In even dimensions all the representations of the gamma matrices are equivalent, in particular $\gamma^\mu$ and $-\gamma^\mu$ are equivalent. Usually the Dirac Lagrangian is \begin{equation} \psi^\dagger \gamma^0 (i \gamma^\mu \partial_\mu - m) \psi. \end{equation} It follows that if this is written considering the $\gamma^\mu$ representation, in terms of the equivalent $-\gamma^\mu$ it reads \begin{equation} \psi^\dagger \gamma^0 (i \gamma^\mu \partial_\mu + m) \psi. \end{equation} Is the sign of the mass in the Dirac action irrelevant?
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