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Sorry, I have a strange way of writing the "curly L" which is used to represent the "Lagrangian density". You are on the right track for what "L" is but its a bit trickier. On mobile device right now so cant explain fully... may have to do another post :) So glad you are looking into it, makes my day.

See near 1 minute 20 seconds in this video for an example of what a Lagrangian density used by a field theorist looks like https://steemit.com/science/@nonlinearone/physics-out-loud-broken-symmetries-and-the-masses-of-gauge-bosons-by-peter-w-higgs-1964

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