full classical configuration space is thus $\big(\prod\limits_{x\in\Lambda^0}\Cn\big)\times
\big(\prod\limits_{\ell\in\Lambda^1}G\big)$, and the local gauge group $\prod\limits_{x\in \Lambda^0}G$ acts on it by $$\label{ClassGTr}
\big(\prod_{x\in\Lambda^0}v_x\big)\times\big(\prod_{\ell\in\Lambda^1}g_\ell\big)\mapsto
\big(\prod_{x\in\Lambda^0}\zeta(x)\cdot v_x\big)\times\big(\prod_{\ell\in\Lambda^1}\zeta(x_\ell)\,g_\ell\,\zeta(y_\ell)^{-1}\big)$$ where $ \ell=(x_\ell,y_\ell)$ and $\zeta \in\prod\limits_{x\in \Lambda^0}G$. Note that the orientation of links in $\Lambda^1$ was used in the action because it treats the $x_\ell$ and $y_\ell$ differently.

This is the basic classical kinematical model for which the quantum counterpart is given below for finite lattices.

The finite lattice model. {#PM}
-------------------------

In this subsection we want to state the model for finite lattice approximation of Hamiltonian QCD in $\R^3$ developed by Kijowski, Rudolph [@KR; @KR1]. A more expanded version of this section is in [@GrRu]. This model is based on the model constructed in the classical paper of Kogut [@K] (which elaborates the earlier one of Kogut and Susskind [@KS]).

Fix a finite connected subgraph $S$, and let $\Lambda_S:=(\Lambda^0_S, \Lambda^1_S, \Lambda^2_S)$. For ease of notation, we will omit the subscript $S$ in this section. Given such a finite lattice $\Lambda^0$, the model quantizes the classical model on $\Lambda^0$ above, by replacing for each lattice site $x\in\Lambda^0$, the classical matter configuration space $\Cn$ with the algebra for a fermionic particle on $\Cn$ (the quarks), and for each 