  We study the non-uniqueness problem of the gauge-invariant angular momentum
separation for the case of QED, which stems from the recent controversy
concerning the proper definitions of the orbital angular momentum and spin
operator of the individual parts of a gauge field system. For the free quantum
electrodynamics without matter, we show that the basic requirement of Euclidean
symmetry selects a unique physical angular momentum separation scheme from the
multitude of the possible angular momentum separation schemes constructed using
the various Gauge Invariant Extentions. Based on these results, we propose a
set of natural angular momentum separation schemes for the case of interacting
QED by invoking the formalism of asymptotic fields. Some perspectives on such a
problem for the case of QCD are briefly discussed.
