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The original poster, naqo, asked: "couldn't one work from the begining with a real wave?". On the basis of the Schroedinger's article, I gave a (qualified) positive reply in post #3 (mind you, I was not trying to answer the question "shouldn't one...?"). I have not seen an objection to that in this thread.

Then there was a discussion of whether a wave can be represented by one real function. I expressed my point of view (it "can"; whether it "should", is a completely different question). I don't want to speculate whether there is a consensus now.

jtbell asked: "Can you replace the time-dependent Schrödinger Equation with a differential equation that does

**not**have an [itex]i[/itex] in it, has a real (not complex) wave function as its solution, and describes all the phenomena that the SE and its solutions can?" Again, using the results of the Schroedinger's article, I gave a qualified positive reply in post #20 and have not seen an objection to that in this thread. Again, I was not trying to answer the question "Should you...?"

Finally, several knowledgeable people asked just that: Why do you need to replace a complex wavefunction with a real one, although it is much more convenient to work in the complex domain? Let me give my reasons here (the details may be found in my article quoted in my post #7).

I believe this opens a way for a different interpretation of quantum mechanics. Namely, in the Klein-Gordon-Maxwell system of equations you can naturally exclude the wavefunction describing the matter and obtain independent evolution of the electromagnetic field. Therefore, in the Bohmian interpretation of quantum mechanics, the electromagnetic field, not the quantum potential, plays the role of the guiding field (unfortunately, the extension of this conclusion to the Dirac-Maxwell system in the article is not satisfactory, and I hope I'll be able to correct this in a few days). Is this worth the trouble? Is the Bohmian interpretation itself worth the trouble? I think so, but I suspect most people in this forum will disagree