# Dirac matrices for generalized metric tensors

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?

• See Michael Brown's answer to Dirac equation in curved space-time and the references it cites. The question is different, but the answer includes relevant material. – Chiral Anomaly Aug 6 '19 at 3:05
• By the way, this only works if the manifold admits a spin structure. This is mentioned in the answer to Does Clifford algebra depend on the topology of manifold?. More references about formulating the Dirac equation in curved spacetime are given in Dirac equation in curved spacetime -- same title as the question I cited in the previous comment, but the content is different. The keywords "spin geometry" will also lead you to more information. – Chiral Anomaly Aug 6 '19 at 3:19
• This is a very broad subject. Can you please check the wikipedia page and try and narrow down your question a bit? – AccidentalFourierTransform Aug 7 '19 at 0:39
• Another reference that might be of interest is Penrose and Rindler, Spinors and Space-time, 2 vols. – Robin Ekman Aug 7 '19 at 0:40