Non-Trivial Effects of Sourceless Forces for Spinors: Toward an Aharonov-Bohm gravitational effect?

TL;DR

This paper explores how spinor fields can experience effects from sourceless forces, similar to the Aharonov-Bohm effect, even in the absence of gravity, by analyzing tensorial connections in quantum systems.

Contribution

It introduces the concept of tensorial connection as a real tensor in spinor fields and demonstrates its non-zero effects in sourceless scenarios like hydrogen and harmonic oscillator.

Findings

Tensorial connection can be non-zero without Riemann curvature.

Spinors can be affected by sourceless forces, analogous to Aharonov-Bohm effects.

Examples include hydrogen atom and harmonic oscillator cases.

Abstract

Spinor fields are written in polar form so as to compute their tensorial connection, an object that contains the same information of the connection but which is also proven to be a real tensor. From this, one can still compute the Riemann curvature, encoding the information about gravity. But even in absence of gravity, when the Riemann curvature vanishes, it may still be possible that the tensorial connection remains different from zero, and this can have effects on matter. This is shown with examples in the two known integrable cases: the hydrogen atom and the harmonic oscillator. The fact that a spinor can feel effects due to sourceless actions is already known in electrodynamics as the Aharonov-Bohm phenomenon. A parallel between the electrodynamics case and the situation encountered here will be drawn. Some ideas about relativistic effects and their role for general treatments of quantum field theories are also underlined.