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Why do we never use the SM equations of motions? We know the Lagrangian and therefore could write down the equations of motions for all fields. In principle we could solve the equations numerically. Why do we never do that?

Would a leading order calculation for example for electron positron scattering be equivalent to solving the equations of motions when considering initial conditions where an electron and positron move towards each other?

I guess also a reason why we do not solve the equations of motions is, that quantum effects are not included, right?

Qmechanic
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    See https://physics.stackexchange.com/q/504526/ and https://physics.stackexchange.com/q/91728/ – Eletie Feb 17 '21 at 22:13
  • Thanks! The first link actually contained useful information! –  Feb 17 '21 at 22:34
  • Ok. I looked at the Schwinger-Dyson equations. I think this is not what I asked for. Let me make it more clear. You have a Lagrangian and derive the equations of motions from it. As I understand these are the classical equations of motions. How can I get the equations of motions which include the quantum effects, which arise due to quantization of the theory? –  Feb 18 '21 at 10:51
  • The path integral approach is the QM version of the action principle essentially, and the Schwinger–Dyson equations specifically are the quantum analogues of the EL equations. I'm not sure what more you mean? – Eletie Feb 18 '21 at 12:20
  • The equations of motion including the quantum effects are the equations of motion of the quantum effective SM lagrangian, if and when you computed that. – Cosmas Zachos Apr 05 '21 at 15:27

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