Hi i am trying to derive the Dirac equation of the form:(adsbygoogle = window.adsbygoogle || []).push({});

[itex] [i\gamma^0 \partial_0 + i\frac{1}{a(t)}\gamma.\nabla +i\frac{3}{2}(\frac{\dot{a}}{a})\gamma^0 - (m+h\phi)]\psi [/itex] where a is the scale factor for expnasion of the universe.

I understand that the matter action is [itex]S=\int d^{4}x e [\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi - V(\phi) + i \bar{\psi}\bar{\gamma}^{\mu}\vec{D}_{\mu}\psi -(m+h\phi)\bar{\psi}\psi)] [/itex] but i don't understand firstly why there is a vierbein and not a [itex]\sqrt{-g}[/itex] term.

I don't really understand why this is the case [itex]D_{\mu}=\frac{1}{4}\bar{\psi}\bar{\gamma}^{\mu} \gamma_{\alpha \beta}\omega^{\alpha \beta}_{\mu}[/itex] and why the arrow above the D is gone.

And lastly I don't understand why [itex]\bar{\gamma}^{i}=\frac{1}{a(t)}\gamma^{i}[/itex]

I understand that one needs to vary the action and i can do that bit but I don't understand some of these conversions, thx. I would appareciate any help that anyone can offer in tis challenge.

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# Deriving the dirac equation

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