A dilemma in representing observables in quantum mechanics
  There are self-adjoint operators which determine both spectral and semispectral measures. These measures have very different commutativity and covariance properties. This fact poses a serious question on the physical meaning of such a self-adjoint operator and its associated operator measures. 
D. A. Dubin, M. A. Hennings, P. Lahti, J.-P. Pellonpaa