Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook