Gauge covariant observables and phase operators

We characterize those Hilbert space operators which are first moments of the gauge covariant positive operator-valued measures introduced by Lahti and Pellonpaa. We consider the suitability of such measures for describing quantum phase, observing that the gauge angle defining the measure is not always closely related to the spectrum of the first moment operator. We take the position that the first moment operator represents the quantal property the measure is (imperfectly) describing. This is in contrast to the viewpoint that positive operator-valued measures are the fundamental quantum observables.