[Physics] How to derive the Lagrangian for the Standard Model


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?


Best Answer

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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