Usually the Dirac equation in curved space is written as $$i\Gamma^{\mu}D _{\mu}\Psi-m\Psi=0,$$ where $\Gamma_{\mu}$ are curved space gamma matrices and $D_{\mu}$ is covariante derivative. This equations comes from a variational principle when only the part including the product of gama matrices with covariante derivatives have the imaginary number $i$ in front. However, DeWitt in his book: The global approach to quantum field theory or in his classic Dynamical theory of groups and fields constructed an action with the imaginary basis $i$ in front of the complete action integral (i.e, also multilying the mass) to obtain Dirac equation in the form $$\Gamma^{\mu}D _{\mu}\Psi+m\Psi=0,$$ that is really the most used form in general relativity applications. What is the correct form of the equation and of the action? In the formulation of DeWitt, is the mass a real quantity (or imaginary)? Any explanation will be welcome..
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$\begingroup$ In the limit where the curved geometry approaches the flat Minkowski spacetime, you need to recover the ordinary Dirac equation. So $\left( i \gamma^{I} e_{I}^{\mu} D_{\mu} - m \right) \Psi = 0$ looks like the correct form. I am not familiar with DeWitt's notation, but note that if you multiply all gamma matrices by $i$ it is equivalent to changing the metric signature, which is a choice of convention. $\endgroup$– Prof. LegolasovCommented Oct 8, 2019 at 13:07
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2 Answers
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check the definitions of $\Gamma^{\mu}$, $D _{\mu}$, and the signature of $g^{\mu\nu}$. I think you will find that the equations are actually the same, possibly modulo an overall sign.
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It may depend on the metric convention in the papers. In the West Coast metric $g_{\mu\nu}= {\rm diag}(+,-,-,-)$ the flat space Dirac equation is usually written as $$ (-i\gamma^\mu\partial_\mu +m)\psi=0. $$ In the East Coast metric $g_{\mu\nu}= {\rm diag}(-,+,+,+)$ it is $$ (\gamma^\mu\partial_\mu+M)\psi=0. $$ In both cases $$ \gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu= 2g^{\mu\nu}, $$ and the inclusion or not of the "$i$" means that the actual equation is the same in both metrics.
Ahhh -- only just saw Aldo's answer.
answered Oct 8, 2019 at 17:22