quantities by enlarging the superblock is used as an estimate for the accuracy of the method.\
It is apparent that only in the case of an exact RGT we are able to calculate global quantities like the total ground state energy shift. Since we are mainly interested in an overall effective coupling determining the RG flow we are looking for exact RGTs.

Conclusions
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We have invented a non perturbative quantum RG method based on the idea of an additional auxiliary space. The work was motivated by the success of the DMRG concerning numerical results and the open question of an underlying general mathematical framework.\
The main objects introduced in this article are the auxiliary space ${\cal H}_{\text{aux}}$ and the two maps $G^+_{{\cal H}_{I}\,\otimes\,{\left({\cal H}_{\text{aux}}\right)}_{I}}$ and $G_{{\cal H}'_{I}\,\otimes\,{\left({\cal H}'_{\text{aux}}\right)}_{I}}$ which generate the RGT. By using these quantities we were able to give the definition of an exact local RGT which is the final result of this work. An exact local RGT involves all the information provided by the physical system.\
In future work we will proceed by applying our abstract formalism presented here to quantum spin chains like the Heisenberg models and compare our results in the context of related 