Fock State Representation of Wilson's Equation

Abstract

We apply Wilson's renormalization group analyses to the two dimensional theory interacting with the string field and derive the Fock state representation of Wilson's equation for the string field. We solve this equation and show that in contrast with the usual string field theories in order to generate the correct S ·matrices this equation requires a summation rule, which can convert the infinite summations arising from contractions of two string field vertices to some definite functions. theory of elementary particles, it has become clear that the study of ·off·shell prop­ erties of the theories is crucial to uncover the dynamical structures of the vacuum. The approaches to the off-shell string theory are classified into two parts; one is the string field l ) and the other is the two dimensional renormalization group (RG) ana­ lyses. 2),3) In the former approach the Riemann surfaces are decomposed into some funda­ mental blocks; string fields, vertices and propagators. The off-shell theory is defined by the string field action. The propagator corresponds to the strip with a finite length and without any cut. The vertices connect these strips with each other. Therefore in the path-integral formalism the vertices are represented by the delta-functionals for the coordinates of. the strings on the common regions of the strips. In the operator formalism;·on the other hand, they are expressed by the direct products of some Fock states to satisfy the particluar overlap conditions corresponding to the delta­ functionals. In the RG analyses we impose the conformal invariance or equivalently the cutoff-independence not on amplitudes but on the partition function of the two dimensional theory with some background fields, which appear as coupling constants . in the· theory. Some constraints to the backgrounds are given by this condition and are identiiied with the equations of motion for the background string fields.