# A Physical interpretation of Pauli-Lubanski pseudo-vector

#### spaghetti3451

$P^{\mu}$ generates translations and extracts the four-momentum of a particle when it acts on the momentum eigenstate of a particle.

$J^{\mu\nu}$ generates rotations and measures the spin angular momentum along the $i$-direction of a particle when it acts on the $i$-th direction spin angular momentum eigenstate of a particle.

The Pauli-Lubanski psuedo-vector is given by $W_{\mu}=\frac{1}{2}\epsilon_{\mu\sigma\rho\tau}J^{\sigma\rho}P^{\tau}$ such that
$W_{\mu}|P,j,j_{z}\rangle = -mJ_{i}|P,j,j_{z}\rangle,$ where $|P,j,j_{z}\rangle$ is a momentum space eigenstate representing a particle of spin $j$ at rest with $P^{\mu}=(m,0,0,0)$ and $m\neq 0$.

What is the physical interpretation of $W_{\mu}$?

To prove that $[J_{\mu\nu},W^{2}]=0$, an explicit form of $[J_{\mu\nu},W_{\rho}]$ is necessary. One way to obtain $[J_{\mu\nu},W_{\rho}]$ is to define $I=\frac{i} {8}\epsilon_{\alpha\beta\gamma\delta}J^{\alpha\beta}J^{\gamma\delta}$ and show that $W_{\rho}=[I,P_{\rho}]$ and $[J_{\mu\nu},I]=0$.

What is an easy way to show that $[J_{\mu\nu},I]=0$ using the epsilon symbol in $I$?

$I$ is a scalar as all the indices are $0$, so why can not say that $[J_{\mu\nu},W^{2}]=0$ trivially?

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#### vanhees71

Science Advisor
Gold Member
The Pauli Lubanski vector is pretty intuitive, when you familiarize yourself with Wigner's analysis of the irreducible unitary representations of the orthochronous proper Poincare group, the space-time symmetry group of special relativity. It provides the infinitesimal generators of the socalled little group associated with the representation. For details, see Appendix B of my lecture notes on QFT,

http://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf

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