# What is the value of $W_\mu W^\mu$ for massless particles?

What is the value of the quantity $$W_\mu W^\mu$$ for massless particles where $$W^\mu$$ is called Pauli-Lubanski vector defined as $$W^\mu=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}P_\nu J_{\alpha\beta}$$.

It is trivial to see that $$W^\mu P_\mu=0$$ and in addition for massless particles $$P_\mu P^\mu=0$$ holds true. For the massless case, we can choose $$P=(p,0,0,p)$$ which gives $$pW^0=pW^3$$ or $$W^0=W^3$$. Therefore, $$W_\mu W^\mu=-(W^1)^2-(W^2)^2$$ which is not necessarily zero. But in order to show $$W^\mu=\lambda P^\mu$$ i.e., $$W^\mu$$ and $$P^\mu$$ are linearly dependent we assume $$W_\mu W^\mu=0$$ or $$W^\mu$$ to be lightlike. See the paragraph above Eq. B.50 here.

What is the actual value of this for massless particles?

Since $$W^2$$ is a Casimir of the Poincaré group that is independent of $$P^2$$, it is not uniquely determined by saying that we have a "massless particle", i.e. $$P^2 = 0$$. It is also not uniquely determined for massive particles, where it has the value $$W^2 = m^2s(s+1)$$ for $$s$$ the spin of the particle, and spin is after all independent from mass.
While the spin representations of massive particles are induced from representations of the massive little group $$\mathrm{SO}(3)$$ with spin $$s$$, the massless little group is $$\mathrm{ISO}(2)$$, the two-dimensional Euclidean group. This group has only one-dimensional and infinite-dimensional unitary representations. The one-dimensional ones are the ones we usually use in physics, and there $$W^\mu = \lambda P^\mu$$ holds. There are two of them that are interchanged by parity transformations, so the "usual" massless particle gets a two-dimensional "helicity representation".
The infinite-dimensional ones, also called "continuous spin representations", have the components of $$W$$ not parallel to $$P$$ not vanish and hence non-trivially represented on the space of states. These representations do not seem to appear ordinarily in nature - like the tachyonic representations with $$P^2 < 0$$ - because they do not lead to consistent causal quantum field theories, though it is hard to find texts that unambiguously state this in modern parlance (see Hirata, "Quantization of Massless Particles with Continuous Spin", Progress of Theoretical Physics 58 (2), 1977).
Therefore, the common assumption that $$W^2 = 0$$ for massless particles is justified by $$W^2 \neq 0$$ producing unphysical representations, but it would be nice if this was stated a bit more explicitly in many cases.
• Good point! I was mislead to believe $W^2=0$ for massless particles can be directed shown by algebra. In fact, it is a physical constraint (no continuous spin) that guarantees $W^2=0$, $W^\mu=\sigma p^\mu$ and the helicity is Lorentz invariant (the operator $\frac{\vec{J}\cdot\vec{p}}{|\vec{p}|}$ commutes with boosts). Commented Feb 29 at 16:42