  The general question, crucial to an understanding of the internal structure
of the nucleon, of how to split the total angular momentum of a photon or gluon
into spin and orbital contributions is one of the most important and
interesting challenges faced by gauge theories like Quantum Electrodynamics and
Quantum Chromodynamics. This is particularly challenging since all QED
textbooks state that such an splitting cannot be done for a photon (and a
fortiori for a gluon) in a gauge-invariant way, yet experimentalists around the
world are engaged in measuring what they believe is the gluon spin! This
question has been a subject of intense debate and controversy, ever since, in
2008, it was claimed that such a gauge-invariant split was, in fact, possible.
We explain in what sense this claim is true and how it turns out that one of
the main problems is that such a decomposition is not unique and therefore
raises the question of what is the most natural or physical choice. The
essential requirement of measurability does not solve the ambiguities and leads
us to the conclusion that the choice of a particular decomposition is
essentially a matter of taste and convenience. In this review, we provide a
pedagogical introduction to the question of angular momentum decomposition in a
gauge theory, present the main relevant decompositions and discuss in detail
several aspects of the controversies regarding the question of gauge
invariance, frame dependence, uniqueness and measurability. We stress the
physical implications of the recent developments and collect into a separate
section all the sum rules and relations which we think experimentally relevant.
We hope that such a review will make the matter amenable to a broader community
and will help to clarify the present situation.
