The wave function y is that function which is a solution to the Schrodinger equation. It's "norm" at x is equal to the probability of finding the subject particle at x. Thus it must be inherently non local since an increase in probability at some point must be accompanied by an instantaneous decrease at some other point. It represents what we know and perhaps can know about the particle's "state". However that does not mean the particle does not have real properties.
The "collapse" of the wave function can be viewed as a measurement providing more detailed boundary and/or initial conditions to Schrodinger's equation.
The wave function includes phase information which is obliterated when y is used with it's complement to find a measurable value as in:
y*x y .
The need for complex terms originates in the fact that must y include phase information to properly represent how the particle's state evolves in time. One might view y as a vector in a plane and the real and imaginary parts as components. The coordinate of this plane on the axis normal to the plane would represent the value of the particle's measurable. Then the magnitude of this vector or the area it sweeps relate directly to the probability density of the particle's measurable quantity.
Solutions to the time dependent Schrodinger include an e-i(Et-px)/h term. Thus, the phase of y changes with action; with change in action equal to E delta t-p delta x, with a change in action of h equal to one full cycle.
Change in action is E delta t-p delta x. This quantity is Lorentz invariant. In its own frame, a particle's change in action is mc2 delta t and it's inherent frequency of phase change is therefore mc2 /h. Several questions arise.
1) Are the phases of particles key to understanding interactions between particles and other particles and fields? This certainly seems to be the case with interference a la the two slit experiments (Ref 2).
2) What is the relationship between particle phase and the phase of y? They seem to change at the same rate.
3) What is the effect of change in potential energy on phase? Why?
