way.

In the representation $\pi=\pi_F\otimes\big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}\pi_{\ell}\big)$ it is also possible to build unbounded observables as gauge invariant operators. For example we can build gauge invariant combinations of the gluonic and the colour electric field generators, and in the finite lattice context, such operators were analyzed in [@KR1; @JKR]. As an example of such a gauge invariant operator, in the context of a finite lattice, we state the Hamiltonian, where we disregard terms by which $H$ has to be supplemented in order to avoid the fermion doubling problem. We first need to add spinor indices, hence take $\Cn = \b W.\otimes\C^k$ where $\b W.\cong\C^4$ will be the spinor part on which the $\gamma$-matrices act. If $\{w_1,\ldots,w_4\}$ is an orthonormal basis of $\b W.$ and $\{e_1,\ldots,e_k\}$ is an orthonormal basis of $\C^k$, then w.r.t. the orthonormal basis $\{w_j\otimes e_n\mid j=1,\ldots,4,\,
n =1,\ldots,k\}$ of $\Cn$, we obtain the indices $$a(w_j\otimes e_n\cdot\delta_x)=:\psi_{jn}(x)$$ for the quark field generators, where the subscript $j$ is the spinor index, and the $n$ is the gauge index. Then the Hamiltonian is $$\begin{aligned}
\label{Hamiltonian}
  H & = & \tfrac{a}{2} \sum_{\ell \in \Lambda^1}
  E_{ij}(\ell) E_{ji}(\ell)
  + \tfrac{1}{2 g^2 a}\sum_{p \in \Lambda^2}
 ( W (p) + W(p)^*) \nonumber \\
  & + & i\tfrac{a}{2} 