Poincare algebra

The Poincare algebra is generated by translations $P^{\mu }$ and Lorentz transformations $M^{\mu \nu }$. Its mixed commutator is $[M^{\mu \nu },P^{\rho }]=i({\eta }^{\nu \rho }{P}^{\mu }-{\eta }^{\mu \rho }{P}^{\nu })$.

The Pauli-Lubanski pseudovector is $W_{\mu }={ϵ}_{\mu \nu \rho \sigma }{M}^{\nu \rho }{P}^{\sigma }/2$, up to an overall sign convention. Its square is a Casimir of the Poincare algebra and measures spin in a massive irreducible representation.

In $W^{\mu }=-{ϵ}^{\mu \nu \rho \sigma }{P}_{\nu }{M}_{\rho \sigma }/2$, contracting with $P_{\mu }$ gives zero because the commuting momentum products are symmetric in $\mu ,\nu$ while the alternating tensor is antisymmetric. Reordering $P$ and $M$ makes no difference because the resulting metric contractions with the alternating tensor vanish.

// Target: special-relativity.bigb

For a massive positive-energy unitary irreducible representation of the Poincaré group with signature $(+---)$, choose a rest frame. The Pauli-Lubanski pseudovector has zero time component and spatial components $m{J}_i$, up to a common sign convention. Its square is therefore $-m^2{\mathbf{J}}^2$. The SU(2) little-group representation has spin $s$ and ${\mathbf{J}}^2=s(s+1)$, proving the invariant value. The mass and spin label the massive induced representation of the Poincare double cover.

With signature $(+---)$ and ${ϵ}_{0123}=+1$, a momentum $(k,0,0,k)$ and helicity $j_3$ give lower components $W_0=k{j}_3$ and $W_3=-k{j}_3$. They obey $W_{\mu }{P}^{\mu }=k(W_0+W_3)=0$. These longitudinal identities require only the momentum and rotation-projection eigenvalues, not a statement about transverse little-group translations.

With signature $(+---)$, ${ϵ}_{0123}=+1$, $J_3=M^{12}$ and the lower-index definition of $W_{\mu }$, a rest momentum $(m,0,0,0)$ gives $W_0=0$ and $W_3=-m{J}_3$ on that momentum eigenspace. A Hermitian spin projection $j_3$ consequently gives eigenvalues $0$ and $-m{j}_3$. Raising the third index changes its sign.