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- Is it mathematically possible to have a working QFT Lagrangian based on operators without canonical quantization? (and without path integrals)

I've learned that in canonical quantization you take a Lagrangian, transform to a Hamiltonian and then "put the hat on" the fields (make them an operator). Then you can derive the equations of motion of the Hamiltonian.

What is the reason that you cannot already put hats in the QFT Lagrangian? Therefore write the Lagrangian with operators and go straight to Euler-Lagrange equations without any additional steps like canonical quantization. You would have to add an operator which transforms the operator expression into a scalar for the Lagrangian. So it it possible to put all that into the Lagrangian alone and do straightforward Euler-Lagrange equations from that? (without introducing additional machinery like path-integrals!)

What is the reason that you cannot already put hats in the QFT Lagrangian? Therefore write the Lagrangian with operators and go straight to Euler-Lagrange equations without any additional steps like canonical quantization. You would have to add an operator which transforms the operator expression into a scalar for the Lagrangian. So it it possible to put all that into the Lagrangian alone and do straightforward Euler-Lagrange equations from that? (without introducing additional machinery like path-integrals!)