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Is there a way to derive the Lagrangian for the Standard Model just like one would for Einstein's field equations for instance? Also, how do you derive the Laganrigans for QCD and QED? Is it possible to do so from first principles?

In addition, what are these Lagrangians for these quantum field theories describing? An action?

Thanks,

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  • $\begingroup$ Comment to the question (v1): Most fundamental models are defined via their action/Lagrangian density (as opposed to their equations of motion). $\endgroup$ – Qmechanic Nov 13 '13 at 2:04
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    $\begingroup$ Get yourself a whiteboard and marker and in a minute done! :) I like many of the minute physics videos, but the way he writes down the Lagrangian and says that's all there is to it! It sounds SOOO trite. (In fairness, he does come back and explain a bit more in later video, but this sounds awfully corny). $\endgroup$ – WetSavannaAnimal Nov 13 '13 at 11:30
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The Lagrangian has many parts that are each guessed at according to symmetry principles, requirements that the theory be well behaved, and reproduce experimental results. It's not something you can do from first principles, because the first principles aren't known. But the aforementioned process took about a 75 years and many Nobel prizes and PhDs were awarded.

The Lagrangian is the thing you integrate over to get the action, but it is used to deduce Feynman rules and calculate scattering amplitudes and cross sections and the like.

It's not too hard to derive the Lagrangian for the electromagnetic field by staring at Maxwell's equations in the manifestly covariant form (i.e. in terms of the Maxwell Tensor $\partial_\alpha F^{\beta\,\alpha}=\mu_0 J^\beta$), and the geometric insights gained allowed people to generalize this action to non-Abelian theories like QCD.

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    $\begingroup$ I changed your "relativistic form" to something more explicit - this could have been confusing since (I'm sure you know this - it's just a matter of finding the right words) Maxwell's equations are always relativistic - just need to be in a form that shows this manifestly. $\endgroup$ – WetSavannaAnimal Nov 14 '13 at 0:48
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If, by "first principles" you mean without any observation or experimental input, I don't think we have the answer yet. We don't know why we have the particles/fields we do. But if we're given those particular particles/fields, then we know that consistency tells us what the Lagrangian must look like and leaves us little room for fixing coefficients of some terms.

In short: Once you know the matter content and the force-carrier content, consistency pretty much tells you what kinds of terms you can have in the lagrangian -- respective kinetic terms and possible interaction terms which respect the gauge symmetry of the force-carriers. Also, every interaction term in the Lagrangian will contribute in a calculable manner to every physical process; so, by calculating the probabilities of physical processes, one can reverse engineer the coefficients of those Lagrangian terms. That's pretty much all there is to it, in principle :-)

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