A phase modulation is simply the addition of phase values: a relative phase offset (phase operator) is added to an absolute phase value (fig.2), as follows:
where W is phase-modulated by phase operator ρ.
Mass-energy acts as a phase operator or modulator on the waves of other bosons, as the sum of the overlapping vacuum operators ρ:
When bosons cross each other, the modulators create an impulsive phase shift in the other bosons at the overlap. Using the terms from fig.1, W is modulated by ρ, which is introduced at t1. Had the bosons not crossed, the quantization condition for wave W would have occurred at t2.
This phase modulation causes solutions to be retarded or advanced, so changing the position (and time) of the solution in the vector of propagation, without violating causality.