Phase operators and phase relations for photon states

For a quantized mode of the radiation field, the operator whose classical analog is the ordinary phase factor of the mode amplitudes has been shown to be nonunitary. A rigorous formulation of the phase P is given on the basis of the canonical factorization theorem. Many of the seemingly complex features of phase operators are found to be simple direct consequences of the general mathematical theory. It can be readily seen that P is a partial isometry but not a unitary operator. In contrast to the amplitude operator, it is found that P is not a spectral operator and that the set of phase eigenstates is not complete. Mathematically precise operator relations are developed, and a complete spectral analysis is given for each of the phase operators.-