---
abstract: 'Based on the original idea of the density matrix renormalization group (DMRG) [@wh93], i.e. to include the missing boundary conditions between adjacent blocks of the blocked quantum system, we present a rigorous and nonperturbative mathematical formulation for the real-space renormalization group (RG) idea invented by L.P. Kadanoff [@ka66] and further developed by K.G. Wilson [@wi71]. This is achieved by using additional Hilbert spaces called auxiliary spaces in the construction of each single isolated block, which is then named a superblock according to the original nomenclature [@wh93]. On this superblock we define two maps called embedding and truncation for successively integrating out the small scale structure. Our method overcomes the known difficulties of the numerical DMRG, i.e. limitation to zero temperature and one space dimension.'
author:
- |
    Andreas Degenhard[^1]\
    Department of Mathematical Physics, University of Bielefeld,\
    Universitätsstraße 25, D-33615 Bielefeld, Germany\
title: |
    A nonperturbative\
    Real-Space Renormalization Group scheme
---

   PACS: 75.10.Jm

Introduction
============

Soon after K.G. Wilson’s dramatic success in applying a momentum space formulation of the renormalization group (RG) method [@ka66] to the Theory of Critical Phenomena and the Kondo Problem [@wi75] there was a considerable amount of efforts in applying the same type of approach 