I'm stuck on a problem. Given a Hamiltonian(adsbygoogle = window.adsbygoogle || []).push({});

[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 [itex] Tr (\alpha) = 0 [/itex] and [itex] Tr( \beta) = 0 [/itex]

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?

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# Working with Dirac matrices

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