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The aim of these lectures is to present an introduction at a fairly elementary level to recent developments in two dimensional field theory, namely in conformal field theory. We shall see the importance of new structures related to infinite dimensional algebras: current algebras and Virasoro algebra. These topics will find physically relevant applications in the lectures by Shankar and Ian Affleck.

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In these lectures some perturbative approaches to the description of high energy scattering processes in QCD are reviewed. It is shown, that the gluon is reggeized and the pomeron is a compound state of two reggeized gluons. We demonstrate, that the equations for compound states of an arbitrary number of reggeized gluons in the multi-colour QCD have remarkable mathematical properties. In the conclusion the effective action describing the gluon-Reggeon interactions is discussed.

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An elementary introduction to Abelian bosonization is provided here. It is shown that although it is applicable to relativistic Dirac fermions, there are many examples of problems in condensed matter theory which are described (with respect to some characteristics) by just such objects. Examples considered here are the uniform and random Ising models, and the Hubbard model.

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Recently, a new approach, based on boundary conformal field theory, has been applied to a variety of quantum impurity problems in condensed matter and particle physics. A particularly enlightening example is the multi-channel Kondo problem. In this review some earlier approaches to the Kondo problem are discussed, the needed material on boundary conformal field theory is developed and then this new method is applied to the multi-channel Kondo problem.

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The perturbative QCD predictions concerning deep inelastic scattering at low \(x\) are summarized. The theoretical framework based on the leading log 1/\(x\) resumation and \(k_t\) factorization theorem is described. The role of studying final states in deep inelastic scattering for revealing the details of the underlying dynamics at low \(x\) is emphasized and some dedicated measurements, like deep inelastic scattering accompanied by an energetic jet, the measurement of the transverse energy flow and deep inelastic diffraction, are briefly discussed.

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Deep-inelastic scattering at small \(x\) is a new field of QCD phenomenology which HERA experiments started to explore. The data show the feature expected by the perturbative Regge asymptotics. The aim of the lectures is to give an introduction to the phenomenological and theoretical ideas of small \(x\) physics as a preparation to the following lectures at this school. We describe the parton picture of structure function evolution both for increasing \(Q^2\) and for decreasing \(x\) and discuss the leading logarithmic approximation of QCD on which this picture is based. The steep rise of the parton density towards smaller \(x\) gets saturated by parton recombination. An essential improvement of the leading logarithmic approximation has to be worked out in order to satisfy the unitarity conditions and in this way to describe the parton recombination in perturbative QCD. We introduce the high-energy effective action, discuss reggeization of gluons and the remarkable symmetry properties of the reggeon interactions. Using this symmetry the solution of the BFKL pomeron equation and also the appearance of the pomeron poles are considered in some detail. We outline the construction of the \(2 \to 4\) reggeon vertex and of amplitudes which can be used to derive predictions about the hadronic final state.

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We present the BFKL equation as a reggeon Bethe–Salpeter equation and discuss the use of reggeon diagrams to obtain 2–2 and 2–4 reggeon interactions at \(O(g^4)\). We then outline the dispersion theory basis of multiparticle \(j\)-plane analysis and describe how a gauge theory can be studied by combining Ward identity constraints with the group structure of reggeon interactions. The derivation of gluon reggeization, the \(O(g^2)\) BFKL kernel, and \(O(g^4)\) corrections, is described within this formalism. We give an explicit expression for the \(O(g^4)\) forward “parton” kernel in terms of logarithms and evaluate the eigenvalues. A separately infra-red finite component with a holomorphically factorizable spectrum is shown to be present and conjectured to be a new leading-order partial-wave amplitude. A comparison is made with Kirschner’s discussion of \(O(g^4)\) contributions from the multi-Regge effective action.

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We give a short review on the application of perturbative QCD to the description of the Regge asymptotics of hadronic scattering amplitude. Considering as an example the small \(x\) behaviour of the structure functions of deep inelastic scattering in the generalized leading logarithmic approximation, we show that the Regge asymptotics are governed in perturbative QCD by the contribution of the color-single compound states of reggeized gluons. The interaction between Reggeons is described by the effective Hamiltonian which in the multi-color limit turns out to be identical to the Hamiltonian of the completely integrable one-dimensional XXX Heisenberg magnet of noncompact spin \(s=0\). We discuss the possibility to find the spectrum of the Reggeon compound states within the Bethe Ansatz approach.

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It is demonstrated that local quark–hadron duality prescription applied to several exclusive processes involving the pion, is equivalent to using an effective \(\bar qq\) (two-body) light-cone wave function \({\mit {\Psi }}^{(LD)}(x,k_{\perp })\) for the pion. This wave function models soft dynamics of all higher \(\bar qG \dots Gq\) Fock components of the standard light-cone approach. Contributions corresponding to higher Fock components in a hard regime appear in this approach as radiative corrections and are suppressed by powers of \(\alpha _s/\pi \).

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We present a pedagogical review of the universal scaling properties displayed by the structure function \(F_2\) at small \(x\) and large \(Q^2\) as measured at HERA. We first describe the derivation of the double asymptotic scaling of \(F_2\) from the leading-order Altarelli–Parisi equations of perturbative QCD. Universal next-to-leading order corrections to scaling are also derived. We explain why the universal scaling behaviour is spoiled when the initial distributions rise too steeply by considering the nonsinglet distribution \(F_2^p\)–\(F_2^n\) as an explicit example. We then examine the stability of double scaling to the inclusion of higher order singularities, explaining how the perturbative expansion at small \(x\) can be reorganized in such a way that each order is given by the sum of a convergent series of contributions which are of arbitrarily high order in the coupling. The wave-like nature of perturbative evolution is then shown to persist throughout almost all the small \(x\) region, giving rise asymptotically to double scaling for a wide class of boundary conditions.

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I discuss various approaches to the high-energy scattering in quantum gravity, which are based on perturbation theory. First, the results for the elastic scattering amplitude, obtained within the eikonal approximation, are reviewed. Then, the effective action approach to the gravity in the multi-Regge kinematics is discussed.

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Composite fermion, which is an electron carrying an even number of vortices of the many body wave function, is a new kind of topological particle, formed in a range of parameters when electrons in two dimensions are subjected to a strong magnetic field. The composite fermions have the same charge and statistics as electrons, but differ from electrons in the important respect that they experience a drastically reduced magnetic field. This article gives an elementary introduction to composite fermions and describes how they help gain a simple understanding of the dramatic phenomena exhibited by two-dimensional electrons in high magnetic fields. It is based on lectures given at the “XXXV Jubilee Cracow School of Theoretical Physics” in Zakopane, Poland.

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We overview the properties of a quantum gas of particles with the intermediate statistics defined by Haldane. Although this statistics has no direct connection to the symmetry of the multiparticle wave function, the statistical distribution function interpolates continuously between the Fermi–Dirac and the Bose–Einstein limits. We present an explicit solution of the transcendental equation for the distribution function in a general case, as well as determine the thermodynamic properties in both low- and high-temperature limits.

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In this talk I would like to report a set of new developments in the Lund Model for Quark and Gluon Interactions. We feel that by the completion of the Linked Dipole Chain Model (developed together with G. Gustafson and a graduate student J. Samuelsson) we have a complete description of all kinds of perturbative QCD interactions (although only a few are reported in my talk) in terms of dipoles. This model describes the evolution of the wave function (or rather the square of it as measured by the partonic structure functions) in terms of space-like cascades of connected dipoles with both a mass and a virtuality. In the same way the Lund Dipole Cascade Model, which has been presented before repeatedly, describes the time-like perturbative cascades in terms of the building of dipoles decaying into smaller dipoles until the fragmentation process into hadrons (“the ultimate dipoles”) sets in.