I read on Wikipedia two different descriptions of the "Husimi-Q representation." One is that it is the Wigner function convolved with a Gaussian, which in particular results in a positive definite function. The other is that it is "essentially" (their words) the density matrix put into normal order. I had some trouble understanding why these are the same.
For instance, if we let $H=\omega a^\dagger a$, then the thermal state at inverse temperature $\beta$ is $$\rho=N\exp(-\beta H),$$ which (if I did everything right) normal-orders to $$:\rho:=N\exp(-\beta'H),$$ where $$\beta'=-\log(1-\beta\omega)/\omega.$$ This seems fairly pathological to me (ignoring the issue that this new density matrix doesn't seem to be normalized): we have $\beta'>\beta$, so the system is at a colder temperature, so I would expect the distribution in phase space to be "less blurry" rather than "more blurry," (certainly classically this is true), and at $\beta\omega=1$ we have singular behavior: for any temperature colder than $\omega$, it seems like the object we'll get out will assign negative probabilities to certain states.
Did I do the computation wrong? Does normal ordering here mean something different than pushing $a$'s to the right of $a^\dagger$'s? Are there other contexts in which we can think of normal ordering as smearing out distribution functions?