Recently, a new functional analytic construction of quasi-free states for a self dual CAR algebra has been presented in [FR13b]. This construction is possible when the so called strong mass oscillation property holds. We provide an example where this is not the case, due to the non vanishing trace of the solutions of the Dirac equation on the horizon of Rindler space, and propose a modification of the construction, in order to weaken this condition. Finally, a connection between the two approaches is built.