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.

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.

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.

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.

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.

Englanniksiko maailmanmaineeseen? : Suomalaisen proosakaunokirjallisuuden kääntäminen englanniksi Isossa-Britanniassa vuosina 1945 2003

This doctoral thesis focuses on the translation of Finnish prose literature into English in the United Kingdom between 1945 and 2003. The subject is approached using translation archaeology, interviews, archival material, detailed text analysis and reception material. The main theoretical framework is Descriptive Translation Studies, and certain sociological theories (Bourdieu s field theory, actor-network theory) are also used. After charting the published translations, two periods of time are selected for closer analysis: an earlier period from 1955 to 1959, involving eight translations, and a later one from 1990 to 2003, with a total of six translations. While these translation numbers may appear low, they are actually rather high in proportion to the total number of 28 one-author literary prose translations published in the UK over the approximately 60 years being studied. The two periods of time, the 1950s and 1990s, are compared in terms of the sociological context of translation activity, the reception of translations and their textual features. The comparisons show that the main changes in translation practice between these two periods are increased completeness (translations in the 1950s group often being shortened by hundreds of pages) and lesser use of indirect translation via an intermediary language (about half of the 1950s translations having been translated via Swedish). Otherwise, translation practices have not changed much: except for large omissions, which are far more frequent in the 1950s, variation within each group is larger than between groups. As to the sociological context, the main changes are an increase in long-term institution-level contacts and an increase in the promotion of foreign translation rights by Finnish publishing houses. This is in contrast to the 1950s when translation rights were mainly sold through personal contacts by individual authors and translators. The reception of translations is difficult to study because of scarce material. However, the 1950s translations were aggressively marketed and therefore obtained far more reviews and reprints than the 1990s translations. Several of the 1950s books, mostly historical novels by Mika Waltari, were mainstream bestsellers at the time, while current translations are frequently made for niche markets. The thesis introduces ample new material on the translation of Finnish prose literature into English in the UK. The results are also relevant to translation from a minority literature into a majority one. As to translation theory, they lead us to question the social nature of translation norms and the assumption of a static target culture. The translations analysed here are located in a very fragmented interculture and gain a stronger position in the Finnish culture than in the British one.

Signifying Angels : Analyses and Interpretations of Rautavaara s Instrumental Compositions

The subject of doctoral thesis is the analysis and interpretation of instrumental pieces composed by Einojuhani Rautavaara (b. 1928) that have been given angelic titles: Archangel Michael Fighting the Antichrist from the suite Icons (1955)/Before the Icons (2006), Angels and Visitations (1978), the Double Bass Concerto Angel of Dusk (1980), Playgrounds for Angels (1981)and the Seventh Symphony Angel of Light (1994). The aim of the work is to find those musical elements common to these pieces that distinguish them from Rautavaara s other works and to determine if they could be thought of as a series. I prove that behind the common elements and titles stands the same extramusical idea the figure of an angel that the composer has described in his commentaries. The thesis is divided into three parts. Since all of the compositions possess titles that refer to the spiritual symbol of an angel, the first part offers a theoretical background to demonstrate the significant role played by angels in various religions and beliefs, and the means by which music has attempted to represent this symbol throughout history. This background traces also Rautavaara s aesthetic attitude as a spiritual composer whose output can be studied with reference to his extramusical interests including literature, psychology, painting, philosophy and myths. The second part focuses on the analysis of the instrumental compositions with angelic titles, without giving consideration to their commentaries and titles. The analyses concentrate in particular on those musical features that distinguish these pieces from Rautavaara s other compositions. In the third part these musical features are interpreted as symbols of the angel through comparison with vocal and instrumental pieces which contain references to the character of an angel, structures of mythical narration, special musical expressions, use of instruments and aspects of brightness. Finally I explore the composer s interpretative codes, drawing on Rilke s cycle of poems Ten Duino Elegies and Jung s theory of archetypes, and analyze the instrumental pieces with angelic titles in the light of the theory of musical ekphrasis.

Makt, konst, elit - konstfältets positioner, relationer och resurser i 1940- och 1950-talens Helsingfors

The dissertation examines the power mechanisms and institutional power hierarchies of the 1940s-1950s era arts elite in Helsinki and their influence on issues of taste in the visual arts. For the purposes of this study, the elite is understood to consist mainly of the board members of the principal elected bodies in the field of the arts. The theoretical framework employed is based on Pierre Bourdieu s field theory and the network perspective. The author has examined what the key, pervasive valuations were that governed the exercising of power by the arts elite in issues of taste, involving determination of who was an acknowledged artist and what was good art. The dissertation demonstrates that this exercising of power was governed by certain collective practices which maintained the illusion that the exercising of power was democratic and based on artistic quality. These practices were the corporate system, using artistic arguments in issues of taste, and using networks in the exercising of power. The struggle in the field of the arts was about who ultimately was entitled to define the value of contemporary art; the issue did not arise regarding historical art. Artists managed to gain a leading position as gatekeepers in issues regarding contemporary art. The author discusses a number of conflicts in the field of the arts that highlight the institutional hierarchies and the capital held by the various players. The structural changes that occurred in administration in the field of cultural production in the 1950s led to the separation of bureaucratic competence on the one hand and aesthetic competence on the other. There was a hierarchy in the field of the arts between institutions, between instruments of legitimisation, and between the symbolic and social capital of players in the field. The hierarchy in the arts ultimately depended on how well the elite could influence tastes through the instruments at their disposal. The various instruments of legitimisation grants, purchases, etc. were ranked differently in the evaluation of acknowledged artists and good art. The dissertation discusses what values, in the form of types of symbolic capital, the arts elite embraced and what role these played in the elite s exercising of power, with particular focus on gender, language, region and economic capital. The aesthetic capital of an artist was of only minor importance in the exercising of power by the arts elite. The dissertation further discusses the points of contact between the arts elite and players in other fields, such as the economic, media and consumer fields. When the arts elite, through the Academy of Fine Arts, became an active player in the art market, this led to a hierarchy where the division between acknowledged and not-acknowledged galleries became sharper.

A sequential analysis of "nú" and "núna" in Icelandic conversation

This thesis is an empirical study of how two words in Icelandic, "nú" and "núna", are used in contemporary Icelandic conversation. My aims in this study are, first, to explain the differences between the temporal functions of "nú" and "núna", and, second, to describe the non-temporal functions of "nú". In the analysis, a focus is placed on comparing the sequential placement of the two words, on their syntactical distribution, and on their prosodic realization. The empirical data comprise 14 hours and 11 minutes of naturally occurring conversation recorded between 1996 and 2003. The selected conversations represent a wide range of interactional contexts including informal dinner parties, institutional and non-institutional telephone conversations, radio programs for teenagers, phone-in programs, and, finally, a political debate on television. The theoretical and methodological framework is interactional linguistics, which can be described as linguistically oriented conversation analysis (CA). A comparison of "nú" and "núna" shows that the two words have different syntactic distributions. "Nú" has a clear tendency to occur in the front field, before the finite verb, while "núna" typically occurs in the end field, after the object. It is argued that this syntactic difference reflects a functional difference between "nú" and "núna". A sequential analysis of "núna" shows that the word refers to an unspecified period of time which includes the utterance time as well as some time in the past and in the future. This temporal relation is referred to as reference time. "Nú", by contrast, is mainly used in three different environments: a) in temporal comparisons, 2) in transitions, and 3) when the speaker is taking an affective stance. The non-temporal functions of "nú" are divided into three categories: a) "nú" as a tone particle, 2) "nú" as an utterance particle, and 3) "nú" as a dialogue particle. "Nú" as a tone particle is syntactically integrated and can occur in two syntactic positions: pre-verbally and post-verbally. I argue that these instances are employed in utterances in which a speaker is foregrounding information or marking it as particularly important. The study shows that, although these instances are typically prosodically non-prominent and unstressed, they are in some cases delivered with stress and with a higher pitch than the surrounding talk. "Nú" as an utterance particle occurs turn-initially and is syntactically non-integrated. By using "nú", speakers show continuity between turns and link new turns to prior ones. These instances initiate either continuations by the same speaker or new turns after speaker shifts. "Nú" as a dialogue particle occurs as a turn of its own. The study shows that these instances register informings in prior turns as unexpected or as a departure from the normal state of affairs. "Nú" as a dialogue particle is often delivered with a prolonged vowel and a recognizable intonation contour. A comparative sequential and prosodic analysis shows that in these cases there is a correlation between the function of "nú" and the intonation contour by which it is delivered. Finally, I argue that despite the many functions of "nú", all the instances can be said to have a common denominator, which is to display attention towards the present moment and the utterances which are produced prior or after the production of "nú". Instead of anchoring the utterances in external time or reference time, these instances position the utterance in discourse internal time, or discourse time.

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.

Bayesian analysis of community dynamics

Elucidating the mechanisms responsible for the patterns of species abundance, diversity, and distribution within and across ecological systems is a fundamental research focus in ecology. Species abundance patterns are shaped in a convoluted way by interplays between inter-/intra-specific interactions, environmental forcing, demographic stochasticity, and dispersal. Comprehensive models and suitable inferential and computational tools for teasing out these different factors are quite limited, even though such tools are critically needed to guide the implementation of management and conservation strategies, the efficacy of which rests on a realistic evaluation of the underlying mechanisms. This is even more so in the prevailing context of concerns over climate change progress and its potential impacts on ecosystems. This thesis utilized the flexible hierarchical Bayesian modelling framework in combination with the computer intensive methods known as Markov chain Monte Carlo, to develop methodologies for identifying and evaluating the factors that control the structure and dynamics of ecological communities. These methodologies were used to analyze data from a range of taxa: macro-moths (Lepidoptera), fish, crustaceans, birds, and rodents. Environmental stochasticity emerged as the most important driver of community dynamics, followed by density dependent regulation; the influence of inter-specific interactions on community-level variances was broadly minor. This thesis contributes to the understanding of the mechanisms underlying the structure and dynamics of ecological communities, by showing directly that environmental fluctuations rather than inter-specific competition dominate the dynamics of several systems. This finding emphasizes the need to better understand how species are affected by the environment and acknowledge species differences in their responses to environmental heterogeneity, if we are to effectively model and predict their dynamics (e.g. for management and conservation purposes). The thesis also proposes a model-based approach to integrating the niche and neutral perspectives on community structure and dynamics, making it possible for the relative importance of each category of factors to be evaluated in light of field data.

Spin and relativistic motion of bound states

The wave functions of moving bound states may be expected to contract in the direction of motion, in analogy to a rigid rod in classical special relativity, when the constituents are at equal (ordinary) time. Indeed, the Lorentz contraction of wave functions is often appealed to in qualitative discussions. However, only few field theory studies exist of equal-time wave functions in motion. In this thesis I use the Bethe-Salpeter formalism to study the wave function of a weakly bound state such as a hydrogen atom or positronium in a general frame. The wave function of the e^-e^+ component of positronium indeed turns out to Lorentz contract both in 1+1 and in 3+1 dimensional quantum electrodynamics, whereas the next-to-leading e^-e^+\gamma Fock component of the 3+1 dimensional theory deviates from classical contraction. The second topic of this thesis concerns single spin asymmetries measured in scattering on polarized bound states. Such spin asymmetries have so far mainly been analyzed using the twist expansion of perturbative QCD. I note that QCD vacuum effects may give rise to a helicity flip in the soft rescattering of the struck quark, and that this would cause a nonvanishing spin asymmetry in \ell p^\uparrow -> \ell' + \pi + X in the Bjorken limit. An analogous asymmetry may arise in p p^\uparrow -> \pi + X from Pomeron-Odderon interference, if the Odderon has a helicity-flip coupling. Finally, I study the possibility that the large single spin asymmetry observed in p p^\uparrow -> \pi(x_F,k_\perp) + X when the pion carries a high momentum fraction x_F of the polarized proton momentum arises from coherent effects involving the entire polarized bound state.

Exotic superfluid states of ultra-cold Fermi gases

This is a study of ultra-cold Fermi gases in different systems. This thesis is focused on exotic superfluid states, for an example on the three component Fermi gas and the FFLO phase in optical lattices. In the two-components case, superfluidity is studied mainly in the case of the spin population imbalanced Fermi gases and the phase diagrams are calculated from the mean-field theory. Different methods to detect different phases in optical lattices are suggested. In the three-component case, we studied also the uniform gas and harmonically trapped system. In this case, the BCS theory is generalized to three-component gases. It is also discussed how to achieve the conditions to get an SU(3)-symmetric Hamiltonian in optical lattices. The thesis is divided in chapters as follows: Chapter 1 is an introduction to the field of cold quantum gases. In chapter 2 optical lattices and their experimental characteristics are discussed. Chapter 3 deals with two-components Fermi gases in optical lattices and the paired states in lattices. In chapter 4 three-component Fermi gases with and without a harmonic trap are explored, and the pairing mechanisms are studied. In this chapter, we also discuss three-component Fermi gases in optical lattices. Chapter 5 devoted to the higher order correlations, and what they can tell about the paired states. Chapter 6 concludes the thesis.