I'm trying to derive this form of the Dirac Lagrangian density in curved space-time:
$$ \mathcal{L}~=~\det\left(e\right)\bar{\Psi}\Bigg (\frac{i}{2}\gamma^{a}\partial_{a}-m+\gamma^{a}\gamma^{5}B_{a}\Bigg)\Psi $$
starting from this form:
$$ \mathcal{L}~=~\sqrt{-g}\Bigg (\frac{i}{2} \bar{\Psi} \gamma^{a} \stackrel{\leftrightarrow}{D}_{a} \Psi-\bar{\Psi}m\Psi \Bigg) $$
I can get the first term, however, in the last term I get a factor of $ \quad – \frac{1}{2} \quad$ out the front, i.e.
$$ \mathcal{L}~=~\det(e)\bar{\Psi}\Bigg (\frac{i}{2}\gamma^{a}\stackrel{\leftrightarrow}{\partial_{a}}-m-\frac{1}{2}B_{a}\gamma^{a}\gamma^{5}\Bigg)\Psi $$
I've tried deriving it in several different ways and always end up with the above expression. The problem is, in all of the papers that I've read on the subject, it is given in the form as presented at the top of this page.
If it helps, here is a link to my derivation in full: http://we.tl/mUgiw0BkMh
Would really appreciate any help in solving this problem.
Best Answer
What does B correspond to here? Is it related to the spin connection? The axial current?
Also the link you propose does not work anymore.
If you want a rather nice demonstration of the Dirac equation in curved space, you can try "Nonlinear spinor equation and asymmetric connection in general relativity" by Hehl and Datta.