Geometrical interpretation of the photon position operator with commuting components

It is shown that the photon position operator \stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{X}}with commuting components can be written in the momentum representation as \stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{X}}=i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{D}} where \stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\stackrel{P\vec}{D}}is a flat connection in the tangent bundle...