This thesis concerns two types of quantum many-body systems in one dimension exactly solved by special functions: firstly, systems with interactions localised at points and solved by the (coordinate) Bethe ansatz; secondly, systems of Calogero-Sutherland type, as well as certain recently introduced deformations thereof, with eigenfunctions given by natural many-variable generalisations of classical (orthogonal) polynomials. The thesis is divided into two parts. The first provides background and a few complementary results, while the second presents the main results of this thesis in five appended scientific papers. In the first paper we consider two complementary quantum many-body systems with local interactions related to the root systems CN, one with delta-interactions, and the other with certain momentum dependent interactions commonly known as delta-prime interactions. We prove, by construction, that the former is exactly solvable by the Bethe ansatz in the general case of distinguishable particles, and that the latter is similarly solvable only in the case of bosons or fermions. We also establish a simple strong/weak coupling duality between the two models and elaborate on their physical interpretations. In the second paper we consider a well-known four-parameter family of local interactions in one dimension. In particular, we determine all such interactions leading to a quantum many-body system of distinguishable particles exactly solvable by the Bethe ansatz. We find that there are two families of such systems: the first is described by a one-parameter deformation of the delta-interaction model, while the second features a particular one-parameter combination of the delta and the delta-prime interactions. In papers 3-5 we construct and study particular series representations for the eigenfunctions of a family of Calogero-Sutherland models naturally associated with the classical (orthogonal) polynomials. In our construction, the eigenfunctions are given by linear combinations of certain symmetric polynomials generalising the so-called Schur polynomials, with explicit and rather simple coefficients. In paper 5 we also generalise certain of these results to the so-called deformed Calogero-Sutherland operators. / QC 20100712

A rigorous formalism for solving the Coulomb scattering problem is presented in this thesis. The approach is based on splitting the interaction potential into a finite-range part and a long-range tail part. In this representation the scattering problem can be reformulated to one which is suitable for applying exterior complex scaling. The scaled problem has zero boundary conditions at infinity and can be implemented numerically for finding scattering amplitudes. The systems under consideration may consist of two or three charged particles. The technique presented in this thesis is first developed for the case of a two body single channel Coulomb scattering problem. The method is mathematically validated for the partial wave formulation of the scattering problem. Integral and local representations for the partial wave scattering amplitudes have been derived. The partial wave results are summed up to obtain the scattering amplitude for the three dimensional scattering problem. The approach is generalized to allow the two body multichannel scattering problem to be solved. The theoretical results are illustrated with numerical calculations for a number of models. Finally, the potential splitting technique is further developed and validated for the three body Coulomb scattering problem. It is shown that only a part of the total interaction potential should be split to obtain the inhomogeneous equation required such that the method of exterior complex scaling can be applied. The final six-dimensional equation is reduced to a system of three dimensional equations using the full angular momentum representation. Such a system can be numerically implemented using the existing full angular momentum complex exterior scaling code (FAMCES). The code has been updated to solve the three body scattering problem. / At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Submitted. Paper 5: Manuscript.

The discovery of integrability on both sides of the duality between planar N=4 super Yang-Mills theory and free type IIB string theory in AdS5 × S5 has lead to great progress in our understanding of the AdS/CFT correspondence. Similar integrable structures also appear in the more recent three-dimensional superconformal N=6 Chern-Simons-matter theory constructed by Aharony, Bergman, Jafferis and Maldacena (ABJM), as well as in its gravity dual, type IIA string theory on AdS4 × CP3. However, new interesting complications arise in the AdS4/CFT3 duality. In the conjectured all-loop Bethe equations by Gromov and Vieira the dispersion relation of the magnons has a non-trivial coupling dependence which is parametrized by a function that is only known to the leading order at weak and strong coupling. In the first part of this thesis I discuss our calculations of the next-to-leading correction to this function at weak coupling. We compute this function from four-loop Feynman diagrams in the SU(2) × SU(2) sector of the ABJM model. As a consistency check we have performed the calculation both in a component formalism and using superspace techniques. At strong coupling the fundamental excitations of the integrable model are the giant magnons. The topic of the second part of this thesis is the spectrum of these giant magnons in CP3. Furthermore, I discuss our analyses of the finite-size corrections beyond the asymptotic Bethe ansatz. At weak coupling we have computed the leading four-loop wrapping diagrams in the ABJM model. At the strong coupling side of the duality I discuss our results for the exponentially suppressed finite-size corrections to the energy of giant magnons.

This thesis presents a brief synopsis of the derivations of the BCS and Eliashberg equations. An analytic formula for the critical temperature $T_c$ in Eliashberg theory is derived, which contains a sum of iterative integral corrections. These iterative integral corrections are the result of an iterative expression for the gap quotient $\Delta(\iw, T)/\Delta(0,T)$, which is derived. At the critical temperature this expression contains no reference to the critical temperature itself due to the gap approaching zero in this limit, $\lim_{T \rightarrow T_c} \Delta(\iw, T) = 0$. This enables explicit calculation of the critical temperature through the aforementioned iterative expression.\\ \\The behaviour of the iterative expression and its corrections are explored numerically with a toy spectral function $\sF$. Through these numerical experiments, this formula is found to be consistent with, though not equal to the successful McMillan formula for the coupling parameter $\lambda$ in the range $0.3 \leq \lambda \leq 1.5$. Below this value, the McMillan formula is found to approach zero critical temperature $T_c$ more rapidly, raising the future question of which of the two expressions is most successful in predicting the critical temperature $T_c$ in this range. \\ \\ For a toy spectral function with a single mode, the zeroth order correction of the iterative expression for the critical temperature $T_c$ is found to be adequate for most practical purposes due to the magnitude of measurement errors in real life measurements of model parameters. / Detta examensarbete går igenom en kort derivation av BCS och Eliashberg ekvationerna. En analytisk formel för den kritiska temperaturen $T_c$ i Eliashbergteori ges. Denna formel innehåller en summa av iterativa integraler som resulterar från ett uttryckt för energigapets kvot. Vid den kritiska temperaturen så kan man explicit lösa ut denna och på så sätt få ett analytiskt uttryck. Den uttrycket för den kritiska temperaturen utforskas numeriskt med en leksaks-spektralfunktion. Genom dessa numeriska experiment visas det hur det iterativa uttrycket sammanstämmer med McMillans formel för kopplingsparametern $0.3 < \lambda < 1.5$, även om dem ej är lika. Under detta intervall så närmar sig McMillans uttryck noll snabbare, vilket höjer frågan vilken utav dem två uttrycken som fungerar bäst i denna gräns.  För en leksaks-spektralfunktion med ett läge så räcker den nollte korrektionen i det iterativa uttrycket för att få godtagbara resultat, med bakgrund av dem relativt stora mätfelen för riktiga parametrar.

Denna rapport introducerar Cliffordalgebra och visar hur det i flera fall kan ersätta tensoralgebran som grundläggande algebraisk struktur för matematisk fysik. Först konstrueras en allmän Cliffordalgebra utifrån ett vektorrum och en kvadratisk form. Sedan studeras de geometriska egenskaperna för reella Cliffordalgebror med icke-degenererad kvadratisk form. Dessa kallas geometriska algebror och är användbara för att generalisera rotationer till högre dimensioner. Efter det utforskas kopplingen mellan generaliserade rotationer i rumtidsalgebran och Lorentztransformationer. Slutligen används rumtidsalgebran som algebraisk grund för att modellera olika fysikaliska fenomen inom speciell relativitetsteori och elektromagnetism.

This thesis explores three different aspects of quantum gravity. First we study D3-brane black holes in Calabi-Yau compactifications of type IIB string theory. Using the OSV conjecture and a relation between topological strings and matrix models we show that some black holes have a matrix model description. This is the case if the attractor mechanism fixes the internal geometry to a conifold at the black hole horizon. We also consider black holes in a flux compactification and compare the effects of the black holes and fluxes on the internal geometry. We find that the fluxes dominate. Second, we study the scalar potential of type IIB flux compactifications. We demonstrate that monodromies of the internal geometry imply as a general feature the existence of long series of continuously connected minima. This allows for the embedding of scenarios such as chain inflation and resonance tunneling into string theory. The concept of monodromies is also extended to include geometric transitions: passing to a different Calabi-Yau topology, performing its monodromies and then returning to the original space allows for novel transformations. All constructions are performed explicitly, using both analytical and numerical techniques, in the mirror quintic Calabi-Yau. Third, we study cosmological topologically massive gravity at the chiral point, a prime candidate for quantization of gravity in three dimensions. The prospects of this scenario depend crucially of the stability of the theory. We demonstrate the presence of a negative energy bulk mode that grows logarithmically toward the AdS boundary. The AdS isometry generators have non-unitary matrix representations like in logarithmic CFT, and we propose that the CFT dual for this theory is logarithmic. In a complementing canonical analysis we also demonstrate the existence of this bulk degree of freedom, and we present consistent boundary conditions encompassing the new mode.

One of the main topics in the modern String Theory are the conjectured string/gauge (AdS/CFT) dualities. Proving such conjectures is extremely difficult since the gauge and string theory perturbative regimes do not overlap. In this perspective, the discovery of infinitely many conserved charges, i.e. the integrability, in the planar AdS/CFT has allowed us to reach immense progresses in understanding and confirming the duality.The first part of this thesis is focused on the gravity side of the AdS5/CFT4 duality: we investigate the quantum integrability of the type IIB superstring on AdS5 x S5. In the pure spinor formulation we analyze the operator algebra by computing the operator product expansion of the Maurer-Cartan currents at the leading order in perturbation theory. With the same approach at one loop order, we show the path-independence of the monodromy matrix which implies the charge conservation law, strongly supporting the quantum integrability of the string sigma-model. We also verify that the Lax pair field strength remains well-defined at one-loop order being free from UV divergences. The same string sigma-model is analyzed in the Green-Schwarz formalism in the near-flat-space (NFS) limit. Such a limit remarkably simplifies the string world-sheet action but still leaving interesting physics. We use the NFS truncation to show the factorization of the world-sheet S-matrix at one-loop order. This property defines a two-dimensional field theory as integrable: it is the manifestation of the higher conserved charges. Hence, we have explicitly checked their presence at quantum level. The second part is dedicated to the AdS4/CFT3 duality: in particular the type IIA superstring on AdS4 x CP3. We compute the leading quantum corrections to the string energies for string configurations with a large but yet finite angular momentum on CP3 and show that they match the conjectured all-loop Bethe Ansatz equations.

In Part I we are dealing with effective description of Yang-Mills theories based on gauge-invarint variables. For pure Yang-Mills we study the spin-charge separation varibles. The dynamics in these variables resembles the Skyrme-Faddeev model. Thus the spin-charge separation is an important intermediate step between the fundamental Yang-Mills theory and the low-energy effective models, used to model the low-energy dynamics of gluons. Similar methods may be useful for describing the Electroweak sector of the Standard Model in terms of gauge-invariant field variables called supercurrents. We study the geometric structure of spin-charge separation in 4D Euclidean space (paper III) and elaborate onconnection with gravity toy model. Such reinterpretation gives a way to see how effective flat background metric is created in toy gravity model by studying the appearance of dimension-2 condensate in the Yang-Mills (paper IV). For Electroweak theory we derive the effective gauge-invariant Lagrangian by doing the Kaluza-Klein reduction of higher-dimensional gravity with 3-brane, thus making explicit the geometric interpretation for gauge-invariant supercurrents. The analogy is then made more precise in the framework of exact supergravity solutions. Thus, we interpret the Higgs effect as spontaneous breaking of Kaluza-Klein gauge symmetry and this leads to interpretation of Higgs field as a dilaton (papers I and II). In Part II of the thesis we study rather simple field theories, called “geometric” or “instantonic”. Their defining property is exact localization on finite-dimensional spaces – the moduli spaces of instantons. These theories allow to account exactly for non-linearity of space of fields, in this respect they go beyond the standard Gaussian perturbation theory. In paper V we show how to construct a geometric theory of chiral boson by embedding it into the geometric field theory. In Paper VI we elaborate on the simplest geometric field theory – the supersymmetric Quantum Mechanics and construct new non-perturbative topological observables that have a transparent meaning both in geometric and in the Hamiltonian formalisms. In Paper VII we are motivated by making perturbations away from the simple instantonic limit. For that we need to carefully define the observables that are quadratic in momenta and develop the way to compute them in geometric framework. These correspond geometrically to bivector fields (or, in general, the polyvector fields). We investigate the local limit of polyvector fields and compare the geometric calculation with free-field approach.

Conservation laws play an important role in science. The aim of this thesis is to provide an overview and develop new methods for constructing conservation laws using Lie group theory. The derivation of conservation laws for invariant variational problems is based on Noether’s theorem. It is shown that the use of Lie-Bäcklund transformation groups allows one to reduce the number of basic conserved quantities for differential equations obtained by Noether’s theorem and construct a basis of conservation laws. Several examples on constructing a basis for some well-known equations are provided. Moreover, this approach allows one to obtain new conservation laws even for equations without Lagrangians. A formal Lagrangian can be introduced and used for computing nonlocal conservation laws. For self-adjoint or quasi-self-adjoint equations nonlocal conservation laws can be transformed into local conservation laws. One of the fields of applications of this approach is electromagnetic theory, namely, nonlocal conservation laws are obtained for the generalized Maxwell-Dirac equations. The theory is also applied to the nonlinear magma equation and its nonlocal conservation laws are computed.

Oil is the black blood that runs through the veins of the modern global energy system. While being the dominant source of energy, oil has also brought wealth and power to the western world. Future supply for oil is unsure or even expected to decrease due to limitations imposed by peak oil. Energy is fundamental to all parts of society. The enormous growth and development of society in the last two-hundred years has been driven by rapid increase in the extraction of fossil fuels. In the foresee-able future, the majority of energy will still come from fossil fuels. Consequently, reliable methods for forecasting their production, especially crude oil, are crucial. Forecasting crude oil production can be done in many different ways, but in order to provide realistic outlooks, one must be mindful of the physical laws that affect extraction of hydrocarbons from a reser-voir. Decline curve analysis is a long established tool for developing future outlooks for oil production from an individual well or an entire oilfield. Depletion has a fundamental role in the extraction of finite resources and is one of the driving mechanisms for oil flows within a reservoir. Depletion rate also can be connected to decline curves. Consequently, depletion analysis is a useful tool for analysis and forecasting crude oil production. Based on comprehensive databases with reserve and production data for hundreds of oil fields, it has been possible to identify typical behaviours and properties. Using a combination of depletion and decline rate analysis gives a better tool for describing future oil production on a field-by-field level. Reliable and reasonable forecasts are essential for planning and nec-essary in order to understand likely future world oil production.