# 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? 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. 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 $$W^\mu = \pm s P^\mu$$.