Finding Eigenvalues of Dirac Matrices with Constraint Conditions

[waht]

I'm stuck on a problem. Given a Hamiltonian

[tex] H_{ab} = cP_j(\alpha^{j})_{ab} + mc^{2} (\beta)_{ab} [/itex]

then

[tex] (H^{2})_{ab} = (\textbf{P}^{2}c^{2} + m^{2}c^{4}) \delta_{ab} [/itex]

holds if

[tex] \left\{\alpha^j,\alpha^k}\right\}_{ab} = 2 \delta^{jk} \delta_{ab} [/itex]

[tex]\left\{\alpha^j, \beta \right\}_{ab} = 0 [/itex]

[tex] \delta_{ab} = (\beta^2)_{ab} [/itex]

I'd like to show that $Tr (\alpha) = 0$ and $Tr( \beta) = 0$

My plan is to find the eigenvalues of alpha and beta and add them up. But how could I find the eigenvalues using the constraint conditions?

 

[tim_lou]

you missed one crucial equation,
$$\alpha_i^2=1$$

a clever use of
$$\alpha_i^2=\beta^2=1$$
and the anti-commutation relation should give you the answer.

also, note that if A,B,C are matrices
$$Tr(ABC)=Tr(CAB)=Tr(BCA)$$

(edit) nevermind using the eigenvalues, i was thinking that those are 2D matrices; their dimensions are not given in this case.

 

[waht]

So would

[tex] Tr(\alpha) = Tr(\alpha\, (\alpha^j)^2 \,\beta^2) [/itex]