# Why is Dirac equation a matrix equation?

According to Wikipedia's Dirac equation article, the Dirac equation can be written in form $$i\hbar\gamma^{\mu}\partial_{\mu}\psi-mc\psi=0,$$ where $$\gamma^{\mu}$$ are gamma matrices which are $$4 \times 4$$ matrices. Now is the wavefunction $$\psi$$ still a scalar like in classical quantum mechanics? If so then isn't this equation a matrix equation meaning that the same scalar function should obey multiple different equations? Why would it be so? Isn't that overdetermination?

• Did you also read the bit where it says $\psi$ is a four component spinor? (That WP page is a mess, it would be fairly easy to do this). – jacob1729 Jun 10 at 10:44
• Well that explains. I tried to see something like that said in that WP page but didn't notice that. – Professor Kirby Jun 10 at 10:47

In this context, $$\psi$$ stands for a set of four wavefunctions, forming a Dirac spinor. Both sides of the equation transform as Dirac spinors.
This is not unusual or overdetermined. It's just like how $$\mathbf{F} = m \mathbf{a}$$ really has "three things" on each side, because it relates two objects that transform as vectors.