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M.Guenin University of Genera, Switzerland 1. Green's Functions Of RELATIVISTIC QUANTUM FIELD TK30RY WITHOUT is still far from complete, specially from ASYMPTOTIC PARTICLES AND QUARKS CONFINEMENT a conceptual point of view, in particular the M.Guenin causal behaviour is rather tricky. University of Genera, Switzerland We can define a field theory by simply giving its vertex and its propagators, this is at least true for the perturbative expansion 1. Green’s Functions of Quantum Field Theories of the ^-matrix and for writing the Bethe-Sal- The бгееп’e functions of a field theory are peter equation. In such a case, it is of course determined Ъу imposing a certain number of bo­ necessary to verify that it corresponds to undary conditions. Among them, is a condition a relativistic, local, causal theory if one on the behaviour at infinity which is usually wants such properties to be true. tacitely assumed, namely that the Green func­ We define what we call a space-like field tion should vanish at infinity in X space. theory for a scalar or vector gluon by the rules: The reason for the existence of this condition 1) Write the propagator inX space, as Bes­ is that one wants asymptotically free particles, sel function (this is necessary because our because a non vanishing propagator function, at propagator does not have a Fourier transform infinity implies a non vanishing potential for in the usual sense). the particles at infinity. 2) T^e space-like theory is obtained from This postulate is therefore perfectly the Feynman form by replacing № by - tyl. justified for standard scattering systems in This gluon field can be coupled with which the particles can be considered as asymp­ a Fermion quark field. Although it is possible totically free. We think, however, that there is to define space-like fields for Fermions as no reason to start from such a theory if one well, we shall treat the quark field as a normal does not want asymptotically free particles, for field, and hence the rule: instance if one wants to explain quark 3) The Fermion propagator (for quarks) are confinement, be specific, we criticize the j' o unchanged. Explicitly stated, we write for approaches based on standard field theory when­ a scalar gluon the propagator ever they try to construct complicated mecha­ nisms to explain the confinement of particles л* (*) ~ fa Щ f 9(*)pi b №) - which were implicitely assumed to be asymptoti­ Mi (-*№ )]- &/'*) К (-toЮ ) cally free to start with. We propose to build where^-X^-X^ end АЯ is the mass of the gluon field. a field theory on solutions of the wave equa­ As the Bessel functions entering the expression tions which are increasing at infinity in space­ of the propagator are multivalued, we define the like directions. The mathematical construction analytical continuation as being the one obtained of the field operators and the construction of by going from#} to-»»)by a path within the upper the Green's functions therefrom is discussed half plane and avoiding the origin. Instead of in a separate paper. The purpose of this note is going down as Z in a space-like to discuss only Green's function aspect and to direction out propagator goes as Z show how such a field theory could eventually behaviour in the time-like direction remaining be useful In describing bound quarks systems. The the same, hence the name "space-like fields" advantages of this approach are that very few for our solution. free parameters enter the theory and that it is Remark that our solution la a solution of the relatively easy to make numerical computations same inhomogeneous Klein-Gordon equation as the to test it. The disadvantage is that the theory usual propagator and hence, that it corresponds to a theory which is formally local in the the esoape of the new quark antiquark pair is interaction term. strongly suppressed in favour of the combina­ tion of the two new constituents with the two 2. THE STATIC APPROXIMATION7"1 Z former one. That means that the diagram showing Working out the observable consequence* connected quark lines is strongly favorized, of this approach In the fully relativistic which is precisely Zweig's rule. framework is difficult and lengthy. We want The static potential shows a minimum to study here what can be learned from the around m"1. Assuming the i-state of the bound very simple static approximation. state of the lightest quark-antiquarks pair We therefore study the solutions of the to correspond to a pion, we get that nfl should equation (Q -*•totx){p(x) = )iJ (X.) be of the order of magnitude of the pion radius. in the static limit ■= - Xj. (x) We remark that the potential is also relatively wherey/ij corresponds to the quark current. flat around the minimum. That means that in this The effective potential is then given by region at least the quarks are behaving as V(x) =_'I {x)d-*0 - fa:> * 'елр(+ m IXI) essentially free which makes this approach and, still in the static approximation, the compatible with the results of the deep inelastic Hamiltonian reduces to scattering experiments. The sign in front of the expression depends 5. Conclusions upon the charges of the quark currents. We shall The ideas presented above are still in consider only the cases where the Hamiltonian a very preliminary stage of development, but is positive, and only the scalar gluon case, detailed numerical computations including the as the vector gluon can be treated in a complete­ relativistic corrections are under way to make ly similar way. a comparison with the experimental results In the WEB approximation, considered in feasible. If the agreement is reasonable, then the potential given above, the wave function it will be time to disouss the more complicated of the quark behaves asymptotically essentially questions, specially causality. The abstract &e etpfimvipfinijjtoi large 2. This implies that theory of space-like fields is, however, etsen- the gluon wave function is going to behave es­ tially complete /2/. sentially in the same way asymptotically. From this we conclude that not only the quark* are References strictly confined , since an infinite energy 1. M.Guenin. Phys.Lett., 62B, 81 (1976). is needed to extract one out of the potential, 2. M.Flato, M.Guenin."On Space-Like Fields", but also that the gluons are confined. We can to appear in Helvetica Fhysica Acta. even draw much stronger conclusions. Indeed, if we consider for instance, a bound quark and antiquark system, a quark can emit a gluon who in turn can emit a new quark- antiquark pair, but as the wave function of the gluon decreases extremely rapidly, the pro­ bability for the event taking place "outside the potential" is exceedingly small. But "in» side", the number of states available is restric­ ted and one sees that the processus leading to.
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