{\Gamma}^*_{\alpha\beta\gamma}$. Note that this equation can be rewritten as $$\label{Hevolution}
\frac{dW_\alpha}{dt} = \sum_\gamma r_{\alpha\gamma} W_\gamma \; ,$$ where $$\label{rdef}
r_{\alpha\gamma} = \frac{2}{\hbar} \sum_{\beta} \hbox{Im}(\Gamma_{\alpha\beta\gamma}) H_\beta .$$ So if we again think of $W_\alpha$ as the probability of the system being at the phase space point $\alpha$, then $r_{\alpha\gamma}$ is playing the role of the probability per unit time that a system at the point $\gamma$ will move to $\alpha$. We will refer to the $r$’s as transition rates, even though, like the $P$’s of the preceding section, they can be negative. In fact, one can see immediately from the definition (\[rdef\]) that $r_{\alpha\gamma}$ is antisymmetric in its two indices. Indeed $r$ must be antisymmetric, since it is the infinitesimal generator of an orthogonal matrix $P$ (Eq. (\[unitaryP\])). So if there is a certain probability $r$ per unit time of going from $\gamma$ to $\alpha$, then the probability per unit time of going from $\alpha$ to $\gamma$ is $-r$.

Another remarkable property of $r_{\alpha\gamma}$ is this: $$\sum_\gamma r_{\alpha\gamma} = 0,$$ as follows directly from Eq. (\[rdef\]). That is, the sum of the transition rates into $\alpha$ from all points in phase space is zero. This does not mean, of course, that 