For a [Wigner-Ville quasi-probability distribution](http://en.wikipedia.org/wiki/Wigner_quasi-probability_distribution) $W(q,p,t)$ and Hamiltonian $H(q,p)$, we can write a quantum Liouville equation:


$\frac{\partial W(q,p,t)}{\partial t} = -\{\{W(q,p,t) , H(q,p )\}\}$


where $\{\{a, b\}\}$ is the [Moyal bracket](http://en.wikipedia.org/wiki/Moyal_bracket). This comes from applying a Wigner transformation to the Von Neumann equation for density matrices. Are there similar, clean-looking, quantum Liouville equations for the evolution of the [Glauber-Sudarshan P](http://en.wikipedia.org/wiki/Glauber_P_representation), the [Husimi Q](http://en.wikipedia.org/wiki/Husimi_Q_representation), or other representations?



Moyal brackets / quantum Liouville equation for non-Wigner representations?
I got an answer to this offline. The answer is that the form looks the same for all representations with a suitable redefinition of the star product. For example, in the Husimi Q representation, we can define a new kind of star product as follows:


$
f\circledast g= f \exp \left ( {{\frac{\hbar }{2}}(\stackrel{\leftarrow }{\partial }\_x
\stackrel{\rightarrow }{\partial }\_{p}+\stackrel{\leftarrow }{\partial }\_{p}\stackrel{\rightarrow }{\partial }\_{x})} \right ) \star g = f \exp \left ( {{\frac{\hbar }{2}}(\stackrel{\leftarrow }{\partial }\_x-i\stackrel{\leftarrow }{\partial }\_{p})(\stackrel{\rightarrow }{\partial }\_x+i\stackrel{\rightarrow }{\partial }\_{p})} \right ) g
$


If $T$ is the map from Wigner to Husimi, then this product has the following property:


$
T(f \star g) = T(f) \circledast T(g)
$


Therefore the quantum Liouville equation in the Husimi representation is


$
\frac{\partial Q}{\partial t} = - \frac{1}{i\hbar} (Q \circledast H - H \circledast Q)
$


See e.g. [this reference](http://www.hep.anl.gov/czachos/a.pdf).


