how this can be done explicitly in two examples in 1D: the Heisenberg-Ising model and the Ising model in a transverse field (ITF).

This paper is organized as follows. In Sect.2 we present a brief introduction to the Block Renormalization Group methods based upon the concept of the intertwiner operator $T$. This will allow us to see the truncation procedure inherent to the BRG method as tensoring representations of the Hamiltonian symmetry algebra, the intertwiner operator $T$ being a Clebsch-Gordan operator. In Sect.3 we use the Heisenberg-Ising model to show how the truncation process pertaining to the real-space RG is nothing but a tensor product decomposition of irreps of the symmetry algebra. The properties of the model are qualitative and quantitative well described by the BRG equations in the massive region $\Delta > 1$, while in the massless region one predicts the massless spectrum [*but not criticality at each value of* ]{} $\Delta > 1$. This latter fact is rather subtle and elusive. In Sect. 4 we set up the foundations of the qRG method using the Heisenberg-Ising model as an example. We derive the new qRG equations and show that the RG-flow diagram obtained in this fashion exhibits the 