In physics, the Pauli–Lubański pseudovector is an operator defined from the momentum and angular momentum, used in the quantum-relativistic description of angular momentum. It is named after Wolfgang Pauli and Józef Lubański. It describes the spin states of moving particles. It is the generator of the little group of the Poincaré group, that is the maximal subgroup (with four generators) leaving the eigenvalues of the four-momentum vector Pμ invariant.
Definition It is usually denoted by W (or less often by S) and defined by:
where
ε μ ν ρ σ {\displaystyle \varepsilon _{\mu \nu \rho \sigma }} is the four-dimensional totally antisymmetric Levi-Civita symbol;
J ν ρ {\displaystyle J^{\nu \rho }} is the relativistic angular momentum tensor operator ( M ν ρ {\displaystyle M^{\nu \rho }} );
P σ {\displaystyle P^{\sigma }} is the four-momentum operator. In the language of exterior algebra, it can be written as the Hodge dual of a trivector,
W = ⋆ ( J ∧ p ) . {\displaystyle \mathbf {W} =\star (\mathbf {J} \wedge \mathbf {p} ).}
Note W 0 = J → ⋅ P → {\displaystyle W_{0}={\vec {J}}\cdot {\vec {P}}} , and W → = E J → − P → × K → {\displaystyle {\vec {W}}=E{\vec {J}}-{\vec {P}}\times {\vec {K}}} where J → {\displaystyle {\vec {J}}} is the generator of rotations and K → {\displaystyle {\vec {K}}} is the generator of boosts. Wμ evidently satisfies
P μ W μ = 0 , {\displaystyle P^{\mu }W_{\mu }=0,}
as well as the following commutator relations,
[ P μ , W ν ] = 0 , [ J μ ν , W ρ ] = i ( g ρ ν W μ − g ρ μ W ν ) , {\displaystyle {\begin{aligned}\left[P^{\mu },W^{\nu }\right]&=0,\\\left[J^{\mu \nu },W^{\rho }\right]&=i\left(g^{\rho \nu }W^{\mu }-g^{\rho \mu }W^{\nu }\right),\end{aligned}}}
Consequently,
[ W μ , W ν ] = − i ϵ μ ν ρ σ W ρ P σ . {\displaystyle \left[W_{\mu },W_{\nu }\right]=-i\epsilon _{\mu \nu \rho \sigma }W^{\rho }P^{\sigma }.}
The scalar WμWμ is a Lorentz-invariant operator, and commutes with the four-momentum, and can thus serve as a label for irreducible unitary representations of the Poincaré group. That is, it can serve as the label for the spin, a feature of the spacetime structure of the representation, over and above the relativistically invariant label PμPμ for the mass of all states in a representation.
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