\sum_{\ell \in \Lambda^1}
  \bar\psi_{jn}(x_\ell)  \big[\underline\gamma\cdot(y_\ell-x_\ell)\big]_{ji}
  \Phi_{nm} (\ell)\psi_{im} (y_\ell)  + h.c.
  \nonumber \\
  & + & ma^3 \sum_{x \in \Lambda^0} \bar \psi_{jn} (x)  \psi_{jn}(x)\,,\end{aligned}$$ where $a$ is the assumed lattice spacing; $W (p)$ is the Wilson loop operator for the plaquette $p= (\ell_1, \ell_2 ,\ell_3, \ell_4)$; the vector $y_\ell-x_\ell$ for a link $\ell=(x_\ell,y_\ell)$ is the vector of length $a$ pointing from $x_\ell$ to $y_\ell$ and h.c. means the Hermitean conjugate. As usual for spinors, $ \bar \psi_{jn} (x)=\psi_{in}(x)^*(\gamma_0)_{ij}$, and we use the standard gamma-matrices. We have omitted the flavour indices. The summands occurring in are either Laplacians, Wilson loop operators or Fermi bilinears, hence they are all gauge invariant, hence are observables, some unbounded.

The above Hamiltonian suffers from the well-known fermion doubling problem (cf. [@FL01]);- the latter can be cured by passing e.g. to Wilson fermions [@Wilson2]. This modification does not affect the arguments below, hence we focus our analysis on the naive Hamiltonian given by (\[Hamiltonian\]). Below in Sect. \[LDA\] we will consider the dynamics produced by this Hamiltonian.

The Fermion algebra for an infinite lattice. {#FAlg}
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It is unproblematic to specify the Fermion field on an infinite lattice $\Lambda=(\Lambda^0, \Lambda^1, \Lambda^2)$:\
[**Assumption:**]{} [*Assume the quantum matter field 