Path integral approach to space-time probabilities: A theory without pitfalls but with strict rules

Following the renewed interest in the topic [Halliwell and Yearsley, Phys. Rev. D 86, 024016 (2012)], we revisit the problem of assigning probabilities to classes of Feynman paths passing through specified space-time regions. We show that by assigning probabilities to interfering alternatives, one already makes the assumption that the interference has been destroyed through interacting with the environment or a meter. Including the effects of the meter allows one to construct a consistent theory, free of logical pitfalls,'' such as those identified in Halliwell and Yearsley. Wherever a meter cannot be constructed, or cannot be set to effect the desired decoherence, formally constructed probabilities have no clear physical meaning and can violate the necessary sum rules. We illustrate the above approach by analysing the three examples considered in Halliwell and Yearsley. DOI: 10.1103/PhysRevD.87.076001