Second quantization and Klein Gordon equation

Question

This is what I understood from Klein Gordon equation :

We start from $$E^2=p^2+m^2.$$

We quantize it replacing $E \rightarrow \partial_t$, $p \rightarrow -ih\nabla$, $m \rightarrow m$

Thus, we get the Klein Gordon equation :

$$ (\Box +m^2) \Psi = 0 $$

But we can't interpret directly $\Psi$ as a wavefunction (it leads to some incoherences).

But we can interpret $\Psi$ as an operator acting on an hilbert space.

It is linked to what we call "second quantization" ? I mean, we first quantized the relation $E^2=p^2+m^2$, and we quantized the solution $\Psi$.

Answer 1

The way I yielded the Klein-Gordon equation is substituting the Einstein energy and momentum relationship into the Hamiltonian of the schrodinger equation since my first intention was modifying the Schrodinger equation a real wave dynamic equation which requires the derivative with respect to time and coordinates of phi to be second order. As for second quantization, I see the word "second" to mean the second type of quantization. What's more, quantization is just a guessing of the classical theory to expand it to more general use. Hope you find it helpful.