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The complementarity between time and energy, as well as between an angle anda component of angular momentum, is described at three different layers ofunderstanding. The phenomena of super-resolution are readily apparent in thequantum phase representation which also reveals that entanglement is notrequired. We modify Schwinger's harmonic oscillator model of angular momentumto include the case of photons. Therein the quantum angle measurement is shownto be equivalent to the measurement of the relative phase between the twooscillators. Two reasonable ways of dealing with degeneracy are shown tocorrespond to: a conditional measurement which takes a snapshot in absolutetime (corresponding to adding probability amplitudes); and a marginalmeasurement which takes an average in absolute time (corresponding to addingprobabilities). The sense in which distinguishability is a "matter of how longwe look" is discussed and the meaning of the general theory is illustrated bytaking the two oscillators to be circularly polarized photons. It is shown thatan odd number of x-polarized photons will never have an angle in correspondencewith the y-axis; but an even number of x-polarized photons always can! Thebehavior of an x-polarized coherent state is examined and the snapshot angulardistributions are seen to evolve into two counter-rotating peaks resulting inconsiderable correspondence with the y-axis at the time for which a classicallinear polarization vector would shrink to zero length. We also demonstrate howthe probability distribution of absolute time (herein a measurable quantity,rather than just a parameter) has an influence on how these snapshot angulardistributions evolve into a quantum version of the polarization ellipse.

A Quantum Theory of Angle and Relative Phase Measurement