Retrieving "Casimir Invariant" from the archives

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1. Angular Momentum Tensor

   Linked via "Casimir invariants"

   where $P^{\nu}$ is the four-momentum of the system.
   For a massive particle, the two Casimir invariants of the Poincaré group are the square of the four-momentum ($P^{\mu}P{\mu} = m^2 c^2$) and the square of the Pauli-Lubanski vector ($W^{\mu}W{\mu}$). The latter invariant, $W^{\mu}W_{\mu}$, determines the intrinsic spin/) $s$ of the particle according to the relationship:
   $$ W^{\mu}W_{\mu} = -m^2 c^2 \hbar^2 s(s+1) $$

2. Minkowski Metric Tensor

   Linked via "Casimir invariant"

   Relation to the Poincaré Group
   The Minkowski metric tensor is intrinsically linked to the structure of the Poincaré group, which encompasses both the Lorentz transformations (rotations and boosts) and the spacetime translations $P^\mu$. The metric defines the quadratic Casimir invariant of the Lie algebra associated with the Poincaré group, demonstrating its role in classifying…