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In this paper we provide a consistent framework to address the notoriousdecomposition of the single-photon total angular momentum (TAM) into a spin(SAM) and an orbital (OAM) component. In particular, we find that the canonicalSAM and OAM, generators of internal and spatial rotations respectively in thespace of physical states, are mutually compatible but unsharp quantumobservables, therefore POVM (Positive Operator-Valued Measures) describe theirjoint measurements. On the other hand, a non-canonical SAM and OAM can beconstructed that is mutually incompatible but represent sharp quantumobservables, thus PVM (Projector-Valued Measurements), reflecting theirconsistency with the transversality condition characterizing singlephotonwavefunctions. We discuss the implementations on joint measurements for bothdecompositions and provide an explicit calculation of all these quantities forcircularly polarized Gaussian single-photon states.

Unified treatment of the total angular momentum of single photons via generalized quantum observables