Stupid question on Dirac alpha and beta matrices

[AuraCrystal]

Homework Statement

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

Homework Equations

The commutation relations I gave above.

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...

 

[mark726]

Thank you for your post. It seems like you are struggling with the problem and are feeling frustrated. Let me try to offer some guidance that may help you approach this problem.

Firstly, it may be helpful to review the properties of matrices, such as trace, Hermitian, eigenvalues, etc. This will give you a better understanding of what these terms mean and how they are related to matrices.

Next, try to break down the problem into smaller parts. For example, you can start by looking at the first commutation relation and see what it implies for the matrices α and β. Then move on to the second commutation relation and see how it adds to your understanding of α and β. You can also try to use the given equations to simplify the expressions and see if you can derive any useful information.

Additionally, you can try to think about the physical implications of these equations. What do the eigenvalues +1 and -1 represent? How do they relate to the energy and momentum of a particle? This may give you some insight into the properties of α and β.

Finally, don't be too hard on yourself. Sometimes it takes time and effort to understand complex concepts. Keep practicing and seeking help when needed. Good luck with your problem!