addition we compute the propagator of the Dirac field in this context.

In the next subsection we briefly survey the Dirac field in order to introduce the notation and conventions that we use in this paper. We stress the uses of the oscillator variables ${\bf b}^*$ and ${\bf b}$ which allowed us to perform the construction [@hatfield].

The Dirac Free Field
--------------------

In this section we discuss the Dirac free field $\psi(x)$ over Minkowski spacetime $M^{3+1}=\mathbb{R}^3 \times \mathbb{R}$ with the signature $(-,-,-,+)$ and $x=(\vec{x},t) \in M$. The action is given by $$I_D[\psi] = \int d^3xdt {\cal L}_D$$ $$= \int d^3xdt \overline{\psi}(\vec{x},t)\cdot \hbar c \big(i
\not \!
\partial -{mc\over \hbar} \big) \psi(\vec{x},t),
\label{diraclagran}$$ where $\not \! \partial=\gamma^{\mu}\partial_{\mu}$, $\gamma^{\mu}$ are the Dirac matrices ($\mu = 0, \dots, 3$), $\overline{\psi}({x}) \equiv \psi^{\dag}({x}) \gamma^0$, $m$ is the mass parameter, $c$ is the speed of light and $\hbar$ the Planck constant. Thus, the field $\psi({x})$ fulfills the Dirac equation $$\big(i\not \! \partial - {mc\over\hbar} \big) \psi (\vec{x},t) =
0. \label{diraceqn}$$ Its [*conjugate momentum*]{} is given by $\pi_{\psi}(\vec{x},t) =
{-\partial {\cal L}_D \over \partial (
\partial \dot{\psi}) } = i \hbar {\psi}^{\dag}(\vec{x},t)$, where $
\dot{\psi}(\vec{x},t) \equiv {\partial \psi(\vec{x},t) \over
\partial t}$ [^1]. Then the Hamiltonian reads 