# Working with Dirac matrices

1. Jul 7, 2008

### waht

I'm stuck on a problem. Given a Hamiltonian

$$H_{ab} = cP_j(\alpha^{j})_{ab} + mc^{2} (\beta)_{ab} [/itex] then [tex] (H^{2})_{ab} = (\textbf{P}^{2}c^{2} + m^{2}c^{4}) \delta_{ab} [/itex] holds if [tex] \left\{\alpha^j,\alpha^k}\right\}_{ab} = 2 \delta^{jk} \delta_{ab} [/itex] [tex]\left\{\alpha^j, \beta \right\}_{ab} = 0 [/itex] [tex] \delta_{ab} = (\beta^2)_{ab} [/itex] I'd like to show that $Tr (\alpha) = 0$ and $Tr( \beta) = 0$ My plan is to find the eigenvalues of alpha and beta and add them up. But how could I find the eigenvalues using the constraint conditions? 2. Jul 7, 2008 ### tim_lou you missed one crucial equation, [tex]\alpha_i^2=1$$

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.

Last edited: Jul 7, 2008
3. Jul 7, 2008

### waht

So would

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