Flavor Oscillations from a Spatially Localized Source: A Simple General Treatment

A unique description avoiding confusion is presented for all flavor oscillation experiments in which particles of a definite flavor are emitted from a localized source. The probability for finding a particle with the wrong flavor must vanish at the position of the source for all times. This condition requires flavor-time and flavor-energy factorizations which determine uniquely the flavor mixture observed at a detector in the oscillation region; i.e. where the overlaps between the wave packets for different mass eigenstates are almost complete. Oscillation periods calculated for ``gedanken'' time-measurement experiments are shown to give the correct measured oscillation wave length in space when multiplied by the group velocity. Examples of neutrinos propagation in a weak field and in a gravitational field are given. In these cases the relative phase is modified differently for measurements in space and time. Energy-momentum (frequency-wave number) and space-time descriptions are complementary, equally valid and give the same results. The two identical phase shifts obtained describe the same physics; adding them together to get a factor of two is double counting.