Three applications of the renormalization group

Three applications of the exact renormalization group (RG) to field theory and string theory are developed. (1) First, $\beta$-functions are related to the flow of the relevant couplings in the exact RG. The specific case of a cutoff $\lambda\phi\sp4$ theory in four dimensions is discussed in detail. The underlying idea of convergence of the flow of effective lagrangians is developed to identify the $\beta$-functions. A perturbative calculation of the $\beta$-functions using the exact flow equations is then sketched. (2) Next, the operator product expansion (OPE) is motivated and developed within the context of effective lagrangians. The exact RG may be used to establish the asymptotic properties of the expansion. Again, the example field theory focused upon is a cutoff $\lambda\phi\sp4$ in four dimensions. A detailed proof of the asymptotics for the special case of the expansion of $\phi$(x)$\phi$(0) is given. The ideas of the proof are sufficient to prove the general case of any two local operators. Although both of the above applications are developed for a cutoff $\lambda\phi\sp4$, the analysis may be extended to any theory with a physical cutoff. (3) Finally, some consequences of the proposal by Banks and Martinec that the classical string field equation can be written as as exact RG equation are examined. Cutoff conformal field theories on the sphere are identified as possible string field configurations. The Wilson fixed-point equation is generalized to conformal invariance and then taken to be the equation of motion for the string field. The equation's solutions for a restricted set of configurations are examined--namely, closed bosonic strings in 26 dimensions. Tree-level Virasoro-Shapiro (VS) S-matrix elements emerge in what is interpreted as a weak component-field expansion of the solution. Then a general argument is given that the solutions of the conformal invariance fixed-point equation contain all the VS S-matrix elements. A simple calculation shows that moduli space is triangulated by the equation of motion.