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

TL;DR

This paper develops a consistent path integral framework for assigning probabilities to space-time regions in quantum mechanics, emphasizing the role of environment-induced decoherence to avoid logical pitfalls.

Contribution

It introduces a method incorporating meter effects to ensure meaningful probabilities, addressing issues of interference and sum rule violations in path integral approaches.

Findings

The approach avoids logical pitfalls identified in previous work.

Probabilities are well-defined when decoherence via environment or meters is present.

Formal probabilities can lack physical meaning without proper decoherence mechanisms.

Abstract

Following the renewed interest in the topic [1], we revisit the problem of assigning probabilities to classes of Feynman paths passing through specified space-time regions. We show that by assigning of probabilities to interfering alternatives, one already makes an assumption that the interference has been destroyed through interaction with an environment, or a meter. Including the effects of the meter allows to construct a consistent theory, free of logical 'pitfalls' identified in Ref. [1]. Wherever a meter cannot be constructed, or cannot be set to perform 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 [1].