# Propagator of the Proca Lagrangian

## Homework Statement

I want to show that the propagator of Proca Lagrangian:

$$\mathcal{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}+\frac{1}{2}M^2A_\mu A^\mu$$

Is given by:

$$\widetilde{D}_{\mu \nu}(k)=\frac{i}{k^2-M^2+i\epsilon}[-g_{\mu\nu}+\frac{k_\mu k_\nu}{M^2}]$$

## Homework Equations

Remember that: $$F_{\mu \nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$$

## The Attempt at a Solution

I tried to use the Euler-Lagrange equation, and I obtained:

$$\partial_{\mu} (\partial^{\mu} A_{\nu} - \partial_{\nu} A^{\mu} ) + M^2 A^{\nu} = 0$$

I suppose I have to do a Fourier Transform in order to express that equation in terms of $$k^\mu$$
but I don't know how to do it. I don't even know if I have started the problem properly, or if there's another way.
Can anyone help me, please?

## Answers and Replies

MathematicalPhysicist
Gold Member
Check it in Problem Book in QFT of Voja's the answers to problems 5.2b, 5.7,6.15.

Muoniex
Check it in Problem Book in QFT of Voja's the answers to problems 5.2b, 5.7,6.15.

Sorry for the late answer, but I wanted to check all the steps with calm.
The problems you told me helped a lot, thanks!