properties of the matrices $\mathbb{N}$ and $\mathbb{M}^{-}$ given by (\[Eq58\]) and (\[Eq59\]). In order to get the right time ordering, observe first that $$\textbf{R}^T(0)(\mathbb{M}^{-})^T \textbf{R}(\tau) =
\textbf{R}^T(\tau) \mathbb{M}^{-}\textbf{R}(0) + \Big[\left(
\mathbb{M}^{-} \textbf{R}(0)\right)_{i},\textbf{X}_{i}(\tau)\Big]
\,,$$ where $$\Big[\left( \mathbb{M}^{-}
\textbf{R}(0)\right)_{i},\textbf{X}_{i}(\tau)\Big]
 = i\hbar \mbox{Tr}\Big[\mathbb{N}(\mathbb{M}^{-})^{T} \Big]
= \frac{i\hbar}{m\Omega} \sin{(2\Omega\tau)}\, .$$ Using the last two equations into (\[Eq63\]) we rewrite the hamiltonian in the desired ordered form, namely, $$\begin{aligned}
\label{Hordenada1}
\mathcal{H}_{\bot}&=&\frac{m\Omega^{2}}{2\sin^{2}{(\Omega\tau)}}\Big[\textbf{R}^{2}(\tau)+\textbf{R}^{2}(0) - 2\textbf{R}^{T}(\tau)\mathbb{M}^{-}\textbf{R}(0)   \nonumber \\
&&  \qquad \qquad - \frac{i\hbar}{m\Omega} \sin{(2\Omega\tau)}
\Big]\ .\end{aligned}$$ For future convenience, let us define $$\begin{aligned}
U(\tau)&=&\cos{(\omega\tau)}\cos{(\Omega\tau)}
 + \frac{\omega}{\Omega}\sin{(\omega\tau)}\sin{(\Omega\tau)}\, , \label{Eq67} \\
V(\tau)&=&\sin{(\omega\tau)}\cos{(\Omega\tau)}
 - \frac{\omega}{\Omega}\cos{(\omega\tau)}\sin{(\Omega\tau)}
\label{Eq68}\end{aligned}$$ and write matrix $\mathbb{M}^{-}$, defined in (\[Eq59\]), in the form $$\mathbb{M}^{-}=U(\tau)\uma + V(\tau)\mathbb{C}. \label{Eq66}$$ Substituting (\[Eq66\]) in (\[Hordenada1\]) we have $$\begin{aligned}
\mathcal{H}_{\bot}&=&\frac{m\Omega^{2}}{2\sin^{2}{(\Omega\tau)}}\Big[\textbf{R}^{2}(\tau)+\textbf{R}^{2}(0)
 - 2U(\tau)\textbf{R}^{T}(\tau)\textbf{R}(0)   \nonumber \\
  &-&   2V(\tau)\textbf{R}^{T}(\tau)\mathbb{C}\textbf{R}(0)-\frac{i\hbar}{m\Omega}
  \sin{(2\Omega\tau)}\Big]\,.
\label{Eq69}\end{aligned}$$ The next step is to compute the classical function $F({\bf r},{\bf r^{\prime}};\tau)$. Using the following identities $$\begin{aligned}
\frac{\Omega U(\tau)}{\sin^{2}{(\Omega\tau)}}
 &=&
 - \frac{d}{d\tau} \Big[\frac{\cos{(\omega\tau)}} {\sin{(\Omega\tau)}}\Big]\, , \label{Eq70} \\
\frac{\Omega V(\tau)}{\sin^{2}{(\Omega\tau)}}
 &=&
 - \frac{d}{d\tau}\Big[\frac{\sin{(\omega\tau)}} {\sin{(\Omega\tau)}}\Big],
\label{Eq71}\end{aligned}$$ into (\[Eq69\]), we write $F(\textbf{r},\textbf{r}^{\prime};\tau)$ in the convenient form $$\begin{aligned}
\!\!\!\! F(\textbf{r},\textbf{r}^{\prime};\tau)
 \!\!&=&\!\!
 \frac{m\Omega^{2}}{2}(\textbf{r}^{2}+\textbf{r}^{\prime^{2}})\mbox{csc}(\Omega\tau)^{2}
\!\!+ m\Omega\textbf{r}\cdot\textbf{r}^{\prime}\! \frac{d}{d\tau}
\! \Big[ \frac{\cos{(\omega\tau)}} {\sin{(\Omega\tau)}}\Big] \nonumber \\
&+& \; m\Omega\textbf{r}\cdot \mathbb{C}\textbf{r}^{\prime}
 \frac{d}{d\tau} \Big[ \frac{\sin{(\omega\tau)}}
 {\sin{(\Omega\tau)}}\Big]-i \hbar \Omega
 \frac{\cos{(\Omega\tau)}}{\sin{(\Omega\tau)}}\, .
\label{Eq72}\end{aligned}$$ Inserting this result into the differential 