Time in relativistic and nonrelativistic quantum mechanics

TL;DR

This paper explores the definition of a time operator in relativistic and nonrelativistic quantum mechanics, highlighting interpretational challenges and discussing the covariant nature of the Bohmian interpretation.

Contribution

It introduces a spacetime-position operator in QM and analyzes its implications, emphasizing the role of the Bohmian interpretation in a relativistic context.

Findings

Time operator eigenstates are unphysical due to dynamics.

Bohmian interpretation is shown to be relativistically covariant.

The role of time in quantum mechanics is clarified through operator formalism.

Abstract

The kinematic time operator can be naturally defined in relativistic and nonrelativistic quantum mechanics (QM) by treating time on an equal footing with space. The spacetime-position operator acts in the Hilbert space of functions of space and time. Dynamics, however, makes eigenstates of the time operator unphysical. This poses a problem for the standard interpretation of QM and reinforces the role of alternative interpretations such as the Bohmian one. The Bohmian interpretation, despite of being nonlocal in accordance with the Bell theorem, is shown to be relativistic covariant.