We analyze the consequences of violation of Lorentz and $CPT$ invariance in the massless neutrino sector by deforming the canonical anti-commutation relations for the fields. We show that, for particular choices of the deformation, oscillation between massless neutrino species takes place when only Lorentz invariance is violated. On the other hand, if both Lorentz and $CPT$ invariances are violated, we show that there is no oscillation between massless neutrino species. Comparing with the existing experimental data on neutrino oscillations, we obtain bounds on the parameter for Lorentz invariance violation.

Several portions of the paper was completely re-written and new references added. To appear in PLB