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  Here particle physics merges with condensed matter theory. Bothdisciplines study problems of infinite number of particles using for thesepurposes statistical description. In both disciplines `elementary excitations'or `particles' emerge not like independent jig-saw pieces, but as waveson a surface of the sea called vacuum. The only difference is that in theparticle theory studies the Ocean - the vacuum of all interactions andcondensed matter one deals with various small vacua - ferromagnetic, superconducting,spin liquid etc. Therefore it is not surprising that particle physics andcondensed matter like to borrow concepts and models from each other. Forexample, the Anderson--Higgs phenomenon of particle theory (screening ofthe weak interactions) appears in condensed matter as the Meissner effect(screening of the magnetic field in superconductors); the concept of `inflation'in cosmology is taken from the physics of first order phase transitions;the hypothetical `cosmic strings' are similar to magnetic field vortexlines in type II superconductors; the Ginzburg--Landau theory of superfluidHe$^3$ has many features common with the theory of hadron-meson interactionetc. When you realize the existence of this astonishing parallelism, itis very difficult not to think that there is something very deep aboutit, that here you come across a general principle of Nature according towhich same ideas are realized on different space-time scales, on differenthierarchical `layers', as a Platonist would put it. So instead of beinga jig-saw the Universe appears as a simphony where the same tune is playedparts and in different modifications. This view puts things in a new perspectivewhere truth is no longer `out there', but may be seen equally well in a`grain of sand' as in an elementary particle. 

  There is an area in theoretical physics where the parallels betweenhigh energy and condensed matter physics are especially strong. This areais the theory of strongly correlated systems. One outstanding problem inthis area is the problem of quark confinement - the fact that individualquarks are non-observable but always exist only inside of other particles.There many other problems however, a lot of them in condensed matter physics.It turns out that reduction of dimensionality may be of a great help insolving models of strongly correlated systems. Most nonperturbative solutionspresently known (and only nonperturbative ones are needed in physics ofstrongly correlated systems) are related to (1 + 1)-dimensional quantumor two-dimensional classical models. There are two ways to relate suchsolutions to reality. One way is that you imagine that reality on somelevel is also two dimensional. If you believe in this you are a stringtheorist. Another way is to study systems where the dimensionality is artificiallyreduced. Such systems are known in condensed matter physics; these aremostly materials consisting of well separated chains, but there are otherexamples of effectively one-dimensional problems such as problems of solitarymagnetic impurities in metallic host (Kondo effect) or of edge states inthe Quantum Hall effect. So if you are a theorist who is interested inseeing your predictions fulfilled during your life time, condensed matterphysics gives you a chance. Curiously enough the Kondo problem has turnedout to be intimately related with the notorious problem of the Schroedingercat. Its solution has greatly helped to resolve the corresponding paradoxes.At present, there are two approaches to strongly correlated systems. Oneapproach operates with exact solutions of many-body theories. Needlessto say not every model can be solved exactly, but fortunately many interestingones can. Some of my research is related to exactly solvable models. 

 The other approach is to try to reformulate complicated interactingmodels in such a way that they become weakly interacting. This is the ideaof bosonization which was pioneered by Jordan and Wigner in 1928 when theyestablished equivalence between the spin S = 1/2 anisotropic Heisenbergchain and the model of interacting fermions. Thus in just two years afterintroduction of the exclusion principle by Pauli it was established thatin many-body systems the wall separating bosons from fermions might becomepenetrable. The example of the spin-1/2 Heisenberg chain has also madeit clear that a way to describe a many-body system is not unique, but isa matter of convenience. The spin S = 1/2 Heisenberg chain has providedthe first example of `particles transmutation'. Here the many-body systemcan be equally well described with bosons and with fermions. The low-energyexcitations in this model differ drastically from the constituent particles.Of course, there are elementary cases when constituent particles are notobservable at low energies, for example, in crystalline bodies atoms donot propagate and at low energies we observe propagating sound waves --phonons; in the same way in magnetically ordered materials instead of individualspins we see magnons etc. These examples, however, are related to the situationwhere the symmetry is spontaneously broken, and the spectrum of the constituentparticles is separated from the ground state by a gap. Despite the factthat continuous symmetry cannot be broken spontaneously in (1 + 1)-dimensionsand therefore there is no finite order parameter even at T = 0, spectralgaps may form. This nontrivial fact, known as dynamical mass generation,was discovered by Vaks and Larkin in 1961. However, one does not need spectralgaps to remove single electron excitations since they can be suppressedby overdamping occuring when T = 0 is a critical point. In this case propagationof a single particle causes a massive emission of soft critical fluctuations.The fact that soft critical fluctuations may play an important role in(1 + 1)-dimensions became clear as soon as theorists started to work withsuch systems. It also became clear that the conventional methods wouldnot work. Bychkov, Gor'kov and Dzyaloshinskii (1966) were the first whopointed out that instabilities of one-dimensional metals cannot be treatedin a mean-field-like approximation. They applied to such metals an improvedperturbation series summation scheme called `parquet' approximation (seealso Dzyaloshinskii and Larkin (1972)). Originally this method was developedfor meson scattering by Diatlov, Sudakov and Ter-Martirosyan (1957) andSudakov (1957). It was found that such instabilities are governed by quantuminterference of two competing channels of interaction -- the Cooper andthe Peierls ones. Summing up all leading logarithmic singularities in bothchannels (the parquet approximation) Dzyaloshinskii and Larkin obtaineddifferential equations for the coupling constants which later have beenidentified as Renormalization Group equations (Solyom (1979)). From theflow of the coupling constants one can single out the leading instabilitiesof the system and thus conclude about the symmetry of the ground state.It turned out that even in the absence of a spectral gap a coherent propagationof single electrons is blocked. 

  The charge--spin separation -- one of the most striking featuresof one dimensional liquid of interacting electrons -- had already beencaptured by this approach. The problem the diagrammatic perturbation theorycould not tackle was that of the strong coupling limit. Since phase transitionis not an option in (1 + 1)-dimensions, it was unclear what happens whenthe renormalized interaction becomes strong (the same problem arises forthe models of quantum impurities as the Kondo problem where similar singularitieshad also been discovered by Abrikosov (1965)). The failure of the conventionalperturbation theory was sealed by P. W. Anderson (1971) who demonstratedthat it originates from what he called `orthogonality catastrophy': thefact that the ground state wave function of an electron gas perturbed bya local potential becomes orthogonal to the unperturbed ground state whenthe number of particles goes to infinity. (Particle transmutation includesorthogonality catastrophy as a particular case.) That was an indicationthat the problems in question cannot be solved by a partial summation ofperturbation series. This does not prevent one from trying to sum the entireseries which was brilliantly achieved by Dzyaloshinskii and Larkin (1974)for the Tomonaga--Luttinger model using the Ward identities. In fact, thesubsequent development followed the spirit of this work, but the changein formalism was almost as dramatic as between the systems of Ptolemeusand Kopernicus. 

  As it almost always happens, the breakthrough came from a changeof the point of view. When Kopernicus put the Sun in the centre of thecoordinate frame, the immensely complicated host of epicycles was transformedinto an easily intelligeble system of concentric orbits. In a similar waya transformation from fermions to bosons (hence the term {\it bosonization})has provided a new convenient basis and lead to a radical simplificationof the theory of strong interactions in (1 + 1)-dimensions. The bosonizationmethod was conceived in 1975 independently by two particle and two condensedmatter physicists -- Sidney Coleman and Sidney Mandelstam, and Daniel Mattisand Alan Luther respectively. (The first example of bosonization was consideredearlier by Schotte and Schotte (1969).) The focal point of their approachwas the property of Dirac fermions in (1 + 1)-dimensions. They establishedthat correlation functions of such fermions can be expressed in terms ofcorrelation functions of a free bosonic field. In the bosonic representationthe fermion forward scattering became trivial which made a solution ofthe Tomonaga--Luttinger model a simple exercise. The new approach had beenimmediately applied to previously untreatable problems. The results byDzyaloshinskii and Larkin were rederived for short range interactions andgeneralized to include effects of spin. It was then understood that low-energysector in one-dimensional metallic systems might be described by a universaleffective theory later christened `Luttinger-' or `Tomonaga--Luttingerliquid'. The microscopic description of such a state was obtained by Haldane(1981), the original idea, however, was suggested by Efetov and Larkin(1975). Many interesting applications of bosonization to realistic quasi-one-dimensionalmetals had been considered in the 1970s by many researches. Another quitefascinating discovery was also made in the 1970s and concerns particleswith fractional quantum numbers. Such particles appear as elementary excitationsin a number of one-dimensional systems, with typical example being spinonsin the antiferromagnetic Heisenberg chain with half-integer spin. Imaginethat you have a magnet and wish to study its excitation spectrum. You doit by flipping individual spins and looking at propagating waves. Naturally,since the minimal change of the total spin projection is $|\Delta S^z|= 1$ you expect that a single flip generates a particle of spin-1. In measurementsof dynamical spin susceptibility $\chi''(\omega, q)$ an emission of thisparticle is seen as a sharp peak. This is exactly what we see in conventionalmagnets with spin-1 particles beeing magnons. However, in many one dimensionalsystems instead of a sharp peak in $\chi''(\omega, q)$, we see a continuum.This means that by flipping one spin we create at least two particles withspin-1/2. Hence fractional quantum numbers. However, excitations with fractionalspin are subject of topological restriction -- in the given example thisrestriction tells us that the particles can be produced only in pairs.Therefore one can say that the elementary excitations with fractional spin(spin-1/2 in the given example) experience `topological confinement'. Topologicalconfinement puts restriction only on the overall number of particles leavingtheir spectrum unchanged. Therefore it should be distinguished from dynamicalconfinement which occurs, for instance, in a system of two coupled spin-1/2Heisenberg chains. There the interchain exchange confines the spinons backto form S = 1 magnons giving rise to a sharp single-magnon peak in theneutron cross section which spreads into the incoherent two-spinon tailat high energies. An important discovery of non-Abelian bosonization wasmade in 1983--4 by Polyakov and Wiegmann (1983), Witten (1984), Wiegmann(1984) and Knizhnik and Zamolodchikov (1984). The non-Abelian approachis very convenient when there are spin degrees of freedom in the problem.Its application to the Kondo model done by Affleck and Ludwig in the seriesof papers (see references in Part III) has drastically simplified our understandingof the strong coupling fixed point. The year 1984 witnessed another revolutionin low-dimensional physics. In this year Belavin, Polyakov and Zamolodchikovpublished their fundamental paper on conformal field theory (CFT). CFTprovides a unified approach to all models with gapless linear spectrumin (1 + 1)-dimensions. It was established that if the action of a (1 +1)-dimensional theory is quantizable, that is its action does not containhigher time derivatives, the linearity of the spectrum garantees that thesystem has an infinite dimensional symmetry (conformal symmetry). The intimaterelation between CFT and the conventional bosonization had became manifestwhen Dotsenko and Fateev represented the CFT correlation functions in termsof correlators of bosonic exponents (1984). In the same year Cardy (1984)and Bl\"ote, Cardy and Nightingale (1984) found the important connectionbetween finite size scaling effects and conformal invariance. Both non-Abelianbosonization and CFT are steps from the initial simplicity of the bosonizationapproach towards complexity of the theory of integrable systems. Despitethe fact that correlation functions can in principle be represented interms of correlators of bosonic exponents, the Hilbert space of such theoriesis not equivalent to the Hilbert space of free bosons. In order to makeuse of the bosonic representation one must exclude certain states fromthe bosonic Hilbert space. It is not always convenient to do this directly;instead one can calculate the correlation functions using the Ward identities.It is the most important result of CFT that correlation functions of criticalsystems obey an infinite number of the Ward identities which have a formof differential equations. Solving these equations one can uniquely determineall multi-point correlation functions. This approach is a substitute forthe Hamiltonian formalism, since the Hamiltonian is effectively replacedby Ward identities for correlation functions. Conformally invariant systemsbeing systems with infinite number of conservation laws constitute a subclassof exactly solvable (integrable) models. After many years of intensivedevelopment the theory of strongly correlated systems became a vast andcomplicated area with many distingushed researchers working in it. Differentpeople have different styles and different interests -- some are concernedwith applications and some with technical developments. There is stilla gap between those who develop new methods and those who apply them, butit is closing fast.