Finding Graviton Propagator from QFT in a Nutshell- Zee's Exercise

[jdstokes]

On page 36 of QFT in a nutshell, Zee sets an exercise to find the propagator for a massive spin 2 particle by assuming that it is a linear combination of terms such as $G_{\mu\nu}G_{\lambda\sigma}$. The only method I can think up for doing this is to write down all 36 products of $G_{\mu\nu},G_{\lambda\sigma},G_{\mu\sigma},G_{\mu\lambda},G_{\nu\sigma},G_{\lambda}$ and apply the symmetry operations to see which sum of terms is left invariant, likewise for the $G_{\mu\nu}k_\lambda k_\sigma$ terms.

Are there any slicker combinatorial methods for figuring this out or am I hoping for too much?

 

[Jhero]

Thank you for bringing up this exercise from Zee's book. Finding the propagator for a massive spin 2 particle can indeed be a challenging task, but there are some alternative methods that may be helpful.

One approach is to use the Feynman diagram technique, where the propagator can be represented as a sum of all possible Feynman diagrams with external lines corresponding to the spin 2 particle. This method can be more efficient than manually writing out all possible combinations of terms.

Another option is to use tensor calculus, where the propagator can be expressed in terms of tensors and their contractions. This can simplify the process and make it more manageable.

Lastly, there may also be some symmetry arguments that can be used to reduce the number of terms to consider. For example, the propagator must be symmetric under the exchange of the two external lines, so some terms may cancel out due to this symmetry.

In summary, there are certainly alternative methods that can make the task of finding the propagator for a massive spin 2 particle more manageable. I encourage you to explore these options and see which one works best for you. Good luck with your calculations!