Revisiting the Schrödinger–Dirac Equation

Author:

Fleury Nicolas1,Hammad Fayçal12ORCID,Sadeghi Parvaneh1ORCID

Affiliation:

1. Department of Physics and Astronomy, Bishop’s University, 2600 College Street, Sherbrooke, QC J1M 1Z7, Canada

2. Physics Department, Champlain College-Lennoxville, 2580 College Street, Sherbrooke, QC J1M 2K3, Canada

Abstract

In flat spacetime, the Dirac equation is the “square root” of the Klein–Gordon equation in the sense that, by applying the square of the Dirac operator to the Dirac spinor, one recovers the equation duplicated for each component of the spinor. In the presence of gravity, applying the square of the curved-spacetime Dirac operator to the Dirac spinor does not yield the curved-spacetime Klein–Gordon equation, but instead yields the Schrödinger–Dirac covariant equation. First, we show that the latter equation gives rise to a generalization to spinors of the covariant Gross–Pitaevskii equation. Then, we show that, while the Schrödinger–Dirac equation is not conformally invariant, there exists a generalization of the equation that is conformally invariant but which requires a different conformal transformation of the spinor than that required by the Dirac equation. The new conformal factor acquired by the spinor is found to be a matrix-valued factor obeying a differential equation that involves the Fock–Ivanenko line element. The Schrödinger–Dirac equation coupled to the Maxwell field is then revisited and generalized to particles with higher electric and magnetic moments while respecting gauge symmetry. Finally, Lichnerowicz’s vanishing theorem in the conformal frame is also discussed.

Funder

Natural Sciences and Engineering Research Council of Canada

Fonds de Recherche du Québec—Nature et Technologies

Bishop’s University

Bishop’s University Research Assistantship program

Publisher

MDPI AG

Subject

Physics and Astronomy (miscellaneous),General Mathematics,Chemistry (miscellaneous),Computer Science (miscellaneous)

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