# Physical meaning of the Casimir operators of Poincarè algebra

If one considers the algebra $$su(2)$$, it is well known that the Casimir Operator is $$C=L_1^2+L_2^2+L_3^2.$$ It corresponds to the total angular momentum and correctly is a conserved quantity.

I would like to know which is the physical meaning of the two Casimir operators of the Poincarè algebra.

• What did you find on Wikipedia and did not understand? Commented Oct 15, 2018 at 10:30
• On Wikipedia you can read "The Casimir invariants of this algebra are $P_\mu P^\mu$ and $W_\mu W^\mu$ where $W_\mu$ is the Pauli–Lubanski pseudovector; they serve as labels for the representations of the group." I've read the definition and their mathematical expression, but I've not understood the physical meaning. Commented Oct 15, 2018 at 10:34

The two Casimir operators are $$P^2 =P^\mu P_\mu$$ where $$P$$ is momentum, and $$W^2 =W^\mu W_\mu$$ where $$W$$ is the so-called Pauli-Lubanski pseudovector.
Evaluating $$P^2$$ in a particle's rest frame, we find that $$P^2 = m^2$$, so the first Casimir labels representations by mass.
The Pauli-Lubanski pseudovector is a bit more complicated but also has a simple interpretation in terms of spin. By definition, it's $$W^\mu = -\frac{1}{2} \epsilon^{\mu\nu\rho\sigma} P_\nu S_{\rho\sigma}$$, where $$S$$ is the relativistic spin angular momentum. So $$W^2$$ has to do with spin. In particular it gives you the particle's spin for massive particles: $$W^2 = -m^2 s(s+1)$$ and helicity for massless particles, $$W^2=0$$ and $$W^\mu = \pm s P^\mu$$.