When Canonically Quantising the Klein-Gordon Field you usually start with the Klein-Gordon Equation, from which you can guess a corresponding Lagrangian Density. Then utilising this information along with assumed Commutation relations for the field and it’s conjugate momentum you “Quantise” the field. From what I’ve read, upgrading the Klein Gordon Field (which is a scalar field) to an operator (or I guess “operator-valued distribution” or so I’m told) is what “Quantises” what I assume to be the classical Klein Gordon Equation.
But it’s also trivially easy to prove that the Klein Gordon Equation can be derived from the relativistic Energy-Momentum Relation by substituting in Energy and Momentum Operators. It would seem then that the Klein Gordon Equation is already “Quantum”. This seems to be a contradiction to me (and I’m guessing may be the historical motivation behind the term “Second Quantisation”?).
If anybody can answer, I’d like to know what the orthodox viewpoint on this. Is the Klein Gordon Equation already “Quantum” or does it only become Quantum when you go through the process of Canonical Quantization? Is it both somehow? If the Klein Gordon Equation is really a classical field then why can I derive it by substituting Quantum Operators into a classical expression from Special Relativity?