Action Principle (Relativistic)

In the previous derivation, we used the continuity equation to show that the action's variation satisfies gauge invariance, corresponding to charge conservation. In the later development of physics, the process was reversed: when we require the action to satisfy gauge invariance, we derive the continuity equation and charge conservation. The variation process is the same, with the only difference being that we enforce: $${1 \over c^2} \delta \int G[\partial _\mu J^\mu ] d^4 x =0$$ to ensure that the variation result is unaffected by the gauge, leading to: $$\partial _\mu J^\mu =0$$ By studying this, we can understand why physicists often say that the action is powerful: writing down the action allows us to derive all experimentally observed equations. However, in the actual development of the action, experiments must first obtain some or all of the equations of motion, and physicists use physical reasoning to guess the form of the action, which then allows them to derive other deeper physics. Since classical electromagnetism is fully developed, all the equations have been obtained experimentally, so deriving the variation of the action to obtain \(F^{\mu \nu}\) or Maxwell's equations seems like reverse-engineering the problem from the answer. Nevertheless, historically, physicists derived charge conservation experimentally and then rewrote it in terms of the action, linking charge conservation and gauge invariance rigorously through Noether's theorem. This is similar to how Newtonian mechanics, through the principle of virtual work, leads to the Hamilton principle in action. Physicists can then use the action to derive Newtonian mechanics, but the action framework can also describe electromagnetism and even later quantum mechanics (Feynman's path integral). The action plays a crucial role in extending physics.

Originally written in Chinese by the author, these articles are translated into English to invite cross-language resonance.