# Commutator of the Dirac Hamiltonian and gamma 5

1. Nov 21, 2017

### Milsomonk

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. Nov 21, 2017

### TSny

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,

$$[-i\gamma^0 \gamma^i \partial_i, \gamma^5] = [\gamma^0 \gamma^i , \gamma^5] \left( -i \partial_i \right)$$

3. Nov 23, 2017

### Milsomonk

Ah thank you! That clears everything up :)