Helicity quantization of massless particles

Question

In Appendix B of QFT in a nutshell by Zee, a review of group theory is given. In the last paragraph of the appendix on page 533, he briefly discusses the helicity quantization of massless particles.

Firstly, he takes a rotation of $4\pi$ and considers possible helicities by using $exp(i4 \pi h) = 1$, where h is the helicity. Why he considers a rotation by $4\pi$ is not clear to me, and also why the path traced by this rotation can be shrunk to a point. Secondly, why is he not quantizing the helicity algebraically as conventionally done as in the case of angular momentum, but instead applying topological arguments. Is there a relation between the two?

Answer 1

1. The path traced out by a rotation by $4\pi$ can be shrunk to a point in $\mathrm{SO}(3)$ because such a rotation corresponds to a closed curve in the double cover $\mathrm{SU}(2)\cong S^3$, which is simply connected and therefore any closed curve in it can be shrunk to a point. Curves which can be shrunk in the cover can also be shrunk in the base since covering maps are injective on the fundamental groups. Contrast this with a rotation by $2\pi$, which in $S^3$ is a path between two antipodal points, hence not closed and cannot be shrunk. This works the same way for the Lorentz group $\mathrm{SO}(1,3)$ and its universal (double) cover $\mathrm{SL}(2,\mathbb{C})$.

2. Helicity is different from angular momentum because the angular momentum algebra is the Lie algebra of the massive little group $\mathrm{SO}(3)$, while helicity is for massless particles and hence must be related to the massless little group, which is a group variously denoted as $\mathrm{ISO}(2),\mathrm{SE}(2)$ or similar - it is the symmetry group of two-dimensional affine Euclidean space, hence consists essentially of $\mathrm{SO}(2)$ together with two-dimensional translations. Since the representations of translations are trivial, the question of the representations of the massless little group is reduced essentially to the representation theory of $\mathrm{SO}(2)\cong\mathrm{U}(1)$.

   This representation theory can be determined algebraically and one arrives at quantized values for helicity , but this cannot be done by the usual physicist's approach of representing the Lie algebra - the Lie algebra of $\mathrm{SO}(2)$ is simply the trivial Lie algebra $\mathbb{R}$, and there appear no quantization conditions in its representation theory - the quantization instead comes from the global structure of $\mathrm{SO}(2)\cong\mathrm{U}(1)\cong S^1$ and the fact that the possible unitary irreducible representations must be one-dimensional, hence completely classified by representation maps $\mathrm{U}(1)\to\mathrm{U}(1)$, since $\mathrm{U}(1)$ is the unitary group of a one-dimensional Hilbert space.

   It is straightforward to see that such maps $\rho_n : \mathrm{U}(1)\to\mathrm{U}(1), \mathrm{e}^{\mathrm{i}\phi}\mapsto\mathrm{e}^{\mathrm{i}n\phi}$ are simply classified by how often they wind the $S^1$ around itself, i.e. by the integers $n\in\mathbb{Z}$.

3. So how does this helicity relate to $4\pi$ rotations? The generator of this $\mathrm{U}(1)\subset\mathbb{R}^4\rtimes\mathrm{SL}(2,\mathbb{C})$ (the r.h.s. is the universal cover of the Poincaré group) is physically the projection of spin onto momentum, $S\cdot p$ (how exactly you work this out depends on how exactly you determined that the little group of the massless particle is $\mathrm{SE}(2)$). So a full $4\pi$ rotation around $p$ acts as $\rho_n(1) = \rho_n(\mathrm{e}^{2\pi\mathrm{i}})$ on our massless particle: $$ \mathrm{e}^{4\pi\mathrm{i}h} = \mathrm{e}^{\mathrm{2\pi\mathrm{i}n}},$$ hence $h=\frac{n}{2}$.