Question

For a Wigner-Ville 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. 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, the Husimi Q, or other representations?

Answer 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 ) \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.