Field Algebra {#FieldAlgebra}
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We consider a model for QCD in the Hamiltonian framework on an infinite regular cubic lattice in $\Z^3.$ For basic notions concerning lattice gauge theories including fermions, we refer to [@Seiler] and references therein.

We first fix notation. For the lattice, define a triple $\Lambda:=(\Lambda^0, \Lambda^1, \Lambda^2)$ as follows:

-   $\Lambda^0:=\{(n,m,r)\in\R^3\,\mid\,n,\,m,\,r\in\Z\}=\Z^3$ i.e. $\Lambda^0$ is the unit cubic lattice and its elements are called sites.

-   Let $\wt\Lambda^1$ be the set of all directed edges (or links) between nearest neighbours, i.e. $$\wt\Lambda^1:= \{(x,y)\in\Lambda^0\times\Lambda^0\,\mid\,y=x\pm \b e._i\;\;\hbox{for some $i$}\}$$ where the $\b e._i\in\R^3$ are the standard unit basis vectors. Let $\Lambda^1\subset\wt\Lambda^1 $ denote a choice of orientation of $\wt\Lambda^1,$ i.e. for each $(x,y)\in\wt\Lambda^1$, $\Lambda^1$ contains either $(x,y)$ or $(y,x)$ but not both. Thus the pair $(\Lambda^0,\Lambda^1)$ is a directed graph, and we assume that it is connected.

-   Let $\wt\Lambda^2$ be the set of all directed faces (or plaquettes) of the unit cubes comprising the lattice i.e. $$\wt\Lambda^2:= \{(\ell_1,\ell_2,\ell_3,\ell_4)\in\big(\wt\Lambda^1\big)^4\,\mid\,Q_2\ell_i=Q_1\ell_{i+1}\;
    \hbox{for}\;i=1,2,3,\;\hbox{and}\; Q_2\ell_4=Q_1\ell_1\}$$ where $Q_i:\Lambda^0\times\Lambda^0\to\Lambda^0$ is the projection onto the $i^{\hbox{th}}$ component. Note that for a plaquette $p=(\ell_1,\ell_2,\ell_3,\ell_4)\in\wt\Lambda^2,$ it has an orientation given by the order of the edges, and the reverse ordering is $\overline{p}=(\overline\ell_4,\overline\ell_3,\overline\ell_2,\overline\ell_1)$ where $\overline\ell={(y,x)}$ if $\ell={(x,y)}\in \wt\Lambda^1$. In 