is described by a standard linear Schrödinger equation with a random Hamiltonian: $$\label{eq:htot}
H_{\text{\tiny TOT}} = H - \hbar \sqrt{\gamma} \sum_j \frac{m_{j}}{m_{0}}
\sum_s\int d\mathbf{y}\, w(\mathbf{y},t)
\psi_{j}^{\dagger}(\mathbf{y},s) \psi_{j}(\mathbf{y},s),$$ where $$\label{eq:sfsoi}
w(\mathbf{y},t) = \int d\mathbf{x}\, g(\mathbf{x-y})\xi_{t}(\mathbf{x}),$$ and $\xi_{t}(\mathbf{x}) = dW_{t}(\mathbf{x})/dt$ is a white noise field, with mean equal to zero and correlation function $\mathbb{E}[ \xi_{t}(\mathbf{x}) \xi_{s}(\mathbf{y})] = \delta(t-\nobreak s) \delta({\bf x-y})$. As such, $w(\mathbf{x},t)$ is a Gaussian noises field, with zero mean and correlation function: $$\label{eq:sdfddas}
{\mathbb E}[w(\mathbf{x},t) w(\mathbf{y},s)] \; = \;
\delta(t-s)F({\bf x} - {\bf y}), \qquad  F({\bf x}) \; = \;
\frac{1}{(\sqrt{4 \pi} r_C)^3} e^{-{\bf x}^2/4 r_C^2}.$$

We are interested in the relativistic generalization of the Hamiltonian (\[eq:htot\]). The most natural choice is: $H_{\text{\tiny TOT}} = H_{\text{\tiny D}} + N(t)$ where (in the case of just one type of particle): $$H_{\text{\tiny D}}=\int d\mathbf{x}\mathcal{H}_{\text{\tiny D}}\left(x\right)=\int d\mathbf{x}\, \psi^{\dagger}\left(x\right)\left[-i\hbar c\,{\vec{\alpha}}\cdot\vec{\nabla}+mc^{2}{\beta}\right]\psi\left(x\right)$$ is the standard Dirac Hamiltonian. Here we have introduced the four-vector notation $x \equiv (ct, {\bf x})$, $\psi\left(x\right)$ is the Dirac spinor field, $c$ the speed of light, $m$ the mass of the particle associated to this field, $\vec{\alpha} \equiv (\alpha^1, \alpha^2, \alpha^3)$ with $\alpha^{i}=\gamma^{0}\gamma^{i}$ and $\beta=\gamma^{0}$ where the $\gamma^{\mu}$ are the Dirac matrices, which we take in their standard representation: $$\gamma^{0}=\left(\begin{array}{cc}
1 