Gauss law and charged localization obstruction

Assume the exact finite-graph model and let $R\subseteq V$. Set $Q_R={\sum }_{x\in R}(n_x-b_{x}I)$ and let $\mathrm{\partial }R$ and its orientation signs be as in Region charge equals boundary electric flux. On ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\cap {\bigcap }_{e\in \mathrm{\partial }R}D(E_e)$, suppose a bounded operator $O$ preserves ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ and, for every boundary link $e$, maps $D(E_e)$ into itself and satisfies $E_{e}O\varphi =O{E}_e\varphi$ for every $\varphi \in D(E_e)$. Then $[Q_R,O]\psi =0$ for every vector in that physical domain. Consequently, a charge-changing operator with $[Q_R,O]\psi =qO\psi$ for a fixed $q\mathrm{\ne }0$ cannot both preserve the physical sector and commute with the boundary electric operators while acting nontrivially on any such physical vector.

There is also the following conditional continuum statement; it does not assert that a continuum representation exists. Let $D$ be a common invariant dense domain in a Hilbert space, and let $\rho (f)$ and $E_k(g_k)$ be linear assignments from real-valued test functions $f,g_k\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ to operators $D\to D$. For a real vector test function $\mathbf{g}=(g_1,g_2,g_3)\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3;{\mathbb{R}}^3)$ set $E(\mathbf{g})={\sum }_{k=1}^3{E}_k(g_k)$. Assume the assignments satisfy on $D$ $\rho (f)=-{\epsilon }_0{\sum }_{k=1}^3{E}_k({\mathrm{\partial }}_{k}f)(f\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)),$ for a fixed ${\epsilon }_0>0$, and let $K\subset {\mathbb{R}}^3$ be compact. Suppose $O$ is bounded, maps $D$ into itself, and satisfies $E(\mathbf{g})O\psi =OE(\mathbf{g})\psi$ for every $\psi \in D$ whenever $\mathrm{supp}\mathbf{g}$ is disjoint from $K$. For each integer $R\ge R_0$, choose a real-valued ${\chi }_R\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ equal to $1$ on a neighborhood of $K$ and with $\mathrm{supp}\mathrm{\nabla }{\chi }_R$ disjoint from $K$, and put $Q_R=\rho ({\chi }_R)=-{\epsilon }_{0}E(\mathrm{\nabla }{\chi }_R)$. Assume there is an operator $Q:D\to D$ such that, for every $\psi \in D$, both $Q_R\psi \to Q\psi$ and $Q_{R}O\psi \to QO\psi$ as integer $R\to \mathrm{\infty }$. Then $[Q,O]\psi =0$ on $D$. A bounded charge-changing operator satisfying $[Q,O]\psi =qO\psi$ for $q\mathrm{\ne }0$ on $D$ must therefore be the zero operator. These continuum conclusions are conditional on the displayed domain, locality, convergence and commutator-passage assumptions.