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!

 

[Entropia]

Hello! It's great to see someone studying classical field theory in general backgrounds. You're right, using differential forms is a beautiful way to approach this topic. The lagrangian density you provided for a massless scalar field and a YM field are both excellent examples of this approach.

As for your question about adding spinors and tetrads into the equation, yes, they do introduce a bit more complexity. In this case, you can write the lagrangian density using tetrads and spin connection by using the vielbein formalism. This allows us to write the lagrangian in terms of the tetrad and spin connection 1-forms, as you have shown in your example for the spinor field.

Using the vielbein formalism, we can express all of the lagrangian densities in a unified way, making it easier to compare and analyze them. This approach is often used in the study of gravity and other theories involving spinors.

I hope this helps answer your question. Keep up the great work in your studies!