The Dirac matrices are defined by the relations $$\left [\gamma^{i},\gamma^{j}\right]_{+}=2\eta^{i,j}\mathbb{1}$$ where $[\cdot,\cdot]_{+}$ is the anti-commutator.
What happens if I replace $\eta^{i,j}$ by a generalized metric tensor $g^{i,j}$ that arises, for instance, from a gravitational field in general relativity?
In particular, if I consider matrices $\gamma^{i}$ that satisfy
$$\left [\gamma^{i},\gamma^{j}\right]_{+}=2g^{i,j}\mathbb{1}$$
and plug these into Dirac's equation (with the derivative replaced by the contravariant derivative), what do the solutions to that "generalized" Dirac equation look like?
