Gauge symmetry and background independence: Should the proton spin decomposition be path independent?

Exploring the similarities between the Chen \emph{et al.} approach, where physical and gauge degrees of freedom of the gauge potential are explicitly separated, and the background field method, we provide an alternative point of view to the proton spin decomposition issue. We show in particular that the gauge symmetry can be realized in two different ways, and discuss the relations between the concepts of path dependence, Stueckelberg dependence and background dependence. Finally, we argue that path/Stueckelberg/background-dependent decompositions of the proton spin are in principle measurable and therefore physically meaningful.