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It is known that in QFT the Euler-Lagrange equations are used to obtain the equations of the quantum fields. Nevertheless, from the path integral's point of view (where you integrate over all $\it{paths}$/possible field configurations, even if they don't satisfy Euler-Lagrange) the use of this equations makes only sense if you are looking for the classical field, since this one satisfies the least action principle.

So, why do we use these equations to obtain the ones for the quantum field?

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marked as duplicate by Qmechanic lagrangian-formalism Jan 3 at 0:30

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    $\begingroup$ "It is known that in QFT the Euler-Lagrange equations are used to obtain the equations of the quantum fields." - No such thing is known to me, so I do not understand what you are trying to ask. What do you mean by "equations of the quantum fields"? The quantum analogue of the classical E-L equations are the Schwinger-Dyson equations $\endgroup$ – ACuriousMind Jan 2 at 21:22
  • $\begingroup$ The Euler-Lagrange equations are classical equations for classical fields. QFT evolves the fields in the same way QM evolves operators: they either do evolve (Heisenberg), don't (Schroedinger) or do but weirdly (interaction). $\endgroup$ – jacob1729 Jan 2 at 21:23
  • $\begingroup$ @ACuriousMind For instance, by E-L eqs. applied to the Lagrangian of a scalar field, you can obtain the Klein-Gordon eq., the eq. of the field. $\endgroup$ – Vicky Jan 2 at 21:38
  • $\begingroup$ ...and the problem with that is? You seem to want to say that we cannot use the KG equation because the fields the path integral integrates over do not obey it in general. That is correct - you cannot claim that the integration variable of the path integral obeys the KG equation. But I very much doubt that anyone else claims that. Please give an explicit example of a case where the KG equation is used and where you think it is not justified. $\endgroup$ – ACuriousMind Jan 2 at 21:44
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    $\begingroup$ Possible duplicate: physics.stackexchange.com/q/234774/50583, see also physics.stackexchange.com/q/83113/50583 for caveats. $\endgroup$ – ACuriousMind Jan 2 at 23:44