in terms of the fermions and the gluonic zero modes. For that we use the hermiticity of $E'_x$ and get $$\begin{aligned}
H'_E &=& \frac{g^2}2\sum_{ix} \left( E^{'i}_x \right)^2=\frac{g^2}2\sum_{ix} E^{'i\dag}_x E^{'i}_x=\frac{g^2}2\sum_{ix} E^{''i\dag}_x E^{''i}_x = \frac{g^2}2\sum_{ip} E^{''i\dag}_p\, E^{''i}_p\nonumber \\
&=&\frac{g^2}2\left( \frac2{L_s}\sum_{ab\atop x,y,p} ^\prime \, \frac{\rho_{x,ab}\, \rho_{y,ba}\, e^{i(y-x)\,p}}{4\sin^2 \left(\frac{p+\varphi_a-\varphi_b}{2}\right)} + \sum_{I=1}^{N-1} E^{''I\dag}_{p=0} E^{''I}_{p=0}\right). \end{aligned}$$ Here, by the primed sum, we mean that one must sum over all $a$, $b$ and $p$ for which the denominator is [*nonzero*]{}. This restriction is a direct consequence of one’s inability to resolve all of the Gauss law constraints and will become the source of the principle value prescription often used in the ‘t Hooft model.

Realization of the zero modes in the Schröedinger picture {#0mode}
=========================================================

The resolution of Gauss law left us in the glue sector with a restricted set of dynamical degrees of freedom – the $N-1$ eigenvalues of the spatial Polyakov loops, $\varphi_a$, and their $N-1$ conjugate momenta $E^{I''}_{p=0}$. In this section we focus on these operators to which we collectively refer as the zero modes. Specifically, we calculate their commutation relations and also find a simple way to realize $E^{I''}_{p=0}$ in the Schröedinger picture, where $\varphi_a$ are c-numbers. For that purpose we first calculate 