What is the exact reason why the solution of the classical Klein-Gordon equation is written as a mode expansion with a Lorentz invariant measure and after that the coefficients are promoted to operators? Why do we want the measure to be Lorentz invariant? In particular, what do we want to be Lorentz invariant and what are the aforesaid Lorentz transformations acting on (4-coordinates,4-momenta,the scalar field itself,...)?