# Fields in general background

1. May 15, 2009

### Dox

Hi everybody!

I'm studing some classical field theory in general backgrounds. Of course the most beautiful way of doing so is using differential forms. For example, the lagrangian density of a massless scalar field would be
$$L_{\phi}=d\phi\wedge * d\phi,$$​
while the lagrangian density for a YM field is
$$L_{YM}=\left<d_A A\wedge *d_A A\right>.$$​

However, once one is interested in adding spinors, tetrads (and spin connection) enter into action...
$$L_{\psi}=\epsilon_{abcd}\bar{\psi}\Gamma^a e^b e^c e^d (d+\omega)\psi,$$​
with $$e^{a}$$ the tetrad 1-form and $$\omega$$
the spin-connection 1-form.

Although all lagrangian densities are coordinate independent, they are written in different ways... Is there a form of writing the first two using tetrads and spin connection?