Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 50 / 1 / b / Solution

Apply $i{\gamma }^{\nu }{\partial }_{\nu }+m$ to the Dirac equation. Since partial derivatives commute, only the symmetric part of the gamma matrix product contributes:Here $\square ={\partial }_t^2-{\nabla }^2$ in the signature fixed above. The remaining operator is scalar in spinor indices, givingThis is the Klein-Gordon equation for each component, although the first-order Dirac equation additionally couples them.