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I know there are many deep issues that come up in attempts to do quantum gravity, but I suspect that the following is not deep but easily explained by someone who knows more about this than I do.

Suppose two neutrons are interacting gravitationally. The Feynman diagram that I would naively draw for this process would show two neutron world-lines with a graviton being exchanged. I.e., it would be exactly like the H-shaped diagram I'd use to show two particles interacting electrically, except that the photon is replaced by a graviton. But the graviton version obviously doesn't work, because at a given vertex, it's impossible to couple the angular momenta correctly; spins 1/2, 1/2, and 2 don't satisfy the triangle inequality. This would seem to prove that no particle with spin less than 1 can interact gravitationally, which is obviously wrong.

So what does the correct Feynman diagram look like?

Suppose two neutrons are interacting gravitationally. The Feynman diagram that I would naively draw for this process would show two neutron world-lines with a graviton being exchanged. I.e., it would be exactly like the H-shaped diagram I'd use to show two particles interacting electrically, except that the photon is replaced by a graviton. But the graviton version obviously doesn't work, because at a given vertex, it's impossible to couple the angular momenta correctly; spins 1/2, 1/2, and 2 don't satisfy the triangle inequality. This would seem to prove that no particle with spin less than 1 can interact gravitationally, which is obviously wrong.

So what does the correct Feynman diagram look like?

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