# Stupid question on Dirac alpha and beta matrices

1. Jan 18, 2012

### AuraCrystal

1. The problem statement, all variables and given/known data
Dirac proposed that a relativistic wave equation that is linear in both space and time (unlike the Klein-Gordon equation, which is second order) has the form
$i\frac{\partial}{\partial t}\Psi = (\mathbf{\alpha} \cdot \mathbf{p)+\beta m)\Psi$
After squaring this, we'd like it to satisfy the equation $E^2=p^2+m^2$. So, after some algebra, you conclude that α and β must be matrices that satisfy:
$[\alpha_{i}, \alpha_{j}]_{+} \equiv \alpha_{i}\alpha_{j}+\alpha_{j}\alpha_{i}=0$
$[\alpha_{i}, \beta]_{+}=0$
$(\alpha_{i})^2=(\beta)^2=1$
Show that α and β are traceless, Hermetian, have eigenvalues +1 and -1, are of even dimension greater than 4
2. Relevant equations
The commutation relations I gave above.

3. The attempt at a solution
I honestly have no idea where to start...
Maybe write out the equations in terms of the trace?

Ugh, I'm such an idiot....