link $\ell\in\Lambda^1$ we replace the classical connection configuration space $G$ by an algebra which describes a bosonic particle on $G$ (the gluons).

Equip the space of classical matter fields $\prod\limits_{x\in\Lambda^0}\Cn=\{f:\Lambda^0\to\Cn\}$ with the natural pointwise inner product ${\langle}f, h {\rangle}=\sum\limits_{x\in\Lambda^0}\big({f(x)},\, h(x)\big)_{\Cn}$, and take for the quantized matter fields the CAR-algebra ${\mathfrak F}_{\Lambda} :=  {\rm CAR}\big(\prod\limits_{x\in\Lambda^0}\Cn)$. That is, for each classical matter field $f\in\prod\limits_{x\in\Lambda^0}\Cn$, we associate a fermionic field $a(f)\in{\mathfrak F}_{\Lambda}$, and these satisfy the usual CAR–relations: $$\{a(f), a(h)^*\}  = {\langle}f, h {\rangle}{\mathbf{1}}\quad \hbox{ and } \quad  \{a(f),a(h)\} = 0
\quad \mbox{ for } \quad f, h\in \prod\limits_{x\in\Lambda^0}\Cn$$ where $\{A,B\} := AB + BA$ and ${\mathfrak F}_{\Lambda}$ is generated by the set of all $a(f)$. As $\Lambda^0$ is finite, ${\mathfrak F}_{\Lambda}$ is a full matrix algebra, hence up to unitary equivalence it has only one irreducible representation.

In physics notation, the quark at $x$ is given by $a(\delta_x{\bf v}_i)=\psi_i(x)$ where ${\{{\bf v}_i\mid i=1,\ldots n\}}$ is an orthonormal basis for $\Cn$ and $\delta_x:\Lambda^0\to\R$ is the characteristic function of $\{x\}$. Further indices may be included if necessary, e.g. by putting $\Cn = \b W.\otimes\C^k$ where $\b W.$ has non–gauge degrees of freedom (such as the spinor part), and $\C^k$ has the gauge degrees of 