# Weyl- Squared Lagrangians

I'm studying conformal gravity theories, in particular I read that if we take $$L=\sqrt{g}C_{abcd}C^{abcd}$$ where $$C$$ is the Weyl tensor the theory we get is not unitary.

What does it means unitary at this level?

Why is important?

And, how do we check if a theory is unitary or not?

• Perhaps you could give us the reference so we can read a bit more? But in principle, the Weyl squared Lagrangian seems like a four-derivative field theory. So perhaps you want to look for issues in field theories like that more generically. – A_user_with_NoName Oct 17 at 6:45
• For example I read here en.wikipedia.org/wiki/Conformal_gravity#Weyl-squared_theories – Alabarda 980 Oct 17 at 6:46
• When you said studying I thought from book, lecture notes, review paper, not just wiki. This is why I asked for the reference. Maybe someone else knows the theory and can help more – A_user_with_NoName Oct 17 at 6:48
• I'm studying this from lecture notes of my university course, but unitarity is not mentioned at all. Reading that page i found unitarity for the first time in this contest, so I linked that page. – Alabarda 980 Oct 17 at 6:52
• There is a paper on Weyl squared supergravity where the authors have some comments on unitarity, ghosts and time propagation. I am posting the link with the hope that it will be a helpful source arxiv.org/abs/1806.10016 – A_user_with_NoName Oct 17 at 7:03