# Casimir operators for Poincare algebra

I have seen at various places the comment that the operator $$P_\mu P^\mu$$ is a Casimir operator of Lorentz algebra and thus it satisfies a on-shell condition like $$P_\mu P^\mu=m^2$$. Given the Poincare algebra \begin{aligned} i\left[M^{\mu \nu}, M^{\rho \sigma}\right] &=g^{\mu \sigma} M^{\nu \rho}+g^{\nu \rho} M^{\mu \sigma}-g^{\mu \rho} M^{\nu \sigma}-g^{\nu \sigma} M^{\mu \rho} \\ i\left[P^{\mu}, M^{\rho \sigma}\right] &=g^{\mu \rho} P^{\sigma}-g^{\mu \sigma} P^{\rho} \\ \left[P^{\mu}, P^{\nu}\right] &=0. \end{aligned} How does one derive its Casimir operators, especially the one $$P_\mu P^\mu$$? Can someone show the crucial steps? Also Does the method works for any other similar algebra? Moreover, if an operator, say, $$A$$ commutes with the generators $$M^{\mu\nu}$$, i.e., $$[A,M^{\mu\nu}]=0$$, can it be said that $$A$$ is a Casimir operator?

Casimir operators commute with all generators. That's what you need to check. $$P^\mu P_\mu$$ does commute with $$M$$ and $$P$$. A fast way to say it is that
1. $$[P^2,P_\mu] \propto [P_\nu,P_\mu] =0$$
2. $$P^2$$ is a scalar and therefore it's annihilated by the rotation generators
But if you don't believe 2. you can just check \begin{aligned} i[M_{\mu\nu},P^2] &= 2\,(g_{\rho\mu}P_\nu-g_{\rho\nu}P_\mu)P^\rho\\ &=2 P_\mu P_\nu - 2 P_\nu P_\mu \\&= 0\,. \end{aligned} For other Casimirs such as $$W^\mu W_\mu\,,\qquad W_\mu := \tfrac12 \varepsilon_{\mu\nu\rho\lambda} M^{\nu\rho}P^\lambda\,,$$ you can do the same. This is just a bit harder. The argument 2. still works because it's a scalar. Then by explicit computation $$[P_\mu,W_\nu] = 0\,,$$ so also its square commutes with $$P$$.
As far as I know, there are no ways of deriving Casimirs. In the case of the Poincare algebra one guesses and checks that $$P_{\mu}P^{\mu}$$ and $$W_{\mu}W^{\mu}$$ are Casimirs, based on physical principles.
There are some ways of deriving the quadratic Casimir of a (simple) Lie algebra. If you calculate its Cartan metric then the quadratic Casimir is $$g_{\mu \nu}x^{\mu}x^{\nu}$$.
You might check the answers to this question: Explicit Quadratic Casimir for $sp(2N)$