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1 Chapter 1 : CiteSeerX â Two-dimensional Quantum Field Theory, examples and applications The second edition of Non-Perturbative Methods in Two-Dimensional Quantum Field Theory is an extensively revised version, involving major changes and additions. Although much of the material is special to two dimensions, the techniques used should prove helpful also in the development of techniques applicable in higher dimensions. The work of Feynman in the late forties provided a powerful tool for the calculation of processes in Quantum Electrodynamics. Second quantization led, however, to new conceptual and technical difficulties. Quantum fields had to be regarded as operator-valued distributions, their local products being ill defined as a result of ultraviolet divergencies, which plagued the higher order computations in perturbation theory. This problem was partially mastered via the techniques of renormalization, and later on completely understood [2]. In the early fifties, there appeared a series of papers concerned with the extraction of general non-perturbative properties of quantum field theory from a perturbative setup. Of particular importance in this respect were the papers of Lehmann, Symanzik and Zimmermann [3] as well as of Wightman, Haag, and others [4]. The so-called LSZ formalism established the relation between fields and particles in terms of the asymptotic conditions for the interpolating fields. The reduction formula provided the connection between expectation values of fields and 5-matrix elements of particle scattering. From the study of the analytic properties of Feynman diagrams dispersion relations were derived, which could be used to obtain non-perturbative information. Here functional analysis played an important role. Some of the non-perturbative results of the LSZ formalism, which had been based on perturbative studies, could thereby be derived from general principles, if the theory has a non-vanishing mass gap. An important consequence of this approach was a theorem connecting spin and statistics. Meanwhile, dynamical calculations in QFT were restricted to perturbation theory. This rendered computations involving strong interactions unreliable, and made information about the bound state spectrum only accessible within approximative non-perturbative â and often non-unitary â schemes e. As a result, QFT had fallen into stagnation, and even discredit, in the late fifties. The predictive power of this theory turned out to be very limited, being entirely based on kinematical principles and analyticity, supplemented by the bootstrap idea. An underlying dynamical framework was lacking. Nevertheless, analyticity in the complex angular momentum plane led to the important concept of duality, expressing the possibility of representing a given scattering amplitude as a sum over poles in crossed channels [6]. An explicit realization of these concepts in terms of a remarkable formula proposed by Veneziano [7] led to a new parallel development in the sixties, summarized under the name of dual models. The high-energy behaviour was however found to be incorrectly described in this approach. Moreover, an analysis of the pole structure of higher order corrections required the introduction of a somewhat mysterious new concept, the Pomeron [8]. The ever increasing number of parameters which were required to describe experiments within these schemes, and the resulting loss of predictive power, eventually led physicists to abandon them, and to turn again to QFT. In the meantime, QFT had scored some remarkable successes in the realm of the weak interactions [9]. This situation led to a revival of QFT in the late sixties and in the seventies, when much attention was given to the non-perturbative aspects. Quantum chromodynamics QCD had been proposed as the fundamental theory of the strong interactions, but reliable calculations confronting QCD with experimental tests were lacking. The high-energy behaviour of QFT was investigated by means of the renormalization-group and Callan-Symanzik equations, which describe the behaviour of the theory under finite renormalizations of the parameters. As a result it was possible to relate the zero mass limit to the high energy behavior. A momentum dependent, running coupling constant turns out to properly characterize the strength of the interaction: In the case of non-abelian gauge theories, perturbation theory turned out to be a good approximation at high energy asymptotic freedom. Furthermore, classical solutions minimizing the action were shown to play a central role in the non-perturbative semi-classical analysis of QFT. Monopole solutions for Minkowski space, and instanton solutions for Euclidean space have been obtained, showing the Page 1

2 important role played by the topology of the manifold on which the fields are defined. In this way, more abstract branches of mathematics such as algebraic topology turned out to be significant in the unraveling of the structural properties of gauge theories. Although the above studies have been weighty for revealing a non-trivial and highly interesting structure, exact non-perturbative results for all correlators are only available for specific models, all in two-dimensional space-time. The first such model, discussed by Thirring [12] in, describes the current-current interaction 19 of massless fermions. It provided an example of a completely soluble quantum field theoretic model obeying the general principles of a QFT. The complete quantum solution was given in a classic paper of Klaiber [13], and was shown to satisfy all Wightman axioms. This long range force was interpreted as being responsible for the confinement of quarks [16], that is, their occurrence in the form of permanently bound states of quark-antiquarks pairs baryonic bound states happen to be absent in QED-i. The problem of confinement and the related phenomenon of screening of charge quantum numbers has been extensively studied [17], and has served as a basis for sharpening these concepts appearing also in higher dimensions. The surprisingly rich structure of two-dimensional quantum electrodynamics was found to describe several important features of non-abelian gauge theories, under investigation in the seventies. As a matter of fact, in the late sixties one learned that the short distance singularities of quantum field theory play a key role in the dynamical structure of the theory [18]. The experimental results on leptonproton scattering at large momentum transfer, required that a realistic theory of the strong interactions be asymptotically free [10, 11]. This promoted QCD to a good candidate for the theory describing strong interactions, since it was shown that no renormalizable theory without non-abelian gauge fields can be asymptotically free [19]. Following these exactly solvable quantum field theoretical models, the study of other two-dimensional quantum field theories has played an important role in the development of a non-perturbative understanding of quantum field theory in general. By the end of the seventies a very comprehensive knowledge had been gathered, revealing an unexpected complexity and richness of the non-perturbative structure of relativistic quantum field theories. Several further developments of growing importance in two-dimensional QFT, followed. Classically exactly integrable models, and the quantization of solitons were extensively studied in two dimensions. Such integrable models were generally characterized by the existence of an infinite number of conservation laws [20]. In the cases where these conservation laws survive quantization, the S-matrices and their associated monodromy matrices could be computed exactly [21]. Some of the results concerning classical integrability have also been generalized to higher dimensions [23]. In the particular case of the sine-gordon theory, the exact results for the Smatrix of fundamental fields could be extended to include the scattering of bound 20 Introduction states and solitons, as well. This equivalence, partially conjectured long ago by Skyrme [24], was proven by Coleman [25] at the level of Green functions, and later obtained by the use of operatorial methods [26]. In both versions bosonic or fermionic the S-matrices could be exactly computed, and turn out to be identical. In the framework of two-dimensional models, the possibility of writing fermions in terms of bosons bosonization has been a powerful tool for obtaining nonperturbative information. One of the features to be stressed in this context is that charge sectors of the fermionic theory are found to correspond to soliton sectors in the purely neutral bosonic theory: The building blocks of the bosonization scheme are the exponentials of bosonic fields, the fermionic and chiral selection rules associated with this composite operator being directly linked to the infrared behavior of the zero mass scalar fields. This leads to a superselection rule [27], which makes the charge sectors appear in a natural way. The bosonization technique becomes cumbersome when applied to non-abelian theories. The reason is twofold: Significant progress in the direction of non-abelian bosonization was provided by the work of Polyakov, Wiegmann and Witten [28]. Although these authors discussed the problem in different contexts, they all arrived at an equivalent bosonic action involving the action of the principal sigma model plus a Wess-Zumino term. In two dimensions, fermionic theories were thus found to exhibit a remarkable universality in the bosonic formulation, where the non-linear sigma model and a topological term seem to play a fundamental role. Non-linear sigma models have a long history. Particularly important has been the class of Page 2

3 the two-dimensional integrable ones. Their geometrical origin makes them very interesting mathematical objects to be studied in their own right [29]. They have also been shown to share several properties with Yang-Mills theories in four dimensions [30]: The non-linear sigma models for symmetric spaces [29], and the Yang-Mills theories for either the self-dual sector, or with extended supersymmetry, share similar integrability properties. When quantized, non-linear sigma models further exhibit features believed to be properties of realistic theories, such as confining long range force generated by quantum fluctuations, when the gauge-group is not simple [32], and dynamical mass generation. These properties make them appealing as toy models for the strong interactions. They are also important the context of string theory, where the D-dimensional target manifold is compactified to four-dimensional space-time [33], the corresponding action being described by a sigma model. The requirement of conformal invariance at the quantum level leads directly to the Einstein equation of general relativity, and predicts its quantum corrections [34]. An unexpected underlying differential geometric significance of such anomalies has thereby been revealed [35]. The exact solubility of two-dimensional chiral QED [36] has played here an important role in opening up a whole new line of developments in the area of chiral gauge theories. The successful quantization of such seemingly inconsistent theories without recourse to an algebraic cancellation of the gaugeanomalies on group-theoretical grounds was of much interest at a time, where the top quark had not yet been found. Although interest in such model studies had levelled off by the end of the seventies, this area of research experienced a remarkable back-come in the eighties, when an almost forgotten acquaintance from the seventies - string theory - experienced itself a dramatic revival as a result of pioneering work due to Green and Schwarz [37]. Ever since, two-dimensional quantum field theory â which had already found applications in statistical mechanics in the past - has become an important subject of elementary particle physics. More recently, it was shown that in two-dimensional quantum field theories, Poincare and scale invariance, alone, imply invariance under the infinite dimensional conformal group in two dimensions [38]. As a result, non-trivial correlators could be exactly computed and were found to be related to solutions of hypergeometric differential equations. The parameters labelling these equations, which are regarded as the critical indices, have been classified and characterize the correlators uniquely. The conformal algebras are realized in terms of the so-called primary fields and their descendents. In Minkowski space, this construction leads naturally to the use of Artin Braids, which relate this problem to the algebraic construction of exact S-matrices, since the star-triangle relations obtained from the infinite local conservation laws have the same structure as the permutation relations of knot theory [39]. The above ideas may be generalized to include the interaction with conformally invariant gravity [40]. In the light-cone-gauge the theory simplifies dramatically, due to a new SL 2, R symmetry [40, 41]. The critical indices of the theory may be computed from a very simple equation relating them to the critical indices of the theory in flat space. The results have also been generalized to the supersymmetric case [42]. To summarize, two-dimensional models have been an extraordinary laboratory to test ideas in quantum field theory. Thus, the Thirring model provided a realization of an exactly soluble quantum field theory, while the Schwinger and the non-linear sigma models were found to exhibit properties of four-dimensional nonabelian gauge theories. However, two-dimensional QFT also plays a direct role in the description of physical reality, having applications in string theories, as well as statistical mechanics. In particular, the methods developed in two-dimensional QFT have been used to extract results concerning the critical behavior of models in statistical mechanics, using conformal invariance alone. An extraordinary amount of physically interesting as well as mathematically elegant concepts have emerged from the study of such theories. Beyond their status as a theoretical laboratory, and their applications in string 22 BIBLIOGRAPHY theories and statistical mechanics, the study of these models has also led to recent developments opening new possibilities for applications of some of the above methods in the study of quantum field theories in higher dimensions. There is a deep relationship between rational conformal invariance in two-dimensional space-time and the Chern-Simons action in three dimensions which is also equivalent to conformal gravity in three dimensions. The Chern-Simons action proves to be a key element in the generalization of the fermion-boson equivalence to Page 3

4 three-dimensional spacetime, and also plays an important role in the discussion of non-abelian anomalies of chiral gauge theories in any dimension. Zimmermann, Nuovo Cimento 1 Veneziano, Nuovo Cimento 57A Salam, in Elementary Particle Theory, ed. Svartholm, Almquist and Wiksells, Stockholm D2 ; K. Gross and F A. Swieca, Annals of Phys. Page 4

5 Chapter 2 : CiteSeerX â Citation Query Non-perturbative methods in 2 dimensional quantum field theory We study non-perturbative aspects of these theories which make them particularly valuable for testing ideas of four-dimensional quantum field theory. The dynamics of confinement and theta vacuum are explained by using the non-perturbative methods developed in two dimensions. Show Context Citation Context As for left-right scattering, we present a class of examples. A series of old and recent theoretical observations suggests that the quanti-zation of gravity would be feasible, and some problems of Quantum Field Theory would go away if, somehow, the spacetime would undergo a dimensional reduction at high en-ergy scales. But an identification of the de But an identification of the deep mechanism causing this dimensional reduction would still be desirable. They turn out to work fine for some known types of cosmological singularities black holes and FLRW Big-Bang, allowing a choice of the fundamental geometric invariants and physical quantities which remain regular. The resulting equations are equivalent to the standard ones outside the singularities. One consequence of this mathematical approach to the singularities in General Relativity is a special, geo metric type of dimensional reduction: A review of the hints that, in various approaches to QG, a dimensional reduction occurs at small scales, is done by Carlip [31,32]. Many of these hints involve the s Our previous constructions of Borchers triples are extended to massless scattering with nontrivial left and right components. A massless Borchers triple is constructed from a set of leftâ left, rightâ right and leftâ right scattering functions. We find a correspondence between massless leftâ We find a correspondence between massless leftâ right scattering S-matrices and massive block diagonal S-matrices. We point out a simple class of S-matrices with examples. We study also the restriction of two-dimensional models to the lightray. Several argu-ments for constructing strictly local two-dimensional nets are presented and possible scenarios are discussed. As for leftâ right scattering, we present a class of examples. Sodano, " We use staggered fermions and the Hamiltonian approach to lattice gauge theories. We show that the one-flavor model is effectively described by the antiferromagnetic Isi We show that the one-flavor model is effectively described by the antiferromagnetic Ising model, whose ground state is the vacuum of the gauge model in the infinite coupling limit; expanding around this ground state we derive a strong coupling expansion and compute the lowest lying hadron masses as well as the chiral condensate of the gauge theory. Our lattice computation well reproduces the results of the continuum theory. Baryons are massless in the infinite coupling limit; they acquire a mass already at the second order in the strong coupling expansion in agreement with the Witten argument that baryons are the QCD solitons. The spectrum and chiral condensate of the two-flavor model are effectively described in terms of observables of the quantum antiferromagnetic Heisenberg model. We explicitly write the lowest lying hadron masses and chiral condensate in terms of spin-spin correlators on the ground state of the spin model. Chapter 3 : The Net Advance of Physics Note: Citations are based on reference standards. However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied. Chapter 4 : Conformal bootstrap - Wikipedia The study of two dimensional models in order to improve the understanding of four dimensional physical systems was found to be fruitful. This can be achieved following two di erent approaches. In the rst one applies non-perturbative methods on the simpler two dimensional model, extract the physical behavior and extrapolate it to four dimensions. Page 5

6 Chapter 5 : [hep-th/] Two-dimensional Quantum Field Theory, examples and applications Even though there are various differences between QCD 4 and QCD 2, this theory may provide interesting insights into the physical four-dimensional world. Generalized sine-gordon model and. Chapter 6 : Non-Perturbative Methods in 2 Dimensional Quantum Field Theory - PDF Free Download Non-perturbative methods in 2 DIMENSIONAL Second Edition Elcio Abdalla Universidade de Sao Paulo, Brazil M. Cristina B. Abdalla. Chapter 7 : Elcio Abdalla (Author of Non-Perturbative Methods in 2 Dimensional Quantum Field Theory) Abstract. A series of old and recent theoretical observations suggests that the quanti-zation of gravity would be feasible, and some problems of Quantum Field Theory would go away if, somehow, the spacetime would undergo a dimensional reduction at high en-ergy scales. Page 6