Quantization of Klein-Gordon Field

[go quantum!]

I was reading the book written by Peskin about QFT when I found that the following equation:
$$(\frac{\partial}{\partial t^2}}+p^2+m^2)\phi(\vector{p},t)=0$$

has as solutions the solutions of an Harmonic Oscillator.

From what I know about harmonic oscillators, the equation describing them should have, for instance in 1-d, a second derivative and squared multiplication term with respect to the same variable, let's say x.

Thanks for your kind help-

 

[matonski]

In this case the variable is $\phi$:
$$\ddot \phi = -\left(p^2 + m^2 \right) \phi$$

 

[go quantum!]

So it's the classical HO's equation. Why then write $$\phi$$ in terms of ladder operators (the famous a's)?

 

[matonski]

It is also the quantum HO equation in the Heisenberg picture.