Weyl-invariant derivation of Dirac equation from scalar tensor fields in curved space-time

TL;DR

This paper derives the Dirac equation in curved space-time from a Weyl-invariant action, revealing geometric quantum effects and small deviations from standard quantum mechanics that could be tested experimentally.

Contribution

It extends the derivation of Dirac's equation to curved space-time using Weyl invariance, introducing new scalar potential terms absent in standard quantum mechanics.

Findings

Correct gyromagnetic ratio $g_e=2$ for the electron.

Coupling to scalar curvature is about 1/4 instead of 1/2.

Additional scalar potential terms appear at small scales.

Abstract

In this work we present a derivation of Dirac's equation in a curved space-time starting from a Weyl-invariant action principle in 4+K dimensions. The Weyl invariance of Dirac's equation (and of Quantum Mechanics in general) is made possible by observing that the difference between the Weyl and the Riemann scalar curvatures in a metric space is coincident with Bohm's Quantum potential. This circumstance allows a completely geometrical formulation of Quantum Mechanics, the Conformal Quantum Geometrodynamics (CQG), which was proved to be useful, for example, to clarify some aspects of the quantum paradoxes and to simplify the demonstration of difficult theorems as the Spin-Statistics connection. The present work extends our previous derivation of Dirac's equation from the flat Minkowski space-time to a general curved space-time. Charge and the e.m. fields are introduced by adding extra-coordinates and then gauging the associated group symmetry. The resulting Dirac's equation yields naturally to the correct gyromagnetic ratio $g_e=2$ for the electron, but differs from the one derived in the Standard Quantum Mechanics (SQM) in two respects. First, the coupling with the space-time Riemann scalar curvature is found to be about 1/4 in the CQG instead of 1/2 as in the SQM and, second, in the CQG result two very small additional terms appear as scalar potentials acting on the particle. One depends on the derivatives of the e.m. field tensor and the other is the scalar Kretschmann term $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$. Both terms, not present in the SQM, become appreciable only at distances of the order of the electron Compton length or less. The Kretschmann term, in particular, is the only one surviving in an external gravitational field obeying Einstein's equations in vacuum. These small differences render the CQG theory confutable by very accurate experiments, at least in principle.