SELF-CONSISTENT CALCULATION OF REAL-SPACE RENORMALIZATION-GROUP FLOWSAND EFFECTIVE POTENTIALS

We show how to compute real space renormalization group flows in latti ce field theory by a self-consistent method which is designed to prese rve the basic stability properties of a Boltzmann factor. Particular a ttention is paid to controlling the errors which come from truncating the action to a manageable form. In each step, the integration over th e fluctuation field (high frequency components of the field) is perfor med by a saddle point method. The saddle point depends on the block sp in. Higher powers of derivatives of the field are neglected in the act ions, but no polynomial approximation in the field is made. The flow p reserves a simple parameterization of the action. In the first part th e method is described and numerical results are presented. In the seco nd part we discuss an improvement of the method where the saddle point approximation is preceded by self-consistent normal ordering, i.e. so lution of a gap equation. In the third part we describe a general proc edure to obtain higher order corrections with the help of Schwinger-Dy son equations. In this paper we treat scalar field theories as an exam ple. The basic limitations of the method are also discussed. They come from a possible breakdown of stability which may occur when a composi te block spin or block variables for domain walls would be needed.