\exp \left[ +i\sum_x \sum_{i=1}^{N^2-1} \theta^i_x  \left( E^i_{x} - \left(U^{\rm Adj.}_{x-1}\right)_{ji} \, E^j_{x-1}\right)\right], \label{gauge_G}$$ where here the matrix of operators $\left(U^{\rm Adj.}_{x-1}\right)_{ji}$ is the link matrix in the adjoint representation, i.e.[^4] $$\left(U^{\rm Adj.}_{x}\right)_{ij}= 2 \, \tr \left( \lambda^i \, U_{x} \, \lambda^j \, U^\dag_{x}\right).$$

Putting , and together, we see that the operator that induces gauge transformations is $${\cal V}=\exp \left[ i\sum_x \sum_{i=1}^{N^2-1} \theta^i_x  \left( \rho^i_G - \rho^i_F \right)_x\right],\label{gauge_trnsf}$$ where $\rho^i_{Fx}$, and $\rho^i_{Gx}$ are the color charge densities of the fermions and of the gauge fields. These two quantities are given by $$\begin{aligned}
\rho^i_{Fx}&=&\psi^{\dag a}_x \lambda^i_{ab} \psi^b_x, \\
\rho^i_{Gx}&=&  E^i_x-\left( U^{{\rm}{Adj.}}_{x-1}\right)_{ji} E^j_{x-1} \equiv D E^i_x, \label{rhoG} \end{aligned}$$

Since the lattice Hamiltonian is gauge invariant, we know that the generators of the gauge transformations commute with the Hamiltonian $$\left[ H , \rho^i_{Gx} - \rho^i_{Fx} \right]=0, \hskip 1cm \forall i \quad \forall x.$$ This means that we can choose to work with a basis that block diagonalizes $\rho^i_G-\rho^i_F$. This breaks the Hilbert space to separate sectors classified by their eigenvalue $\rho^i_{\rm external,x}\equiv \rho^i_G-\rho^i_F$ on the lattice. Each set of values $\rho^i_{{{\rm}external},x}$ describes a different physical case, with a different [*external*]{} distribution of color charge (that can correspond, for example, to infinitely heavy quarks 