2 fixed star symbol

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:

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

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

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

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.

1

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:

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

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

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

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.