the traceless generators of $SU(N)$ in the fundamental representation, and $f^{ijk}$ are the structure constants of $SU(N)$. We choose a normalization where $$\begin{aligned}
\tr \left(\lambda^i\lambda^j\right) &=& \frac12 \delta^{ij},\label{norm1}\\
\sum_{i=1}^{N^2-1} \, \lambda^i_{ab} \, \lambda^i_{cd} &=& \frac12 \left(\delta_{ad}\, \delta_{cb} - \frac1{N} \, \delta_{ab}\, \delta_{cd}\right).\label{norm2}\end{aligned}$$

Let us now discuss the lattice Hilbert space. A general state $|\Omega \rangle$ is the direct product on all lattice sites, $$|\Omega \rangle = | \Omega \rangle_G \otimes | \Omega \rangle_{\Psi}.$$ Here the first factor is the projection of the state $|\Omega\>$ to the gauge field sector, while the second factor describes the fermionic sector. Any state $|\Omega\>_{\Psi}$ is the following direct product $$|\Omega\>_{\Psi} = \prod_{\otimes x} \left( |\Omega\>_{\Psi} \right)_x.$$ Concentrate on the Hilbert space of each site : the “lowest” state is the no-quantum “drained” state $|\textrm{dr}\rangle_x$, defined as $$\psi^{a\alpha}_x|\textrm{dr}\rangle_x = 0. \label{drained}$$ Applications of various $\psi^{\dag a\alpha}_x$ create the corresponding quarks on that site, $$|a\alpha \rangle_{x\Psi} = \psi^{\dag a\alpha}_x |\textrm{dr}\rangle_x.$$ For the free theory, $\alpha=1$ correspond to creation of positive energy excitations and $\alpha=2$ corresponds to creation of negative energy excitations. To use the usual quark–anti-quark language we write for each site and color $$\psi=\left( \begin{array}{c} b \\ d^{\dag} \end{array} \right).$$ $b^{\dag}$ creates a quark and 