Gravitation field in the Dirac equation

Since neutrons are spin half particles, one may consider that, in the absence of a magnetic field, their behavior should be correctly described by the Dirac equation. In the case with a gravitational field, described by a curved space-time, the Dirac equation is usually modified to the form derived independently by Weyl and by Fock in 1929, hereafter the Dirac-Fock-Weyl (DFW) equation. We can try to study the physical consequences of the DFW equation, naturally in the framework of its weak-field and/or non-relativistic limit, but the corrections to the non-relativistic Schrödinger equation in the Newtonian gravity potential are usually very small. For instance, in the experiments on gravitational stationary states, one uses ultra-cold neutrons in the Earth’s gravitational field . It has been shown recently that, in this particular case, the corrections brought by the DFW equation to the non-relativistic Schrödinger equation in the gravity potential are quite hopelessly negligible. Nevertheless, one may expect that, in the future, experiments (possibly using lighter neutral particles: massive neutrinos?) should be able to check this kind of corrections and, therefore, to distinguish between possible competing gravitational extensions of relativistic quantum mechanics.