Recently, a new functional analytic construction of quasi-free states for a self-dual CAR algebra has been presented in Finster and Reintjes (Adv Theor Math Phys 20:1007, 2016). This method relies on the so-called strong mass oscillation property. We provide an example where this requirement is not satisfied, due to the nonvanishing trace of the solutions of the Dirac equation on the horizon of Rindler space, and we propose a modification of the construction in order to weaken this condition. Finally, a connection between the two approaches is built.
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