  The theory of entropic gravity conjectures that gravity emerges
thermodynamically rather than being a fundamental force. One of the main
criticisms of entropic gravity is that it would lead to quantum massive
particles losing coherence in free fall, which is not observed experimentally.
This criticism was refuted in [Phys. Rev. Res. 3, 033065 (2021)], where a
nonrelativistic master equation modeling gravity as an open quantum system
interaction demonstrated that in the strong coupling limit, coherence could be
maintained and reproduce conventional free-fall dynamics. Moreover, the
nonrelativistic master equation was shown to be fully compatible with the
qBounce experiment for ultracold neutrons. Motivated by this, we extend these
results to gravitationally accelerating Dirac fermions. We achieve this by
using the Dirac equation in Rindler space and modeling entropic gravity as a
thermal bath thus adopting the open quantum systems approach as well. We
demonstrate that in the strong coupling limit, our entropic gravity model
maintains quantum coherence for Dirac fermions. In addition, we demonstrate
that spin is not affected by entropic gravity. We use the Foldy-Wouthysen
transformation to demonstrate that it reduces to the nonrelativistic master
equation, supporting the entropic gravity hypothesis for Dirac fermions. Also,
we demonstrate how antigravity seemingly arises from the Dirac equation for
free-falling antiparticles but use numerical simulations to show that this
phenomenon originates from zitterbewegung thus not violating the equivalence
principle.
