(GRG) and refer to the corresponding RG transformation as GRGT.

The construction of the local GRG transformation {#constr}
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So far we have discussed different types of quantum decompositions and types of auxiliary spaces. We now turn to the question how to construct the embedding map $G_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}$ and the truncation map $G^+_{{\cal H}\otimes{\cal H}_{\text{aux}}}$. In (\[transf2\]) we used the functional dependence ${\cal O}$ to introduce physical constraints within the RG transformation. To determine $G_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}$ and $G^+_{{\cal H}\otimes{\cal H}_{\text{aux}}}$ we introduce another constraint. In addition to keeping the structure of the operator ${\cal O}$ we relate ${\cal O}$ to a physical quantity $\cal Z(\cal O)$ which acts as a physical invariant $\!\!$

[^2]. Equating the original physical quantity $\cal Z(\cal O)$ calculated from the original quantum lattice and the effective physical quantity $\cal Z(\cal O')$ for the reduced lattice we obtain $G^+_{{\cal H}\otimes{\cal H}_{\text{aux}}}$ and $G_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}$ from $$\label{inveq}
{\cal Z}\left[{\cal O}_{{\cal H}\otimes{\cal H}_{\text{aux}}}
  \left({\bf K}\right)\right]\,=\,{\cal Z}\left[
  G^+_{{\cal H}\otimes{\cal H}_{\text{aux}}}\,\circ\,
  {\cal O}_{{\cal H}\otimes{\cal H}_{\text{aux}}}\left({\bf K}\right)\,\circ\,
  G_{{\cal H}\otimes{\cal H}_{\text{aux}}}\right]
\,=\,{\cal Z}\left[{\cal O}_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}
  \left({\bf K}'\right)\right]\;.$$ We refer to equation (\[inveq\]) as the [*[invariance relation]{}*]{} for the RGT. Finally we have to decompose $G^+_{{\cal H}\otimes{\cal H}_{\text{aux}}}$ and 