Recently, there have been many attempts to extend the notion of proper time
to quantum mechanics with the use of quantum clocks. Using a similar idea
combined with the relativistic mass-energy equivalence, we consider an
accelerating massive quantum particle with an internal clock system. We show
that the ensuing evolution from the perspective of the particle's internal
clock is non-Hermitian. This result does not rely on specific implementations
of the clock. As a particular consequence, we prove that the effective
Hamiltonian of two gravitationally interacting particles is non-Hermitian from
the perspective of the clock of either particle.Comment: 6+1 pages, 2 figures. Comments are welcome
