Following the notation used in David J. Griffiths Introduction to Elementary Particles, recall that $a^{\mu}b_{\mu}:=a^{0}b^{0}-a^{1}b^{1}-a^{2}b^{2} - a^{3}b^{3}$. Recall the relativistic Dirac equation $ i\hbar\gamma^{\mu}\partial_{\mu}\psi = mc \psi $ In this case, our notational definition above does not hold, in that $ \gamma^{\mu}\partial_{\mu} = \gamma^{0}\partial_{0} + \gamma^{1}\partial_{1} + \gamma^{2}\partial_{2} + \gamma^{3}\partial_{3}. $ Why is this?
Note that the Dirac equation as given above derives from $ \gamma^{\mu}p_{\mu}\psi = mc \psi $ where the convention above does hold, in that $ \gamma^{\mu}p_{\mu} = \gamma^{0}p^{0} - \gamma^{1}p^{1}-\gamma^{2}p^{2}-\gamma^{3}p^{3} $ then the quantum substitution $p_{\mu}\to i\hbar\partial_{\mu}$ (is this right?) is given.