Multiple-event probability in general-relativistic quantum mechanics

TL;DR

This paper explores defining multi-event quantum probabilities in timeless general-relativistic quantum mechanics, showing they can be reduced to single-event probabilities by considering the quantum nature of measuring devices.

Contribution

It introduces a method to express multi-event probabilities as single-event probabilities using the quantum apparatus, clarifying the role of wave function collapse in relativistic quantum contexts.

Findings

Multi-event probability can be reduced to single-event probability.

Quantum apparatus can be modeled to unify measurement sequences.

Wave function collapse results can be recovered in this framework.

Abstract

We discuss the definition of quantum probability in the context of "timeless" general--relativistic quantum mechanics. In particular, we study the probability of sequences of events, or multi-event probability. In conventional quantum mechanics this can be obtained by means of the ``wave function collapse" algorithm. We first point out certain difficulties of some natural definitions of multi-event probability, including the conditional probability widely considered in the literature. We then observe that multi-event probability can be reduced to single-event probability, by taking into account the quantum nature of the measuring apparatus. In fact, by exploiting the von-Neumann freedom of moving the quantum classical boundary, one can always trade a sequence of non-commuting quantum measurements at different times, with an ensemble of simultaneous commuting measurements on the joint system+apparatus system. This observation permits a formulation of quantum theory based only on single-event probability, where the results of the "wave function collapse" algorithm can nevertheless be recovered. The discussion bears also on the nature of the quantum collapse.