evaluated along a straight line between ${\bf r}^{\prime}$ and ${\bf r}$. If for some reason we want to choose another path, instead of integral $\int_{{\bf r}^{\prime}}^{\bf r}
     {\bf A}(\mbox{{\mathversion{bold}${\xi}$}})\cdot
    d\mbox{{\mathversion{bold}${\xi}$}}$, we must evaluate $\int_{{\bf r}^{\prime}}^{\bf r}[
     {\bf A}(\mbox{{\mathversion{bold}${\xi}$}})-(1/2){\bf B}\times
    (\mbox{{\mathversion{bold}${\xi}$}}-{{\bf r}}^{\;\prime})]\cdot
     d\mbox{{\mathversion{bold}${\xi}$}}$.

2.  Since we solved the Heisenberg equations for the gauge invariant operators ${\bf R}_\bot$ and $\mbox{{\mathversion{bold}${\pi}$}}_\bot$, our final result is written for a generic gauge. Note that the gauge-independent and gauge-dependent parts of the propagator are clearly separated. The gauge fixing corresponds to choose a particular expression for $\bf A(\mbox{{\mathversion{bold}${\xi}$}})$. Besides, from (\[PropagadorFinalBUniforme\]) we imediately obtain the transformation law for the propagator under a gauge transformation ${\bf A} \rightarrow {\bf A} +
      \mbox{{\mathversion{bold}${\nabla}$}}\Lambda$, namely, $$\label{GaugeTransformation}
     K({\bf r},{{\bf r}}^{\;\prime};\tau)\longmapsto
    \mbox{\large $ e^{\frac{ie}{\hbar}\,\Lambda({\bf r})}$}\, K({\bf
    r},{{\bf r}}^{\;\prime};\tau)\, \mbox{\large $
    e^{-\frac{ie}{\hbar}\,\Lambda({{\bf r}}^{\;\prime})}$}\; .\nonumber$$ Although this transformation law was obtained in a particular case, it can be shown that it is quite general.

3.  It is interesting to show how the energy spectrum (Landau levels), with the corresponding degeneracy per unit area, can be extracted from propagator (\[PropagadorFinalBUniforme\]). With this purpose, we recall that the partition function can be obtained from the Feynman propagator by taking $\tau=-i\hbar\beta$, with $\beta=1/(K_BT)$, and taking the 