Klein-Gordon equation for electrically charged particles with new energy-momentum operator

TL;DR

This paper proposes a new energy-momentum operator for charged particles in electromagnetic fields, addressing negative probability density issues in the Klein-Gordon equation by redefining the momentum operator.

Contribution

It introduces a novel energy-momentum operator that replaces the canonical one, solving the negative probability density problem in the Klein-Gordon equation for charged particles.

Findings

Negative probability solutions are eliminated using the new operator.

The new operator incorporates mechanical and electromagnetic momenta.

Implications for quantum theory of charged particles are discussed.

Abstract

We address the Klein-Gordon equation for a spinless charged particle in the presence of an electromagnetic (EM) field, and focus on its known shortcoming, related to the existence of solutions with a negative probability density. We disclose a principal way to overcome this shortcoming, using our recent results obtained in the analysis of quantum phase effects for charges and dipoles, which prove the need to abandon the customary definition of the momentum operator for a charged particle in an EM field through its canonical momentum, and to adopt the more general definition of this operator through the sum of mechanical and electromagnetic momenta for the charged particle in an EM field. We show that the ap-plication of the new energy-momentum operator to the Klein-Gordon equation actually eliminates solutions with negative probability density. Some implications of the obtained results are discussed