applicability of the quantum logic recipe to defining the projector $\Pi_\Omega$.

Perhaps it is even impossible to make consistent assignments of all projectors $\Pi_\Omega$, in view of the fact that alternative attempts in consistent histories interpretations run into contradictions ([Kent]{} [@Ken]). (But possibly these contradictions are due to the non-projector nature of these histories; cf. [Isham]{} [@Ish].)

However, we can be more modest and only look at special sets of paths for which we may define proper projectors. For example, we may define the distance of two paths $\xi^k:[\alpha,\omega]\to \Rz^3$ ($k=1,2$) by d(\^1,\^2)=. Then, for each specific path $\xi$, we can use the position operators $x(t)=\exp(iHt)x(0)\exp(-iHt)$ to define the observable $\Delta_\xi:=d(x,\xi)$ that measures the distance of the particle path from $\xi$. Suitable projectors are now the projectors $\Pi_{\xi,\eps}$ projecting to the subspace spanned by the wave functions corresponding to states in which $\Delta_\xi$ has with certainty a value not exceeding $\eps$. Then, for a system in an arbitrary pure state $a$, P(d(x,))=a|\_[,]{}|a is, according to the most orthodox interpretation of quantum mechanics, a well-defined quantum mechanical probability for the event that the particle path is, in the root mean square sense, within $\eps$ of the path $\xi$. Note that, in 