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What are the properties which support this?
1) QM has been enormously successful over the past years for predicting all kinds of phenomena. Any successor theory must give the same results within the valid regime of QM. The overall phase does not change the probabilistic predictions of QM.
Let |S> = eia |s>, then <S|S> = <s|e-ia eia |s> = <s| e-ia+ia |s> = <s| e0 |s> = <s|s>
Hence, as stated in so many QM books, the overall phase plays no role in the probabilistic predictions of QM. However, this does not rule out the possibility that the overall phase could play some role in which eigenstate is selected in a measurement process. It simply puts a condition on the allowed values that the phase might take - it must obey a statistical law if one repeats "identical up to a phase factor" experiments.
2) The overall phase is a relativistic invariant. Thus, all inertial observers will agree on phase values.
Phi = K·R = wt - k·r
Also, this relation gives the phase values as the particle moves through spacetime. Other relativistic invariants include such things as c-speed of light, m0-rest mass, hbar-Planck's constant, q-electric charge, etc. Note that in the particle's own rest-frame, the overall phase will change according to wt. Thus the phase is always "spinning" around the time axis. If a particle has a constant K, one can show that:
Phi(K=K0) = (m0c2 / hbar) T + K0·R0
This description shows how the phases of separate particles could maintain a phase entanglement. If the particles are of equal rest mass and if they begin at the same spacetime event point, then their respective phases change at the same rates according to each one's proper time.
3) When one considers free particle scattering theory, the result is that the only change in the wave function at large distances is a change in the phase of the outgoing wave. Think about this. We are repeating "identical up to a phase factor" experiments with two particles. We have a target particle at a certain location. We shoot a moving particle at it with the same velocity from the same location in each setup. QM can not predict the exact outcome of this simple experiment, just the scattering cross-section probabilities. What can determine the actualized outcome? There is apparently some sort of phase interaction that occurs in the collision. Perhaps the differing overall initial phases of the particles involved in the same experimental configuration are what account for the different possible outcomes of the same experiment.
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