Square of modified Dirac equation

[cbetanco]

If I take a modified Dirac Eq. of the form (i\gamma^\mu \partial_\mu -M)\psi=0 where M=m+im_5 \gamma_5, and whish to square it to get a Klein-Gordon like equation would I multiply on the left with (i\gamma^\nu \partial_\nu +m+im_5\gamma_5) or (i\gamma^\nu \partial_\nu +m-im_5\gamma_5)?
I was under the impression that to take the square, you put a minus sign on the mass term and multiply with that expression on the left, but I am unsure if the im_5\gamma_5 term should also get the appropriate sign change, since its not the tradition mass term. Any thoughts?

 

[Bill_K]

I say use iγ_μ∂_μ + m - imγ₅. This will eliminate the cross terms.

 

[cbetanco]

Thanks. I was getting a little worried about the m_5 \gamma^\nu\partial_\nu\gamma_5+m_5\gamma_5 \gamma^\mu \partial_\mu in my expression but \gamma_5 anti commutes with \gamma^\mu, so it goes away in the end. Thanks again.