- #1

- 649

- 3

before I begin this exercise for myself I want to make sure I have a few things right. Would the Lagrangian for low energy quantum gravity be

[tex]

\mathcal{L}=\frac{1}{2}\partial^\mu \bar{\psi}\partial_\mu \psi +\frac{1}{2}m^2\bar{\psi}\psi-\frac{1}{2}g_{\alpha\beta}R^{\alpha\beta}

[/tex]

or would it be

[tex]

\mathcal{L}=\frac{1}{2}\nabla^\mu \bar{\psi}\nabla_\mu \psi +\frac{1}{2}m^2\bar{\psi}\psi-\frac{1}{2}g_{\alpha\beta}R^{\alpha\beta}

[/tex]

[tex]

=\frac{1}{2}\left( \partial^\mu \bar{\psi}\partial_\mu \psi+\bar{\psi}\partial^\mu \partial_\mu \psi +\bar{\psi}g^{\alpha\beta}\Gamma^{\gamma}_{\alpha \beta}\partial_\gamma \psi \right)+\frac{1}{2}m^2\bar{\psi}\psi - \frac{1}{2}g_{\alpha\beta}R^{\alpha\beta}

[/tex]

That is, do the covariant derivatives act solely on the scalar functions or does the covariant derivative on the left act on both [itex]\bar{\psi}[/itex] and [itex]\partial_\alpha \psi[/itex]?

If I use the first Lagrangian my equations of motion are incorrect , but I was also not under the impression that the covariant derivative was acting in the way [itex]\nabla^\mu (\bar{\psi}\partial_\mu \psi )[/itex]. Or perhaps I am using the wrong Euler-Lagrange equations. Should the Euler-Lagrange equations have covariant derivatives in them or just normal partials like in the rest of QFT?

Thanks,