Can Scalar and YM Lagrangians Be Written Using Tetrads and Spin Connection?

[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?

Thanks in advance!

 

[azntoon]

</code>Yes, there is a way to write the first two lagrangian densities using tetrads and spin connection. The lagrangian density of a massless scalar field can be written as:L_{\phi} = e^{a} \wedge *(e_a \wedge d\phi)where e^{a} is the tetrad 1-form and * denotes the Hodge dual.The lagrangian density for a YM field can be written as:L_{YM}=\left<e^{a}\wedge *F_{ab}\wedge e^b\right>, where F_{ab} is the curvature 2-form associated with the spin connection 1-form, \omega. Hope this helps!