- #1
Milsomonk
- 96
- 17
Homework Statement
Show that in the chiral (massless) limit, Gamma 5 commutes with the Dirac Hamiltonian in the presence of an electromagnetic field.
Homework Equations
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 :)