# How a Lagrangian can represent a theory?

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• hugo_faurand
In summary, the Lagrangian is used to calculate the particle behavior in quantum field theory. It is determined by the symmetries and order of equations of motion of the theory.f

#### hugo_faurand

Hello everyone !

I recently read an article about Standard Model's Lagrangian. And it made me remember another article (that I read a long time ago) which said that a theory's Lagrangian "represent" the theory.

But How ?

Maybe I didn't get the sense of "represent".
Also I know that there is something called the Lagrangian density so maybe it refers to this.

Regards

The lagrangian is determined by the symmetries and the order of equations of motion you want. You could call this "representing".

The lagrangian is determined by the symmetries and the order of equations of motion you want. You could call this "representing".
Could you develop what is equatuond of motion's symmetries ?

The Lagrangian is used in the path integral formulation of quantum field theory. Basically, there is a direct relation between terms in the Lagrangian and particle behavior. Google up Feynman diagrams.

https://en.wikipedia.org/wiki/Feynman_diagramhttps://simple.wikipedia.org/wiki/Feynman_diagram
Each term in the Lagrangian corresponds to something represented by a Feynman diagram, which represents something happening in the particle theory. Some of the symmetries are represented explicitly. Relativistic in-variance is built in by using the mu-nu 4-derivatives, and building things up out of fields that are relativistic vectors, tensors, etc. Gauge symmetry is built up from the symmetries of the gauge field, usually called A-sub-mu or some such.

To get into this more you will need a lot of math. You could start with learning the Hamiltonian-Lagrange formulation of classical mechanics. The intro to that was a 4 month course for me. After that you could step over to the Hamiltonian-Lagrange formulation of quantum mechanics, then of quantum field theory. That was another two 4-month classes for me.