# Tetrad choice for Pauli-Lubanski in the massless case

The Pauli-Lubanski pseudovector coincides with intrinsic spin in the rest frame of the particle. In a more general frame, one defines a tetrad and projects the PL vector on it to define intrinsic spin components. I am trying to understand how this works for the massless case, but I cannot understand the choice of tetrad member s at equation 10.53 p.117 here: http://staff.science.uva.nl/~msnoek/GT/LectNotes_GrTh_2011_2.pdf: it does not verify the transversality condition of p.115 since $s^0>|s^3|$. I basically don't understand the whole massless discussion p.117 in comparison with the massive case where steps were clear and state label $\sigma$ was decomposed using the Casimir operator $\vec{S}^2$ and an additional operator $S^3$.

For the massless case, one needs to show that $W^\mu = \lambda P^\mu$. Equation (10.53) provides a basis for an arbitrary four-vector and then expands $W^\mu$ in that basis. Imposing the two conditions $W\cdot P=0$ and $W \cdot W=0$ completes the proof by showing that all other "components" in that basis vanish.
• Thanks for your help suresh. However, the basis they provide does not correspond to their definition of tetrad since $s_\mu p^\mu \neq 0$, i.e. the fourth four-vector fails to be transversal. I am suspecting that maybe the definition of tetrad should change in the massless case but this is not explicitly said in the document. – Issam Ibnouhsein Oct 6 '14 at 13:49
• I see. I guess the 4 conditions in the massive case were here to ensure that the boosted tetrad coincides with the rest frame tetrad. However, I don't understand why we impose the two conditions $W\cdot P=0$ and $W\cdot W=0$. I don't get the explanation of the latter with finite dim reps. Furthermore, $\lambda$ is defined as the helicity operator, so $\lambda P^\mu$ makes no sense, indeed we are projecting $W^\mu$ on the tetrad so it should be $\lambda p^\mu$. – Issam Ibnouhsein Oct 6 '14 at 23:30
• The condition $W\cdot P=0$ follows from the definition of the Pauli-Lubanski tensor. The condition $W\cdot W=0$ is something that needs extra input. This comes from representation theory as worked out by Wigner. If you require a finite-dimensional unitary irrep in the massless case, then $W \cdot W=0$. You either accept it or work through Wigner's paper (a worthy thing!). – suresh Oct 7 '14 at 2:15
• Thanks a lot for all the details! One last question: don't you agree with my comment on the distinction between $p^\mu$ and $P^\mu$? I can understand they used notation $\lambda$ for two distinct things, the helicity operator p.117 when they equate it to $W^0$, and a scalar in the decomposition 10.54 of $W$. However, even if we assume that $\lambda$ is now a scalar, why do we choose four-vector $P^\mu$ to decompose $W$ whereas our tetrad was defined by the four-vector of eigenvalues $p=(1,0,0,1)$? – Issam Ibnouhsein Oct 7 '14 at 14:48