I am interested in deriving the anti-commutation relations of a Dirac field for a $(1+1)$D Eddington-Finkelstein metric given by $$ ds^2 = f(r) dt^2 - 2 dt dr, \quad f(r) = 1 - \frac{r_s}{r}.$$
where $r_s$ is the Schwarzschild radius of the black hole. In order to do this I will first derive the classical Poisson bracket and then quantising by replacing Poisson brackets with anti-commutation relations. I will provide some defintions and background below.
Dirac action on curved space
The Dirac action on a curved spacetime is given by
$$ S = \int d^2 x |e| i \gamma^\mu D_\mu \psi \equiv \int d^2 x \mathcal{L} $$
where $D_\mu = \partial_\mu + \omega_\mu$ is the covariant derivative; $\omega_\mu$ is the spin connection; $\gamma^\mu = e^\mu_a \gamma^a$, where $e^\mu_a$ is the tetrad related to the metric via $g_{\mu \nu} = e_\mu^a e_\nu^b \eta_{ab}$ for the Minkowski metric $\eta_{ab} = \mathrm{diag}(+1,-1),$ and where the gamma matrices obey $\{ \gamma^a,\gamma^b \} = 2\eta^{ab}$ and $\{ \gamma^\mu,\gamma^\nu \} = 2g^{\mu\nu}$; $|e| = \det[e_\mu^a]$; and $\bar{\psi} = \psi^\dagger \gamma^0$. See the book by Nakahara for a reference.
Canonical Poisson bracket
The canonical momentum is defined as
$$ \pi = \frac{\partial \mathcal{L}}{\partial \dot{\psi}} = i|e| \psi^\dagger \gamma^0 \gamma^t . $$
Therefore, when we substitute this into the canonical Poisson bracket $ \{ \psi_\alpha(x), \pi_\beta(y) \} = \delta_{\alpha \beta} \delta(x-y)$, we arrive at
$$ \{ \psi_\alpha(x), \psi^\dagger_\beta(y) \} = \frac{-i(\gamma^0 \gamma^t)^{-1}_{\alpha \beta}\delta(x-y)}{|e|}.$$
However, we have the result that $(\gamma^0 \gamma^t)^{-1} = \frac{1}{g^{tt}} \gamma^t \gamma^0$ which can be derived from the Clifford algebra. This is a problem, as for the Eddington-Finkelstein metric $g^{tt} = 0$, which can be seen as this is obtained from the inverse metric:
$$ g_{\mu \nu} = \begin{pmatrix} f & - 1 \\ -1 & 0 \end{pmatrix} , \quad g^{\mu \nu} = \begin{pmatrix} 0 & - 1 \\ -1 & -f \end{pmatrix}$$
in which case the Poisson bracket diverges.
My question
It appears the canonical Poisson bracket for the Dirac field in an Eddington-Finkelstein coordinate system diverges. Does this mean that we cannot quantise the Dirac field in this coordinate system?
I am aware there are many subtleties with canonically quantising the Dirac field as the classical fields are Grassmann-valued and the canonical momentum actually defines a second class constraint due to $\pi \propto \psi^\dagger$, in which case the machinery of Dirac brackets must be used instead of Poisson brackets, however the calculation yields the same result - a diverging bracket.
