From Newsgroup: sci.physics.research

In his Quantum Physics book (1985), Robert Eisberg writes,

|The wave function is complex. That is, it contains the imaginary
|number i. Recall that this behavior was forced upon us. We first
|tried to find a way of satisfying our four assumptions concerning
|the Schroedinger equation by using a purely real free particle wave |function, (5-1), and we found that there was no reasonable way of
|doing this.

There is a global invariance on the phase of this complex
wave function, and when one requires this invariance to be
local, one needs to introduce a gauge field A to get correct
values for the momentum operator.

So, when we describe an electron, we get its charge and its
coupling to the electromagnetic field from this requirement.

Does this mean that as soon as we begin to describe a particle,
electromagnetism becomes necessary?

But what about neutral particles then? In a worlds with only neutral
particles and no electromagnetism, the wave function still needs to
be complex. But what about local gauge invariance in such a world?

It would be easier for me to understand when neutral particles
would have a real wave function!

And there seems to be something like this,

|It is worth remarking here that phi, has been assumed to be
|complex. If phi, is taken as real, then rho in (2.20)
|vanishes, and so does j. It will be shown in the next chapter
|that the correct interpretation of complex phi, is for the
|description of charged particles. Real phi, corresponds to
|electrically neutral particles, and rho and j are then the
|charge and current densities, rather than the probability and
|probability current densities.
from 2.2 Klein Gordon Equation in Quantum Field Theory by Lewis H. Ryder

But that phi Ryder is talking about is not exactly the
traditional wave function, its a scalar quantum field,
solution of the Klein-Gordon equation.

However, such a field would make much more sense to me as a basis
for the U(1) invariance, because it is now clear that it needs
to be complex only for charged particles. What do you think?
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