Time in quantum physics: From an external parameter to an intrinsic observable

TL;DR

This paper explores how time can be treated as an intrinsic observable in quantum physics by enlarging the system to include time as a degree of freedom, leading to a new formalism for time observables.

Contribution

It introduces a formalism where time is an intrinsic observable in quantum systems by extending the algebra of observables and applying completely positive maps, with applications to quantum cosmology.

Findings

Reformulation of the Schrödinger equation with time as an internal degree of freedom.

Development of positive operator valued maps as time observables.

Application of the formalism to Wheeler-DeWitt quantum cosmology.

Abstract

In the Schroedinger equation, time plays a special role as an external parameter. We show that in an enlarged system where the time variable denotes an additional degree of freedom, solutions of the Schroedinger equation give rise to weights on the enlarged algebra of observables. States in the associated GNS representation correspond to states on the original algebra composed with a completely positive unit preserving map. Application of this map to the functions of the time operator on the large system delivers the positive operator valued maps which were previously proposed by two of us as time observables. As an example we discuss the application of this formalism to the Wheeler-DeWitt theory of a scalar field on a Robertson-Walker spacetime.