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I don't know if this will help you, but I'll try to give you a possible way out. It's in line with Feynman's famous "shut up and calculate" advice on QM. Field theory and QFT in themselves are mathematical frameworks, not physics; we can think of it as a piece of applied mathematics. It becomes physics when we plug in a specific Lagrangian, and then apply the framework, eg. we draw Feynman diagrams, regularize, renormalize, all that jazz, and we get eg. a cross-section out of it. So, when you ask "why should L be minimized?", the answer is, because this whole construction works. If you follow the complicated (and convoluted) playbook of QFT, you will be able to calculate physical quantities (like cross-sections, or the fine structure constant), and then when you do an experiment, you find that the numbers match. This doesn't work for all Lagrangians. Most L(x, p, t) or L(phi, phi', ..) functions we come up with don't correspond to physics, and the numbers the framework emits do not line up with experiments. You may be dissatisfied by this, this is a black-box picture. Over time physicists have developed a lot of intuition for parts of the theory, and come up with heuristic explanations what the parts mean. This is also how the whole thing was constructed, by analogy from Lagrangians in classical mechanics, where things can be reduced to Newton's equation of motion, which we know works. But in the end, the reason eg. L should be minimized, is because that's how nature is and that's what works, with specific Ls. |