  Over the past thirty years there has been an ever-increasing interest in the
so-called orbital angular momentum (OAM) of light, driven in large part by its
diverse range of applications, from new imaging techniques to optical tweezers.
However, there are fundamental theoretical issues with splitting the photon
angular momentum operator into spin (SAM) and orbital angular momentum parts.
While such an SAM-OAM splitting is simple and unambiguous for massive
particles, there are many different proposed splittings for massless photons
and no consensus about which is correct. Moreover, it has been explicitly shown
that most of the proposed SAM and OAM operators do not satisfy the defining
commutation relations of angular momentum operators and are thus not legitimate
splittings. Here we prove that it is generally impossible to split the total
angular momentum operator of massless bosons, like photons and gravitons, into
spin and orbital parts. We show that the obstruction to such a splitting can be
understood as both geometric and topological. We prove two additional no-go
theorems which generalize this result, in particular showing that there are no
SAM-OAM splittings even if (1) the SAM operator is allowed to represent
non-internal symmetries or (2) if one allows the SAM and OAM operators to
generate non-$\mathrm{SO}(3)$ symmetries. We show how gauge redundant
descriptions of massless particles can suggest SAM-OAM splittings which are
invalid once gauge-invariance is imposed.
