When I read Mukanov's book in "Quantum effects in gravity", I found the following interesting remark at p. 57.
Remark: Lorentz invariance.To quantize a field theory, we use the Hamiltonian formalism which explicitly separates the time coordinate $t$ from the spatial coordinate $x$. However, if the classical theory is relativistic (Lorentz-invariant), the resulting quantum theory is also relativistic.
This is a statement. I would like to ask: is there any proof, or any simple way to see that it is true? Namely, why by a recipe that is not relativistically invariant (at least it looks so), one can be sure that all the consequences are relativistically variant.
I can't see why it should be true.
PS: This question hits my scar. When I asked exactly that question in my university, I could not get any answer, neither from the lecturer, nor from any book I found. I felt so bad that I even abandoned the quantum field theory course after that.