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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.

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  • $\begingroup$ What did you find on Wikipedia and did not understand? $\endgroup$
    – DanielC
    Commented Oct 15, 2018 at 10:30
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    $\begingroup$ 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. $\endgroup$
    – AndreaPaco
    Commented Oct 15, 2018 at 10:34

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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$.

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