Why are QM wave functions complex?

[psmt]

It is not quite clear why I would need "the field itself"

As i understand it, you haven't shown that the other field components are zero, you've just eliminated them from the Dirac equation - they will still contribute to the current etc (i'd have thought... can't see why not).

A couple of other things: firstly, your paper appears to be on classical field theory... can any conclusions drawn from that paper necessarily be carried over to the quantum theory? Secondly, when vanhees71 says you need a complex field to have charged quanta, he means the field must have the the freedom to be complex, even if it happens to have been made real by a gauge transformation. Although it doesn't appear to realized in Nature, there's nothing inconsistent about a free field theory with charged particles, and the definition of charge as the Noether charge of a global phase symmetry is still good - no gauge fields needed.

 

[vanhees71]

In answer to #35 (so that's clear what I'm referring too). Of course, you can translate everything formulated in terms of complex fields to coupled equations of its real and imaginary parts, giving real equations. That's not very convenient, but if you insist, you can do it.

Further, it's not clear to me in which context Schrödinger made this remark. I'm sure that he is right for his example, but it would be good to have a reference to check, what he was referring to, before discussing this issue.

 

[akhmeteli]

As i understand it, you haven't shown that the other field components are zero, you've just eliminated them from the Dirac equation - they will still contribute to the current etc (i'd have thought... can't see why not).

Of course, the other field components are not zero, but you can just forget about them: the theory can be completely rewritten in terms of just one real function (the former component of the Dirac spinor). We may declare that this "lonely" real function is in fact our wave function, and this new "wave function" is everything we need to describe all phenomena described by the Dirac equation.

A couple of other things: firstly, your paper appears to be on classical field theory... can any conclusions drawn from that paper necessarily be carried over to the quantum theory?

Yes. Please see the following articles: http://akhmeteli.org/akh-prepr-ws-ijqi2.pdf (published in the International Journal of Quantum Information; this paper is discussed in the following thread: https://www.physicsforums.com/showpost.php?p=3413745&postcount=741) and http://arxiv.org/pdf/1108.1588.pdf (accepted for publication in the European Physical Journal C, except for the last two paragraphs in Conclusion, which were written later). If you are interested, it's better to start with the second article, as it largely supersedes and improves the former. However, the part of the second article related to spinor electrodynamics does use complex numbers, but I am not sure it is strictly necessary.

Secondly, when vanhees71 says you need a complex field to have charged quanta, he means the field must have the the freedom to be complex, even if it happens to have been made real by a gauge transformation. Although it doesn't appear to realized in Nature, there's nothing inconsistent about a free field theory with charged particles, and the definition of charge as the Noether charge of a global phase symmetry is still good - no gauge fields needed.

Again, this does not mean that wave function must be complex.

 

[akhmeteli]

In answer to #35 (so that's clear what I'm referring too).

Thank you :-)

Of course, you can translate everything formulated in terms of complex fields to coupled equations of its real and imaginary parts, giving real equations.

This is quite obvious, but this is not what I had in mind. There is just one real wave function after the gauge transform.

Further, it's not clear to me in which context Schrödinger made this remark. I'm sure that he is right for his example, but it would be good to have a reference to check, what he was referring to, before discussing this issue.

I gave the reference in post 4 in this thread.