Wigner's quantum distribution function in spherical coordinates

The possibility is examined of applying the general method for introducing quantum distribution functions (QDF) proposed by Moyal to the concrete system of dynamical spherical variables $\widehat{\varphi}$, $\widehat{\theta}$, $\widehat r$, $\widehat p_{\varphi}$, $\widehat p_{\theta}$, $\widehat p_r$. The Wigner quantum distribution fimction found in this way in spherical coordinates determines the state of a system of particles only if the system is spherically or cylindrically symmetric. The changes are pointed out which must be made in Moyal's expressions for the basis operators $\widehat{\varphi}$, $\widehat{\theta}$, $\widehat r$, $\widehat p_{\varphi}$, $\widehat p_{\theta}$, $\widehat p_r$ if the distribution in the phase space $\varphi$, $\theta$, $r$, $n\hbar$, $m\hbar$, $p_r$ is to describe the state of any system of particles.