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Working with Dirac matrices

  1. Jul 7, 2008 #1
    I'm stuck on a problem. Given a Hamiltonian

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


    [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?
  2. jcsd
  3. Jul 7, 2008 #2
    you missed one crucial equation,

    a clever use of
    and the anti-commutation relation should give you the answer.

    also, note that if A,B,C are matrices

    (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
  4. Jul 7, 2008 #3
    So would

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