$\nabla \times \bm{A}_{pure} = 0$. This property was claimed to be essential for its physical interpretation as an OAM. However, this is not necessarily true as is clear from the papers [@VanEN94A],[@VanEN94B], which treats a similar problem in QED. It was shown there that the spin and OAM operators of the photons do not satisfy the ordinary commutation relation of angular momentum ($SU(2)$ algebra) separately. This is not surprising at all. In fact, it is true that the total momentum as well as the total angular momentum operators of a composite system must satisfy the Poincare algebra, because, in quantum field theory, a physical state of a composite particle must be one of the irreducible representations of the Poincare group. However, it is not an absolute demand of the Poincare symmetry that the momentum and the angular momentum of each constitute of a composite particle satisfies the Poincare algebra separately. Then, the claimed superiority of the decomposition (II) over (I) is not actually present. Nevertheless, since the decomposition (II) is also gauge-invariant, there still remains a possibility that it can be related to observables.

Recently, Hatta made an important step toward this direction [@Hatta12] based on his decomposition formula of 