I would like to know how to quantize the Dirac field in curved space-time. The Dirac equation in curved space-time has the following form:

$i \gamma^a e_a^{\mu} D_\mu \Psi(x) -m \Psi(x) =0$

where $\gamma^a$ are the well-known Gamma-matrices $\gamma^a\gamma^b + \gamma^b\gamma^a =2 \eta^{ab}$, $e_a^{\mu}$ is the Vielbein-field, $D_\mu$ is the covariant derivative and $\Psi$ the Dirac-field. The covariant derivative is defined as

$D_\mu =\partial_\mu -\frac{i}{4}\omega_\mu^{ab}\sigma_{ab}$

where $\sigma_{ab} = \frac{i}{2}[\gamma_a, \gamma_b]$ and $\omega_\mu^{ab}$ is the spin connection.

As the Dirac field couples to the spin connection, in curved space-time the Dirac equation turns out to be nonlinear. So is it still possible to develop the Dirac field $\Psi(x)$ in annihilation and creation operators, or is this not possible anymore to the nonlinearity of the Dirac-equation ? If the latter is the case, could the Dirac field still be treated similar to an interacting quantum field (as for example like a Dirac field in flat space coupled to a EM-field )? If the latter is the case could one-particle states for a Dirac field in curved space-time be still defined ?


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