古典物理的牛頓 interpretation 是以力、加速度等微分和 local 特性為主。
Lagragian interpretation 相反的是以 action, path integral, 等 global 特性為主。我在學 Lagragian 時認知的主要優點是操作面的 (i) scaler instead of vector; (ii) generalized coordinates is a lot easy to operate.
其實 Lagragian 更大的影響是物理哲學，指向更 simple、更 fundmental 、以及更有美感的 interpretation.
Simple: scaler vs. vector; generalized coordinates
Fundmental : Lagragian interprestation 是基於 Least action principle。如同 Fermat 的幾何光學 interpreation. 就我而言，似乎是大自然更基本的 principle. 這在後來的電磁學，量子力學，量子場論都成立。
美感 : 這是最有趣的一點。什麼是更有美感? 當然更 simple or fundmental 也可說是美感。但我 (以及大多數物理學家)認為更有美感顯現在對稱性上。對稱和 least action principle forever change 物理學家的認知。對稱也許更為深遠。
Action 的對稱性首先表現在守恆律上。這是由 Emmy Noether 首先 lay a firm ground:
Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action.
Einstein 是第一個認知而且 explore 對稱性在
(i) Maxwell EM theory is based on
gauge 對稱性 (local symmetry): conservation of charge (because it’s fermion?)
Lorentz symmetry:
(ii) Einstein special relativity is based on Lorentz space-time symmetry
(iii) Einstein general relativity is based on .. symmetry
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Progress in Quantum Mechanics
(i) QED : First quantum field theory based on U(1) gauge symmetry, photon is the boson
Yang-Mills theorem explore gauge symmetry (generalized from Maxwell equation)
It’s a non-Abelian symmetry group --> predict similar gauge bosons (massless) like QED to convey force/information –> No such particle, especially for short range weak/strong force!!
Broken symmetry –> mass in boson (Higgs boson) –> fill the hole in Y-M theory.
Does this mean conservation is also breaking? (no physic law is still symmetry; it’s initial state is asymmetry?)
