The motivation is the following:
For each particle of mass m, we could write :
$- (E/m)^2 + (p_x/m)^2 + (p_y/m)^2 + (p_z/m)^2 = -1$,
which is nothing than a equation for a point in the hyperbolic space $\mathbb H3$, which can be seen, with Poincaré representation (Poincaré ball), as a (full) sphere (with hyperbolic distance)
For photons, we may give us a very tiny mass, so photons could be considered, in the limit where this tiny mass is zero, as living at the boundary of $\mathbb H3$, which is $\mathbb CP1$ (Riemann sphere $S^2$)
With these prescriptions, we cannot distinguish, photons with proportionnal momenta, but if the theory is momentum-scale invariant, it seems OK.
So could we use Hyperbolic Space $H3$ as representation space for momenta, in momentum-scale invariant theories?