Helicity for massless particles

Question

The little group for massless particles is $ISO(2)$, with the following Lie algebra: $$[A,B]=0, \; [J^3,A]=iB, \; [J^3,B]=-iA,$$ where $A,B$ generate translations and $J^3$ generates rotations. To obtain the massless projective representations of the Poincaré group, we first look at projective representations of the little group, thus $ISO(2)$, then use the method of induced representations, i.e. Mackey theory. How can one conclude, that the phases appearing in the projective representation are actually just signs, and not arbitrary phases at this point? Or conversely: how can one conclude that massless particles have integer or half integer helicity? For massive particles, integer or half integer spin is clear, because we look at projective representations of the little group $SO(3)$, which are in 1-1 correspondence with ordinary representations of its universal cover $SU(2)$, which are labelled by integer or half integer numbers. However, apparently the statement is true nevertheless even for massless particles, but I don't see why.

I've read something about this in Weinberg Vol. 1 p.89-90, but I am kind of confused by the statement made above equation (2.7.43), to which I was trying to find an alternative explanation.

How does Weinberg conclude that the loop given can be contracted to a point in the massless case?

Answer 1

Abstractly the group $ISO(2)\supset SO(2)$ has fundamental group $\mathbb{Z}$.

However, the main point is that the massless little group is imbedded inside the restricted Lorentz group $SO^+(3,1)$, where we know for physical reasons that even winding number is contractable, and hence should be identified in the little group, corresponding to helicity $\in\mathbb{Z}/2$ irreps$^1$.

References:

1. S. Weinberg, Quantum Theory of Fields, Vol. 1, 1995; p. 89-90.

2. Wikipedia.

3. N. Straumann, arXiv:0809.4942; p. 7.

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$^1$ Concerning the "continuous spin" representation, see Refs. 2 & 3.