How to solve the Klein Gordon Complex Field?

[sadness]

In what sense is it not "decent"? Note, I forgot some factors of 1/2, went and fixed them.

The first expression you have for the Hamiltonian is correct (with both $\pi \partial_t \phi$ and $\pi^* \partial_t \phi^*$). Try working the problem in detail to see why.

I mean, you can not write the derivative to z* without the help of splitting into x and y. I see your point now. Thanks for pointing out my mistake.

 

[Ben Niehoff]

Or alternatively, you cannot write a derivative w.r.t. x without splitting into z and z*. It's a matter of using different coordinate systems to represent the same thing.

$$\frac{\partial}{\partial x} = \frac{\partial z}{\partial x} \frac{\partial}{\partial z} + \frac{\partial \bar z}{\partial x} \frac{\partial}{\partial \bar z}$$