Different pictures in QFT I am studying the different pictures of quantum mechanics in QFT, but I have not understood one aspect of it.When we have a lagrangian, and we obtain the equations of the field, taking into account that we are considering that the field has time evolution, are we necesarily working in Heisenberg picture?.
 A: 1)  The equations of motion come before you have quantized the fields. You write down the Lagrangian(which is a classical object), find the equations of motion. Solve them, and THEN quantize your fields. Before this quantization procedure, your field is just a field-you haven't yet promoted it to an operator and therefore cannot talk about quantum mechanical pictures.
2) Quantization usually proceeds by promoting the fields and conjugate momenta to operators, defining commutation relations at a given time slice, mode expanding at THAT time slice, and then evolving in time in the usual way. The field, as quantized on the original time slice, lies in the Schrodinger picture, and is the same as the Heisenberg field at that time slice. Time evolution allows you to move between different time slices-this is the Heisenberg picture. This emphasizes why it is not useful to talk about 'pictures' at the level of the Lagrangian- time evolution at the classical level(as dictated by the Poisson Brackets-read 'commutator') has no meaningful analogue like the schrodinger picture. Because there is no notion of a 'state' where the time dependence can alternatively go into.
2) But, as a matter of convenience, we usually work in the Heisenberg picture when doing QFT-they(their correlation functions) are our objects of interest. We may introduce intermediate pictures(interaction picture), but its always the Heisenberg fields we are after.
