mean-field models, in which large-scale magnetic fields and differential rotation arise due to the interaction of rotation and small-scale turbulence, have been enormously successful in reproducing many of the observed features of the Sun. In the meantime, new observational techniques, most prominently helioseismology, have yielded invaluable information about the interior of the Sun. This new information, however, imposes strict conditions on mean-field models. Moreover, most of the present mean-field models depend on knowledge of the small-scale turbulent effects that give rise to the large-scale phenomena. In many mean-field models these effects are prescribed in ad hoc fashion due to the lack of this knowledge. With large enough computers it would be possible to solve the MHD equations numerically under stellar conditions. However, the problem is too large by several orders of magnitude for the present day and any foreseeable computers. In our view, a combination of mean-field modelling and local 3D calculations is a more fruitful approach. The large-scale structures are well described by global mean-field models, provided that the small-scale turbulent effects are adequately parameterized. The latter can be achieved by performing local calculations which allow a much higher spatial resolution than what can be achieved in direct global calculations. In the present dissertation three aspects of mean-field theories and models of stars are studied. Firstly, the basic assumptions of different mean-field theories are tested with calculations of isotropic turbulence and hydrodynamic, as well as magnetohydrodynamic, convection. Secondly, even if the mean-field theory is unable to give the required transport coefficients from first principles, it is in some cases possible to compute these coefficients from 3D numerical models in a parameter range that can be considered to describe the main physical effects in an adequately realistic manner. In the present study, the Reynolds stresses and turbulent heat transport, responsible for the generation of differential rotation, were determined along the mixing length relations describing convection in stellar structure models. Furthermore, the alpha-effect and magnetic pumping due to turbulent convection in the rapid rotation regime were studied. The third area of the present study is to apply the local results in mean-field models, which task we start to undertake by applying the results concerning the alpha-effect and turbulent pumping in mean-field models describing the solar dynamo.

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.