# Complex numbers from unitarity from information conservation?

1. Jun 3, 2012

### friend

I doing some reading on why the wave-function is complex. From what I can tell, it's due to its evolution by unitary operators. But unitary operators seem to have something to do with information conservation. So I wonder if these idea have been developed somewhere in a concise fashion that prove that complex wave-functions come from unitary operators which in turn come from information conservation, or something like that. Thanks.

2. Jun 3, 2012

### akhmeteli

Your assumption (that the wave function is complex) is widely accepted, but not quite obvious. While the following is not well-known, at least in some general and important cases the wave function can be made real - https://www.physicsforums.com/showpost.php?p=3799168&postcount=9 . In those cases, "evolution by unitary operators" peacefully coexists with a real wave function.

3. Jun 3, 2012

### friend

If examples of non-complex forms can be derived from complex forms, BUT complex forms cannot be derived from non-complex forms, then it sounds like the complex formulation is necessary. Thus the question remains.

4. Jun 3, 2012

### akhmeteli

First, it does not look obvious that "complex forms cannot be derived from non-complex forms", as far as quantum theory is concerned.

Second, as "evolution by unitary operators" does not exclude real wave functions, this evolution probably cannot be the reason "why the wave-function is complex", so the question seems to need at least some reformulation.

5. Jun 4, 2012

### friend

Now I'm thinking that the complex nature of QM is responsible for the particle/wave duality. For without the complex i in the lagrangian you can get the kinetic energy of a particle. But when you put an i in front of it, then since it's in the exponent of the path integral, you get a wave nature and thus superpositions. I wonder what other quantum mechanical effects are due to the complex numbers.