Why must the γ^μ in the Dirac equation be matrices?

[raintrek]

Yep, another quick question on the Dirac Equation!
I've become slightly more clued about the use of the DE now in illustrating the negative energy problem in relativistic QM as well as the existence of spin, however one thing is still puzzling me.

I've read this excerpt in a text:

$$(i\gamma^{\mu}\partial_{\mu} - m)\psi(x) = 0$$

where the four coefficients $$\gamma^{\mu}$$ are constants. We shall see immediately that these coefficients cannot commute with each other. They must therefore be square matrices rather than simple numbers, so the wavefunction $$\psi(x)$$ must be a column matrix.

I'm not sure I understand how this is proven. Is it something to do with

$$\gamma^{\mu}\gamma^{\nu} + \gamma^{\nu}\gamma^{\mu} = 2\eta^{\mu\nu}$$

...and if so, is there a way I can show that this condition can't be satisfied if the $$\gamma^{\mu}$$ are ordinary numbers?

 

[Meir Achuz]

The Dirac equation should become the Klein-Gordon equation if the DE is multiplied by
$$i\gamma_\mu\partial^\mu+m$$. It won't if the gammas commute.