I am reading this paper by Tatarskii, which serves as an introduction to the Wigner representation of quantum mechanics.
There is a step in the paper involving the Weyl transform that does not seem valid to me. Define the ordered operator function assigned to the function $f(p,q)$ as $$ \begin{align} \left\{ f(\hat{p}, \hat{q}) \right\} &\triangleq f\left(\frac{1}{i} \frac{\partial}{\partial \lambda}, \frac{1}{i} \frac{\partial}{\partial \mu} \right) \hat{F}(\lambda, \mu) \vert_{\lambda, \mu=0} \\ &= \frac{1}{4\pi^2} \iiiint\!\! d\lambda d\mu dp dq ~~f(p,q)\exp\left(-i(\lambda p + \mu q)\right) \hat{F}(\lambda, \mu) . \end{align} \tag{2.1} $$ This is equation 2.1 in the above paper for reference. The author goes on to explain that our choice of function $\hat{F}$ will determine the actual ordering of the resultant operator equation, but that, no matter what, it should satisfy some listed properties. One of these is that we should have $\hat{F}(0, \mu) = e^{i \mu \hat{q}}$. He then claims that
If $\hat{F}$ satisfies these conditions, then in constructing by means of Eq 2.1 the functions $\{f(\hat{p}, \hat{q}) \}$, functions of the form $f(\hat{p})$ or $f(\hat{q})$, which depend only on one variable, are obtained from the corresponding functions $f(p)$, $f(q)$ by the simple replacement $q \rightarrow \hat{q}$ and $p \rightarrow \hat{p}$.
I am willing to believe this is the case, but I fail to see how it works operationally. If we assume we just have a function of one variable, and that $\hat{F}$ satisfies the above condition, then we have $$ \begin{align} \left\{f(\hat{q}) \right\} &= \frac{1}{4\pi^2}\int \int \int \int d\lambda d\mu dp dq f(q)\exp\left(-i\mu q)\right) \exp(i \mu \hat{q}) \\ &= \frac{1}{4\pi^2} \int \int d\mu dq f(q) \exp(-i\mu(q-\hat{q})) \\ &= \frac{1}{2\pi} \int dq f(q) \delta(q-\hat{q})~, \end{align} $$ where I have used the fact that $(1/2\pi) \int \exp(ik(x-y)) dk = \delta(x-y)$.
This final equation seems reasonable, in the sense that it is telling me to replace my $q$s with $\hat{q}$s and be done with it. But I do not understand how to interpret $\delta(q-\hat{q})$, since we usually have that $\delta(0) = \infty$, but this implies $q=\hat{q}$ which doesn't even make sense since one is a c-number and the other is an operator. Is there a way to interpret a delta-function like this, or have I made a mistake somewhere in my derivation? Any pointers are greatly appreciated.
