& 0\\
0 & -1\end{array}\right),\qquad\gamma^{i}=\left(\begin{array}{cc}
0 & \sigma^{i}\\
-\sigma^{i} & 0\end{array}\right)\qquad\textrm{with}\;\;\; i=1,2,3.$$ where $1$ is the identity matrix in two dimensions and $\sigma^{i}$ are the Pauli matrices. The noise term instead is given by: $$\label{eq:dfgdf}
N\left(t\right)=\int d\mathbf{x}\,\mathcal{N}\left(x\right)=-\hbar\sqrt{\gamma_{m}}\int d\mathbf{x}w\left(x\right)\overline{\psi}\left(x\right)\psi\left(x\right), \qquad \gamma_{m}=\gamma\left(\frac{m}{m_{0}}\right)^{2}$$ It can be shown[^3] that the Hamiltonian $H_{\text{\tiny TOT}} = H_{\text{\tiny D}} + N(t)$ defined in this way, in the non relativistic limit, reduces to the Hamiltonian of Eq. (\[eq:htot\]).

We chose to treat neutrinos as Dirac particles, despite the fact that is not yet known if they are Dirac or Majorana particles. We expect that the size of the collapse effect does not change significantly if Majorana fields are used in place of Dirac fields. As a matter of fact, in [@Kaons] the effect of the CSL model on kaon oscillations formula was studied. Despite the fact that kaons are different from neutrinos and that in [@Kaons] they were studied in the non-relativistic regime, the result is the same as that of this paper: a damping factor in front of the oscillating term, with a decay rate equivalent of to Eq.  when the non-relativistic limit is taken.

Working with plane waves gives rise to unphysical infinities, since they are 