are lagrangians based on physical observation or mathematical reasoning?
In the same question, "Why do we use THIS Lagrangian, or THAT, and HOW did we derive it?", I have been told that "it just suits the corresponding problem, it just works out!".
I guess that the first one who came up with a working Lagrangian had tried a lot, before he got there. One may be able to recognize several proper terms in a Lagrangian, such as the kinetic terms, or mass terms, or interactions between the sources and the fields, which have imply physical meaning. But I don't think there is a "standard" way to create a working Lagrangian, or a step by step proof of the procedure to get there!
in classical mechanics L = T-V, kinetic energy - potential energy of the system.
in QM, and QFT we just push this idea further
I am not sure what you are driving at when you say "it just works out".
It is not a difficult matter to prove the equivalance of newtonian and lagrangian dynamics. You can find this proof in just about all undergraduate classical mechanics texts, see Marion. Given this proof, it doesn't just work out for some mysterious reason, rather it is simply a reformulation of mechanics in terms of energy instead of forces. If anything about the langrangian is mysterious, I would consider it to be the fact that the langrangian formulation implies that nature always minimizes the time integral of action. Which is a topic that has been beat to death on this forum with no real consensus.
When I answered the original question posted, the thread was in the "Quantum Mechanics" section, not the "Classical Mechanics" that has been moved now.
Anyway, in C.M., Lagrangian is always L= T - V, and one can derive it easily. In QFT, one cannot use a similar path, since, as far as I know, it doesn't exist.
The fact that the Lagrangian minimizes the time integral action does not push one further to find the correct corresponding Lagrangian to his problem.
Say, i.e., we need to find the Lagrangian from which Klein-Gordon equation arises. We cannot "prove" the satisfying Lagrangian by means of a "law", such as T-V, as in C.M., but only by means of minimizing the action time integral. Which means that the procedure requires a lot of effort, trying and fantasy to get the proper result!
i am curious to know how can you couple the field A with the dirac field with the use of a coupling constant. I am not quite sure how that works out?
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