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Physical Science International Journal

On the Significance of Density of an Electrically Charged Quantum Particle

E. Comay

Published Jan 29, 2018• 0 citations
Publisher

Abstract

Expressions of a conserved 4-current and its density are examined in cases of electrically charged quantum particles. It is proven that the Noether theorem cannot be regarded as a sufficient condition that yields a consistent expression of a conserved 4-current, because electromagnetic interactions pose further constraints. A mathematical analysis shows that the Dirac linear equation yields a consistent expression for a 4-current. In contrast, second order quantum equations, such as the Klein-Gordon equation of an electrically charged scalar particle and the electroweak equation of the W± vector particles do not provide a consistent expression for a conserved 4-current that adequately describes electromagnetic interactions.

DOI
10.9734/psij/2018/38853

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Expressions of a conserved 4-current and its density are examined in cases of electrically charged quantum particles. It is proven that the Noether theorem cannot be regarded as a sufficient condition that yields a consistent expression of a conserved 4-current, because electromagnetic interactions pose further constraints. A mathematical analysis shows that the Dirac linear equation yields a consistent expression for a 4-current. In contrast, second order quantum equations, such as the Klein-Gordon equation of an electrically charged scalar particle and the electroweak equation of the W± vector particles do not provide a consistent expression for a conserved 4-current that adequately describes electromagnetic interactions.
It is proved by a functional method that the conventional expression for the Dirac current is the only conserved 4-vector implied by the Dirac equation that is a function of just the quantum state. The demonstration is extended to derive the unique conserved currents implied by the coupled Maxwell-Dirac equations and the Klein-Gordon equation. The uniqueness of the usual Pauli and Schrödinger currents follows by regarding these as the non-relativistic limits of the Dirac and Klein-Gordon currents, respectively. The existence and properties of further conserved vectors that are not functions of just the state is examined.
Abstract It is proved by a functional method that the conventional expression for the Dirac current is the only conserved 4‐vector implied by the Dirac equation that is a function of just the quantum state. The demonstration is extended to derive the unique conserved currents implied by the coupled Maxwell‐Dirac equations and the Klein‐Gordon equation. The uniqueness of the usual Pauli and Schrödinger currents follows by regarding these as the non‐relativistic limits of the Dirac and Klein‐Gordon currents, respectively. The existence and properties of further conserved vectors that are not functions of just the state is examined.
The paper examines the Dirac-Pauli differences concerning the order of the primary differential equation of an elementary massive quantum particle.The analysis relies on several self-evident constraints that an acceptable quantum theory must satisfy, like conservation laws, compatibility with Maxwellian electrodynamics and the correspondence principle.The dimension of the quantum function that is used in the Lagrangian density of a given quantum theory together with the corresponding differential equations play an important role in the reasoning procedure.The paper proves that Dirac was right and that second-order quantum theories like the Klein-Gordon equation and the electroweak theory of the W ± bosons do not satisfy fundamental constraints.This outcome is inconsistent with the Standard Model of Particle Physics.
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 EM momenta for the system ‘charged particle in an EM field’. We show that the application 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.