not normalizable. To avoid potential problems, we use the box normalization, i.e. we confine our fields in a box of length $L$, and we impose periodic boundary conditions: $\psi\left(t,\mathbf{x}\right)=\psi\left(t,\mathbf{x}+\mathbf{L}\right)$, where $\mathbf{L}$ is a vector with all the components equal to $L$. In turn, the momentum is discretized: ${\bf p}=\frac{2\pi\hbar}{L}{\bf k}$ with $k^{i}\in\mathbb{Z}$ and $i=1,2,3$ labels the spatial components. Then the Dirac field, in the interaction picture where we choose $H_{\text{\tiny D}}$ as the unperturbed Hamiltonian and $N\left(t\right)$ as the perturbation (the noise coupling $\sqrt{\gamma_{m}}$ is very small), takes the usual expression[^4]: $$\label{eq:wave}
\psi_{I}\left(x\right)=\sum_{s=1}^{2}\sum_{\mathbf{k}=-\infty}^{+\infty}\frac{1}{\sqrt{L^{3}}}\sqrt{\frac{mc^{2}}{E_{p}}}\left[b\left(p,s\right)u\left(p,s\right)e^{-\frac{i}{\hbar}E_{p}t+i\frac{2\pi}{L}\mathbf{k}\cdot\mathbf{x}}+d^{\dagger}\left(p,s\right)v\left(p,s\right)e^{\frac{i}{\hbar}E_{p}t-i\frac{2\pi}{L}\mathbf{k}\cdot\mathbf{x}}\right],$$ where $u$ and $v$ are the usual Dirac spinors, $E_{p}=\sqrt{\mathbf{p}^{2}c^{2}+m^{2}c^{4}}$ is the energy and $b$ and $d$ are operators satisfying the standard anti-commutation relations. We also recall the relation between the evolution operator $U(t)$ in the Schrödinger picture and $U_{I}\left(t\right)$ in the interaction picture [@Sakurai]: $$\label{evolution}
U\left(t\right)=e^{-\frac{i}{\hbar}H_{\text{\tiny D}}t}U_{I}\left(t\right),$$ and we set the initial time to 0.

One can question why we use a relativistic Hamiltonian in a model which is not relativistic, since the correlation function of the noise is not Lorentz-invariant. Our approach to this issue, is that collapse models are phenomenological models emerging from a pre-quantum theory yet to be discovered. The noise field is 