Hermiticity of the Dirac Hamiltonian in Curved Spacetime

Understand

In previous work on the quantum mechanics of an atom freely falling in a general curved background spacetime, the metric was taken to be sufficiently slowly varying on time scales relevant to atomic transitions that time derivatives of the metric in the vicinity of the atom could be neglected.

• However, when the time-dependence of the metric cannot be neglected, it was shown that the Hamiltonian used there was not Hermitian with respect to the conserved scalar product.
• This Hamiltonian was obtained directly from the Dirac equation in curved spacetime.
• This raises the paradox of how it is possible for this Hamiltonian to be non-hermitian.