(adsbygoogle = window.adsbygoogle || []).push({}); 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 :)

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# Homework Help: Commutator of the Dirac Hamiltonian and gamma 5

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