Elements of reality,Lorentz invariance,and the product rule

Recently various gedankenexperiments have been formulated which argue that the assumption that elements of reality are Lorentz invariant cannot be reconciled with standard quantum mechanics. Two of these gedankenexperiments were subsequently analyzed using the notion of pre- and postselected quantum systems, and it was claimed that elements of reality can be made Lorentz invariant if the product rule of standard quantum mechanics is abandoned. In this paper we show that the apparent violations of the product rule in these gedankenexperiments are not as significant as they appeared to be in the previous analysis. We conclude that the problems with Lorentz invariance which arise in these gedankenexperiments are essentially unrelated to the product rule violations.