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Homework Help: Commutator of the Dirac Hamiltonian and gamma 5

  1. Nov 21, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that in the chiral (massless) limit, Gamma 5 commutes with the Dirac Hamiltonian in the presence of an electromagnetic field.

    2. Relevant equations


    3. The attempt at a solution
    My first question is whether my Dirac Hamiltonian looks correct, I constructed it by separating the temporal derivative from the spatial part from the Dirac equation:

    $$ i \gamma^\mu (\partial_\mu +iqA_\mu)\psi=0 $$
    $$-i\gamma^0 \partial_t \psi=(i \gamma^i \partial_i -q\gamma^\mu A_\mu)\psi$$
    $$H\psi=i\partial_t \psi=(-i\gamma^0 \gamma^i \partial_i +q \gamma^0\gamma^\mu A_\mu)\psi$$

    I don't have huge confidence that this Hamiltonian is correct so if anyone has any comments I'd be very grateful :)

    My second sticking point is how to compute the commutator:

    $$[H,\gamma^5]$$

    I see that I can just work out the sum of the commutators of each section:

    $$[-i\gamma^0 \gamma^i \partial_i, \gamma^5] + [q \gamma^0\gamma^\mu A_\mu, \gamma^5]$$

    But I'm not sure how to work out how gamma 5 commutes with the partial_i term, or the A_mu term, any advice would be awesome :)
     
  2. jcsd
  3. Nov 21, 2017 #2

    TSny

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    2017 Award

    I think your Hamiltonian is correct.

    When considering commutators, you only need to worry about the commutation of the matrices. Everything else can be "pulled out". For example,

    $$[-i\gamma^0 \gamma^i \partial_i, \gamma^5] = [\gamma^0 \gamma^i , \gamma^5] \left( -i \partial_i \right)$$
     
  4. Nov 23, 2017 #3
    Ah thank you! That clears everything up :)
     
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