state must incorporate a factor that depends on an interaction, but the list of published papers that accept this premise is much longer than the salient examples cited here. The underlying problem is the assumption that a gauge transformation transforms the Hamiltonian unitarily. That problem exists in all of the references just cited, although it is usually submerged in complicated manipulations. It is especially clear in Ref. [@jbauer], where it is specified that all operators $O$ transform under a gauge transformation according to the unitary-transformation rule$$\widetilde{O}=UOU^{-1},\qquad U^{-1}=U^{\dag}. \label{ap}$$ That specification is applied to $H$, in violation of the condition (\[am\]), and to the interaction Hamiltonian $H_{I}$, with no explanation for how it is possible to gauge-transform from the length-gauge interaction to any other gauge in view of the absence of operators in the scalar potential $\mathbf{r\cdot E}$. In the scheme proposed in Refs. [@yang; @kobesmirl; @beckerss; @lss; @jbauer], if the problem is initially formulated in the context of the $\mathbf{r\cdot E}$ potential, it is never possible to transform to any other gauge. This explains the use of the phrase gauge-invariant formulation with respect to $\mathbf{r\cdot E}$ to be found in some published works.

Locality and nonlocality
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Fields are derived 