dimension of ${\cal H}$ is simply

$$dim{ \cal H} = k^N
\label{2}$$

As a matter of illustration we cite the following examples: $k=4$ (Hubbard model), $k=3$ (t-J model), $k=2$ (Heisenberg model) etc.

When $N$ is large enough the eigenvalue problem (\[1\]) is out of the capability of any human or computer means unless the model turns out to be integrable which only happens in some instances in $d=1$.

These facts open the door to a variety of approximate methods among which the RG-approach is one of the most relevant. The main idea of the RG-method is the mode elimination or thinning of the degrees of freedom followed by an iteration which reduces the number of variables step by step until a more manageable situation is reached. These intuitive ideas give rise to a well defined mathematical description of the RG-approach to the low lying spectrum of quantum lattice hamiltonians.

To carry out the RG-program it will be useful to introduce the following objects:

-   ${\cal H}  $ : Hilbert space of the original problem.

-   ${\cal H }' $: Hilbert space of the effective degrees of freedom.

-   $H $: Hamiltonian acting in $\cal H$.

-   $H'$: Hamiltonian acting in 