Invariant Class Operators in the Decoherent Histories Analysis of Timeless Quantum Theories

TL;DR

This paper develops a method for constructing class operators in the decoherent histories approach to reparametrization invariant quantum models, enabling the calculation of probabilities without a preferred time parameter.

Contribution

It introduces a new proposal for class operators using continuous infinite temporal products that commute with the Hamiltonian constraint in timeless quantum theories.

Findings

The proposed class operators respect reparametrization invariance.

Application to simple models demonstrates the method's viability.

Comparison with evolving constants method shows consistency.

Abstract

The decoherent histories approach to quantum theory is applied to a class of reparametrization invariant models, which includes systems described by the Klein-Gordon equation, and by a minisuperspace Wheeler-DeWitt equation. A key step in this approach is the construction of class operators characterizing the questions of physical interest, such as the probability of the system entering a given region of configuration space without regard to time. In non-relativistic quantum mechanics these class operators are given by time-ordered products of projection operators. But in reparametrization invariant models, where there is no time, the construction of the class operators is more complicated, the main difficulty being to find operators which commute with the Hamiltonian constraint (and so respect the invariance of the theory). Here, inspired by classical considerations, we put forward a proposal for the construction of such class operators for a class of reparametrization-invariant systems. They consist of continuous infinite temporal products of Heisenberg picture projection operators. We investigate the consequences of this proposal in a number of simple models and also compare with the evolving constants method.