# Investigations into a heuristic Lagrangian of graviton field

The following is taken from page 40 of Matthew Schwartz's "Introduction to Quantum Field Theory."

The Lagrangian for the graviton is heuristically ##\mathcal{L}=-\frac{1}{2}h\Box h + \frac{1}{3}\lambda h^{3}+Jh,## where ##h## represents the gravitational potential. We are ignoring spin and treating gravity as a simple scalar field theory. The ##h^3## term represents a graviton self-interaction, which is present in general relativity and so ##\lambda \sim \sqrt{G_N}##. The equations of motion are ##\Box h -\lambda h^{2}-J=0##.

Why the ##h^{3}## term represent the graviton self-interaction? What does self-interaction mean anyway? Does it mean the interaction among the various excitations of the graviton field?

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haushofer
Well, draw the Feynmandiagrams for this theory, and you see that the h^3 term brings together three gravitons. As such it describes an interaction among gravitons themselves, hence "self-interaction". So to answer your last question: yes. You can compare it with a simple phi to the fourth theory :)

Thanks!

I was also wondering why the fact that the ##h^3## term is present in general relativity implies that ##\lambda \sim \sqrt{G_N}##?

Where in general relativity is the ##h^3## term present anyway?

bummp!

haushofer