Spacetime Coarse Grainings in the Decoherent Histories Approach to Quantum Theory

TL;DR

This paper explores assigning consistent probabilities to spacetime histories in quantum systems, addressing issues like the arrival time problem and quantum Zeno effect within the decoherent histories framework, with implications for quantum gravity.

Contribution

It introduces a decoherence condition for spacetime coarse grainings that is independent of the explicit propagator form, aiding applications in quantum cosmology and related fields.

Findings

Derived decoherence conditions for spacetime histories.

Applied formalism to arrival time and tunneling problems.

Discussed implications for quantum gravity and measurement theory.

Abstract

We investigate the possibility of assigning consistent probabilities to sets of histories characterized by whether they enter a particular subspace of the Hilbert space of a closed system during a given time interval. In particular we investigate the case that this subspace is a region of the configuration space. This corresponds to a particular class of coarse grainings of spacetime regions. We consider the arrival time problem and the problem of time in reparametrization invariant theories as for example in canonical quantum gravity. Decoherence conditions and probabilities for those application are derived. The resulting decoherence condition does not depend on the explicit form of the restricted propagator that was problematic for generalizations such as application in quantum cosmology. Closely related is the problem of tunnelling time as well as the quantum Zeno effect. Some interpretational comments conclude, and we discuss the applicability of this formalism to deal with the arrival time problem.