A dynamical time operator in Dirac's relativistic quantum mechanics

TL;DR

This paper introduces a self-adjoint dynamical time operator in Dirac's relativistic quantum mechanics, establishing new commutation relations and linking quantum mechanics' foundational elements with modern phenomena.

Contribution

It presents a novel dynamical time operator in Dirac's framework, resolving Pauli's objection and connecting quantum foundations with contemporary physical effects.

Findings

Time-energy uncertainty relation involving wave packet passage time.

The rate of change of position expectation equals phase velocity.

Potential relevance to interference, Zitterbewegung, and cosmology.

Abstract

A self-adjoint dynamical time operator is introduced in Dirac's relativistic formulation of quantum mechanics and shown to satisfy a commutation relation with the Hamiltonian analogous to that of the position and momentum operators. The ensuing time-energy uncertainty relation involves the uncertainty in the instant of time when the wave packet passes a particular spatial position and the energy uncertainty associated with the wave packet at the same time, as envisaged originally by Bohr. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie's hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli's objection is shown to be resolved or circumvented. Possible relevance to current developments in interference in time, in Zitterbewegung like effects in spintronics, grapheme and superconducting systems and in cosmology is noted.