The problem of time in quantum mechanics

TL;DR

This paper explores the longstanding issue of time in quantum mechanics, examining how Dirac's relativistic quantum mechanics permits a self-adjoint time operator, challenging traditional views and implications for quantum theory.

Contribution

It demonstrates that within relativistic quantum mechanics, a self-adjoint time operator can be introduced, acting as a generator of transformations and an observable, addressing the problem of time.

Findings

Dirac's RQM allows a self-adjoint time operator.

The time operator can generate unitary transformations.

The time operator is an observable subject to uncertainty.

Abstract

The problem of time in quantum mechanics concerns the fact that in the Schr\"odinger equation time is a parameter, not an operator. Pauli's objection to a time-energy uncertainty relation analogue to the position-momentum one, conjectured by Heisenberg early on, seemed to exclude the existence of such an operator. However Dirac's formulation of electron's relativistic quantum mechanics (RQM) does allow the introduction of a dynamical time operator that is self-adjoint. Consequently, it can be considered as the generator of a unitary transformation of the system,as well as an additional system observable subject to uncertainty. In the present paper these aspects are examined within the standard framework of RQM.