  The complementarity between time and energy, as well as between an angle and
a component of angular momentum, is described at three different layers of
understanding. The phenomena of super-resolution are readily apparent in the
quantum phase representation which also reveals that entanglement is not
required. We modify Schwinger's harmonic oscillator model of angular momentum
to include the case of photons. Therein the quantum angle measurement is shown
to be equivalent to the measurement of the relative phase between the two
oscillators. Two reasonable ways of dealing with degeneracy are shown to
correspond to: a conditional measurement which takes a snapshot in absolute
time (corresponding to adding probability amplitudes); and a marginal
measurement which takes an average in absolute time (corresponding to adding
probabilities). The sense in which distinguishability is a "matter of how long
we look" is discussed and the meaning of the general theory is illustrated by
taking the two oscillators to be circularly polarized photons. It is shown that
an odd number of x-polarized photons will never have an angle in correspondence
with the y-axis; but an even number of x-polarized photons always can! The
behavior of an x-polarized coherent state is examined and the snapshot angular
distributions are seen to evolve into two counter-rotating peaks resulting in
considerable correspondence with the y-axis at the time for which a classical
linear polarization vector would shrink to zero length. We also demonstrate how
the probability distribution of absolute time (herein a measurable quantity,
rather than just a parameter) has an influence on how these snapshot angular
distributions evolve into a quantum version of the polarization ellipse.
