EDIT: After doing some digging, I am convinced that the approach taken in this paper was simply an incorrect approach to deriving a quantum version of Hamilton's equations (also related to Ehrenfest's theorem). Perhaps a better way of tackling this issue would be to begin from the fundamental Heisenberg equation of motion, which is easy enough to derive from the Schrödinger equation, $$\frac{d \hat{A}_H}{dt} = \frac{i}{\hbar} [\hat{H}_H, \hat{A}_H] + \left( \frac{\partial \hat{A}}{\partial t} \right)_H$$ where the subscript $H$ denotes that the operator is in the Heisenberg picture, $\hat{A}_H = \hat{U}^\dagger \hat{A} \hat{U}$ where $\hat{U}$ is the unitary time evolution operator. From there, you can, for instance, use the fact that the momentum operator $\hat{p} = -i\hbar \nabla$ has no explicit time dependence to show $$\frac{d\hat{p}_j}{dt} = \frac{i}{\hbar}[\hat{H},\hat{p}_j]$$ where $\hat{p}_j$ is a component of the momentum. Then the quantum Hamilton's equation follows by just using the somewhat straightforwardly derived equation, $$\frac{\partial \hat{A}}{\partial x_j} = -\frac{i}{\hbar} [\hat{A}, \hat{p}_j] $$ $$\implies \frac{\partial \hat{H}}{\partial x_j} = -\frac{i}{\hbar} [\hat{H}, \hat{p}_j] $$ $$\implies \frac{d\hat{p}_j}{dt} = -\frac{\partial \hat{H}}{\partial x_j} $$ which is just the quantum version of Hamilton's equation of motion.
OLD QUESTION CONTEXT:
TLDR; I was attempting to rationalize the expansion of arbitrary quantum operators solely in terms of the position variables using a multi-variable chain rule, i.e. $$ \frac{d\hat{A}}{dt} = \frac{\partial \hat{A}}{\partial t} + \sum_k \frac{\partial \hat{A}}{\partial x_k} \frac{\partial x_k}{\partial t} $$ I suppose that this may be equivalent to the idea that because the position and momentum operators obey the relation $[\hat{x},\hat{p}_x] = i\hbar$, then other operators can necessarily only depend on one of them as a variable. But I have no idea how one would justify such a statement.
Here is the context for this question. A while back I was reading through Victor Stenger's ArXiv paper (PDF) about the role of symmetries in fundamental physics. Overall, the paper is a very nice introduction to these ideas, but I am stuck trying to resolve one aspect of his argument. After showing that the Schrödinger equation is the result of the special relativistic 4-momentum's time component being elevated to a quantum mechanical operator, he goes on to motivate the identification of the operators $\hat{p} = -i\hbar \nabla$ or $\hat{p}_j = -i\hbar \frac{\partial}{\partial x_j}$ with the momentum and $\hat{H} = i \hbar \frac{\partial}{\partial t}$ with the Hamiltonian, each obtained by considering infinitesimal generators of spacetime translations of the wave function. He motivates this identification by deriving the quantum operator analogues to Hamilton's equations from classical physics, $$\frac{\partial p_k}{\partial t} = -\frac{\partial H}{\partial x_k} \rightarrow \frac{\partial \hat{p}_k}{\partial t} = -\frac{\partial \hat{H}}{\partial x_k} $$ and $$\frac{\partial x_k}{\partial t} = \frac{\partial H}{\partial p_k} \rightarrow \frac{\partial \hat{x}_k}{\partial t} = \frac{\partial \hat{H}}{\partial p_k} $$ During these derivations (equation 4.26, though the momentum space equivalent is easy to derive), he asserts that we can expand the total differential of an arbitrary quantum operator $\hat{A}$ in terms of the position space coordinates or the momentum space coordinates only (depending on which representation you are using; he only explicitly treated position variables), giving the following formulae, $$\frac{d\hat{A}}{dt} = \frac{\partial \hat{A}}{\partial t} + \sum_k \frac{\partial \hat{A}}{\partial x_k} \frac{\partial x_k}{\partial t} $$ where $x_k$ refers to the various spatial coordinates, and, $$\frac{d\hat{A}}{dt} = \frac{\partial \hat{A}}{\partial t} + \sum_k \frac{\partial \hat{A}}{\partial p_k} \frac{\partial p_k}{\partial t} $$ where $p_k$ are the momentum components with respect to the spatial variables. What is the justification for this expansion of the total derivatives of the operator in each representation? Given that the phase space in classical mechanics corresponds to both the position and momentum variables, why can we justify truncating this expansion in terms of "half" of the variables? My gut says that it has to have something to do with the fact that we manifestly have $[\hat{x}_k, \hat{p}_j] = i\hbar \delta_{jk}$ from the definition of the momentum and position operators, which means that the corresponding components of position and momentum have incompatible eigenbases: $\langle x|p_x\rangle \neq 0$. Or, framed another way, if we project into the position basis and $| x \rangle$ becomes infinitely localized, then the momentum basis becomes infinitely delocalized. But it is completely unclear to me why that would translate into an inability to expand operators using both sets of variables. This restriction is necessary, however, as if you do expand in terms of both sets of variables you cannot derive the quantum Hamilton's equations.
