, X_{i}(\tau)]\, .$$ Using equation (\[Eq22\]), the usual commutator $[X_{i}(0),{\pi}_j(0)]=i\hbar\delta_{ij}\uma$ and the properties of matrix $\mathbb{C}$ it is easy to show that $$\label{Eq25}
\sum_{i=1}^{2}[X_{i}(0) , X_{i}(\tau)]
 =
\frac{2i\hbar\sin(\omega\tau)\cos(\omega\tau)}{m\omega}\, ,$$ so that hamiltonian $\mathcal{H}_{\bot}$ with the appropriate time ordering takes the form $$\begin{aligned}
\mathcal{H}_\perp&=&\frac{m\omega^2}{2\sin^2{(\omega\tau)}}
\biggl\{{\bf R}^{2}(\tau)+{\bf R}^{2}(0)-2{\bf R}^{T}(\tau){\bf R}(0)\biggr\} \, \nonumber \\ &-& i
 \hbar \omega\cot(\omega\tau).
\label{Eq27}\end{aligned}$$ Substituting this hamiltonian into equation (\[EqDiferencialPropagador2D\]) and integrating in $\tau$, we obtain $$\label{ProtoPropagador}
\langle{\bf r},\tau\vert{\bf r}^{\prime},0\rangle =
 \frac{C({\bf r},{\bf r}^{\prime})}{\sin{(\omega\tau)}}
\exp\biggl\{{\frac{im\omega}{ 2\hbar}}\cot(\omega\tau)({\bf r}-{\bf
r}^{\prime})^2\biggr\} ,$$ where $C({\bf r},{\bf r}^{\;\prime})$ is an integration constant to be determined by conditions (\[CondicaoP(tau)\]), (\[CondicaoP(0)\]) and (\[CondicaoInicial\]), which for the case of hand read $$\begin{aligned}
\langle{\bf r},\tau\vert\pi_j(\tau)\vert{\bf r}^{\prime},0\rangle
\!\!&=&\!\!
 \left(-i\hbar\frac{\partial}{\partial x_j}-eA_j({\bf r})\right)
\langle{\bf r},\tau\vert{\bf r}^{\prime},0\rangle
\label{Eq30}\\
\langle{\bf r},\tau\vert\pi_j(0)\vert{\bf r}^{\prime},0\rangle
 \!\!&=&\!\!
\left(i\hbar\frac{\partial}{\partial x^\prime_j}-eA_j({\bf
r}^{\prime})\right) \langle{\bf r},\tau\vert{\bf
r}^{\prime},0\rangle\ ,
\label{Eq31}\\
\lim_{\tau\rightarrow 0^+}\langle{\bf r},
 \tau\vert{\bf r}^{\prime},0\rangle
 \!\!&=&\!\! \delta^{(2)}({\bf r}-{\bf
r}^{\prime}).\hspace{3.45cm} \label{Eq32}\end{aligned}$$ In order to compute the matrix element on the l.h.s. of (\[Eq30\]), we need to express $\mbox{{\mathversion{bold}${\Pi}$}}(\tau)$ in terms of ${\bf R}(\tau)$ and ${\bf R}(0)$. From equaitons (\[Eq23\]) and (\[Eq24\]), we have $$\mbox{{\mathversion{bold}${\Pi}$}}(\tau)=
\frac{m\omega}{\sin{(\omega\tau)}}\mbox{\large
$e^{\omega\tau\mathbb{C}}$} \biggl({\bf R}(\tau)-{\bf R}(0)\biggr),
\label{Eq33}$$ which leads to the matrix element $$\begin{aligned}
\langle{\bf r},\tau\vert\pi_j(\tau)\vert{\bf r}^{\prime},0\rangle
 &=&
 m\omega[\cot(\omega\tau) \left(x_j-x^\prime_j \right) \nonumber\\
&+& \mbox{\large $\epsilon_{jk3}$}\left(x_k-x^\prime_k\right)]
\langle{\bf r},\tau\vert{\bf r}^{\prime},0\rangle\ , 