  We study the quantum dynamics of fractional excitations in quantum spin ice.
We focus on the density of states in the two-monopole sector, $\rho(\omega)$,
as this can be connected to the wavevector-integrated dynamical structure
factor accessible in neutron scattering experiments. We find that
$\rho(\omega)$ exhibits a strikingly characteristic singular and asymmetric
structure which provides a useful fingerprint for comparison to experiment.
$\rho(\omega)$ obtained from exact diagonalisation of a finite cluster agrees
well with that from the analytical solution of a hopping problem on a Husimi
cactus representing configuration space, but not with the corresponding result
on a face-centred cubic lattice, on which the monopoles move in real space. The
main difference between the latter two lies in the inclusion of the emergent
gauge field degrees of freedom under which the monopoles are charged. This
underlines the importance of treating both sets of degrees of freedom together,
and presents a novel instance of dimensional transmutation.
