\label{Eq34a}\end{aligned}$$ where we used the properties of matrix $\mathbb{C}$ and Einstein convention for repeated indices is summed. Analogously, the l.h.s. of equation (\[Eq31\]) can be computed from (\[Eq24\]), $$\begin{aligned}
\label{Eq34b}
\langle{\bf r},\tau\vert\pi_j(0)\vert{\bf r}^{\prime},0\rangle &=& m\omega[\cot(\omega\tau) \left(x_j-x^\prime_j \right) \nonumber\\
&-&\mbox{\large $\epsilon_{jk3}$}\left(x_k-x^\prime_k\right)] \langle{\bf r},\tau\vert{\bf r}^{\prime},0\rangle\ .\end{aligned}$$ Substituting equations (\[Eq34a\]) and (\[Eq34b\]) into (\[Eq30\]) and (\[Eq31\]), respectively, and using (\[ProtoPropagador\]), we have $$\begin{aligned}
\label{Equacao1ParaC}
\Big[i\hbar\frac{\partial }{\partial x_j} +
 eA_j({\bf r})\! + {1\over 2} e
F_{jk}(x_k \!-\! x^\prime_k)\Big] C({\bf r},{\bf r}^{\;\prime})\! \!\!&=&\!\! 0,\\
\Big[i\hbar\frac{\partial }{\partial x^\prime_j} -eA_j({\bf
r}^{\;\prime}) \! + {1\over 2} e F_{jk}(x_k \! -\!x^\prime_k)\Big]\,
 \!C({\bf r},{\bf r}^{\;\prime}) \!\!\!&=& \!\!0,\label{Equacao2ParaC}\end{aligned}$$ where we defined $F_{jk}={\mbox{\large $\epsilon$}}_{jk3}\, B$.

Our strategy to solve the above system of differential equations is the following: we first equation (\[Equacao1ParaC\]) assuming in this equation variables ${{\bf r}}^{\;\prime}$ as constants. Then, we impose that the result thus obtained is a solution of equation (\[Equacao2ParaC\]). With this goal, we multiply both sides of (\[Equacao1ParaC\]) by $dx_j$ and sum over $j$, to obtain $$\label{dlnC}
{1\over C}\left({\partial C\over\partial x_j}\; dx_j \right) =
{ie\over\hbar} \biggl[ A_j({\bf r})+{1\over 2}F_{jk}\left(
x_k-x^\prime_k\right)\biggr] \; dx_j\; .$$ Integration of the previous equation leads to $$\label{integracaoCrr'1}
C({\bf r},{\bf r}^{\;\prime}) = C({{\bf r}}^{\;\prime},{{\bf
r}}^{\;\prime})\;
 \mbox{\Large$ e^{\{ {ie\over \hbar}_{\;\;\Gamma}
\int_{\;\;\atop{{{\bf r}}^{\;\prime}}}^{\;{\bf r}}
 [ 