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Experimental data on high-energy interaction of elementary particles produced by present-day accelerators give still more convincing evidence in favor of the description of hadron physics by quantum chromodynamics (QCD). 1 -3 Fundamental objects of that theory are spinor fields associated with quarks that interact with non-Abelian gauge fields of massless vector gluons. Hadrons are regarded as bound states of the quarks. The QCD is capable of explaining the basic peculiarity in the quark behavior, i.e., that they do not interact with each other at short distances (the asymptotic freedom). However, whether the quark can exist in a free state or not is still an open problem in QCD.It may happen that at a distance between quarks as small as the hadron size, favored from the energy standpoint are those configurations of gluon fields that do not fill the whole space (as in electrodynamics) but rather concentrate along the lines connecting quarks.." 7 The energy of two quarks coupled by a gluon-field tube is proportional to the distance between the two quarks. The forces of attraction between quarks thus do not decrease as the distance increases; they instead remain constant. Therefore, no external agent of any sort can break that coupling and produce a free quark. This line of reasoning in the modern quantum chromodynamics is postulated as a hypothesis of quark confinement. An important evidence for this hypothesis, besides qualitative arguments in the framework of QCD, is the absence of experimental indications of the existence of free quarks. 1 Introduction to the Relativistic String Theory Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 10/04/15. For personal use only.The first term is the classical energy of the gluon field inside the tube; for simplicity, it is considered homogeneous. The second term is the energy opposite in sign to that of the vacuum fields expelled by the gluon-field tube. The strength of the chromoelectric field |E a | is given by the field flux $ generated by a quark-antiquark pair(1.2)The transverse dimensions of the tube are determined from the require ment for the energy per unit length to be a minimum, de/dR = 0. As a result,1.3) Introduction to the Relativistic String Theory Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 10/04/15. For personal use only. Action Functional for a Relativistic String...The above consideration is purely classical. There does not, of course, exist a complete quantum theory of that phenomenon. One-loop calcula tions 18 show that this picture in outline is preserved at a quantum level, as well. If the flux of the chromoelectric field inside the gluon tube is small compared to the critical value, the quantum fluctuations are also small and they only reduce the energy density per unit length of the gluon tube.Configurations of gluon fields localized along the lines connecting quarks are simulated by relativistic strings with point masses at the ends. The rel ativistic string model is much simpler than...

Fundamental difficulties that arose in constructing quantum theory of the relativistic string (nonphysical dimensionality of space-time, tachyon states) stimulated a search for nonstandard approaches in that model. One of the approaches, a geometrical approach, describes the world surface of the relativistic string with the differential forms obeying the integrability conditions, the Gauss-Petersson-Codazzi-Ricci nonlinear partial differential equations. 149 In the geometrical approach, these equations are considered as equations of motion giving the string dynamics." A remarkable property of those equations is the possibility to construct their general solution. 150 The simplest equations of that sort are the Liouville nonlinear equation and a simplified system of two Lund-Regge nonlinear equations. The Liouville equation appears, besides in the relativistic string, also in the study of instanton solutions in non-Abelian gauge theories; whereas the Lund-Regge system, in a nonlinear two-dimensional sigma-model with the symmetry group SO (4).Before we proceed to quantization of the relativistic string within the geometrical approach, it is necessary to study, at a classical level, the non linear equations describing the string dynamics in this approach, which is just the subject of the present chapter. "First such a consideration of the string model has been proposed in the paper: R. Omnes, Nucl. Phyt. B149 (1979) 269. 121 Introduction to the Relativistic String Theory Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 10/04/15. For personal use only.(in (18.4) there is no summation over a), where the first m vectors e?, i = 1,... , m are tangents to surface (18.1), and the remaining n-m vectors c£, a = m + 1,... , n are normals to the same surface. The factors c M and e a in (18.4) allow for the metric signature of the enveloping space. We shall put the origin of basis (18.3) at the point x^fu 1 , u 2 ,... , u m ). Recall that the set of m vectors tangent to surface (18.1) is given by partial derivatives a^, t' = 1,2,... , m. In the general case, these vectors are not orthonormal, Introduction to the Relativistic String Theory Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 10/04/15. For personal use only. Geometrical Approach in the RdativiiUc String Theory 123 but owing to (18.2) they are certainly independent at every point of the surface. Therefore, one can always construct basis (18.3).It turns out that if a basis like that is known at every point of a surface, the surface may also be reconstructed from it. If we are interested in the lo cal properties of the surface, then it suffices to find the differential equations that define the change of basis (18.3) when its origin ^(u 1 ,!! 9 ,... ,u m ) moves along the surface. These equations describe the change of a radiusvector of the surface, (fa* =«*«?, t=l,...,m, (18.5) and of the basis unit vectors e£, < = n«$ (18.6) when the basis {xM, e£, a = 1,... , n} moves along the surface. Here u>> and D,*'...

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The requirement of reparametrization invariance does not allow us to distribute the mass, as well as the charge, along the string; the only pos sibility is to put the masses at the string ends. If, however, we give up the reparametrization invariance, we cannot consistently introduce into the theory the constraints on dynamical variables (gauge conditions) with the use of which we could eliminate the states with negative norm. 91 As mentioned in the Introduction, the model of a relativistic string with point-like masses at ends provides a clear picture of the quark confinement in hadrons. That picture of quark interaction is supported by rather rea sonable considerations based on quantum chromodynamics. 4 " 7 A good deal of works 39 -92 " 102 are devoted to the study of the relativis tic string with massive ends. Even at the classical level that problem faces difficulties of principle, such as, it is impossible to resolve the nonlinear boundary conditions in an explicit form. Only particular results have been obtained along this line. Some types of motion of the relativistic string with massive ends (e.g., its rotation as a whole) was investigated 92 , 93100 ; classi cal dynamics of that string was thoroughly studied in the two-dimensional space-time 97 ; an exhaustive analysis was made of the nonrelativistic limit of the problem and it was shown how there arises a potential linearly growing with distance. 39 ' 95 Some modifications of the relativistic string model with massive ends were considered that admit quantization of the 69 Introduction to the Relativistic String Theory Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 10/04/15. For personal use only. Intmduction to the Relativiitic String TheoryThe action of the relativistic string with massive ends is chosen as follows, 92 " 94 S = -7 / dr day/{xx>Y -i 2 *' 2 where the constant 7 is of dimensionality of the mass squared; m 1 and m 2 are masses at the string ends; x"(r,f| i as(r| x * = dx»/du i ,i = 0,l, u ± =«°±u 1 .When the equations describing the string dynamics are derived, the string coordinates x"(r, a) and functions a,(r) are to be varied in the action S. Variation of x"(r,cr) in (12.1) results in the equations of motion and boundary conditions. As in the free-string theory, on the world surface X I '(T,(l-