the appropriate time ordering.

Examples {#SectionExamples}
========

Charged particle in an uniform magnetic field {#cpumf}
---------------------------------------------

As our first example, we consider the propagator of a non-relativistic particle with electric charge $e$ and mass $m$, submitted to a constant and uniform magnetic field ${\bf B}$. Even though this is a genuine three-dimensional problem, the extension of the results reviewed in the last section to this case is straightforward. Since there is no electric field present, the hamiltonian can be written as $$\label{HamiltonianaBUniforme}
\mathcal{H}=\frac{\left({\bf P} - e{\bf A}\right)^{2}}{2m}
 = \frac{\mbox{\mathversion{bold}${\pi}$}^2}{2m}\ ,$$ where ${\bf P}$ is the canonical momentum operator, ${\bf A}$ is the vector potential and $\mbox{\mathversion{bold}${\pi}$}={\bf P}-e{\bf A}$ is the gauge invariant mechanical momentum operation. We choose the axis such that the magnetic field is given by ${\bf B}=B{\bf e_3}$. Hence, the hamiltonian can be decomposed as $$\mathcal{H} = \frac{\pi_{1}^{2}+\pi_{2}^{2}}{2m} +
 \frac{P_3^2}{2m} \; =\; \mathcal{H}_{\bot} +
 \frac{P_{3}^{2}}{2m}\ ,$$ with an obvious definition for $\mathcal{H}_{\bot}$. Since the motion along the ${\cal OX}_3$ direction is free, the three-dimensional propagator $K({\bf x},{\bf x}^{\prime};\tau)$ can be written as a product of a two-dimensional propagator, $K_{\bot}({\bf r},{\bf r}^{\prime};\tau)$, related to the magnetic field and a one-dimensional free propagator, $K_{3}^{(0)}(x_3,x_3^{\prime};\tau)$: $$\label{DecomposicaoPropagador}
K({\bf x},{\bf x}^{\prime};\tau)=K_{\bot}({\bf r},{\bf
r}^{\prime};\tau)
 