Action Principle (Relativistic)

While this content mainly discusses classical mechanics, here we briefly touch on how \(A_\mu A^\mu\) corresponds to photon mass in quantum field theory. The Klein–Gordon equation describes spin-0 particles and is derived simply from relativity, based on the 4-momentum: $$P_\mu P^\mu =m^2 c^2$$ In quantum field theory, physical quantities are rewritten as operators acting on the wave function \(\phi\): $$\hat{P} _\mu \hat{P} ^\mu \phi =m^2 c^2 \phi $$ In quantum field theory, the momentum operator is: $$\hat{P} ^\mu =i\hbar \partial ^\mu $$ Substituting: $$(i\hbar \partial _\mu )(i\hbar \partial ^\mu )\phi =m^2 c^2 \phi $$ $$-\hbar ^2 \partial _\mu \partial ^\mu \phi =m^2 c^2 \phi $$ This yields the Klein–Gordon equation: $$(\hbar ^2 \partial _\mu \partial ^\mu +m^2 c^2 )\phi =0$$ If we set \(\hbar =c=1\) (Planck units or natural units): $$(□+m^2 )\phi =0$$ This is the common form. In quantum field theory, the Lagrangian density for a spin-1 massive particle is: $$L=-{1 \over 2} \partial _\mu A_\nu \partial ^\mu A^\nu +{1 \over 2} m^2 A_\nu A^\nu (with \hbar =c=1)$$ Varying \(\delta A^\nu\) satisfies: $$\partial ^\mu \left({\partial L \over \partial \left(\partial ^\mu A^\nu \right)} \right)-{\partial L\over \partial A^\nu }=0$$ This yields: $$-\partial ^\mu (\partial _\mu A_\nu )-m^2 A_\nu =0\to \partial ^\mu \partial _\mu A_\nu +m^2 A_\nu =0$$ $$(□+m^2 ) A_\nu =0$$ This is the Klein–Gordon equation describing a spin-1 massive particle's equation of motion. We can see that \(A_\nu A^\nu\) corresponds to the photon's mass. However, the correct Lagrangian density to describe a spin-1 massless particle is not \(\mathcal{L}=-{1 \over 2} \partial _\mu A_\nu \partial ^\mu A^\nu\), but rather: $$\mathcal{L}=-{1\over 4} F_{\mu \nu} F^{\mu \nu} $$ Further details involve technical aspects of eliminating two degrees of freedom in the 4-potential to correspond to the two polarization states of the photon, but this discussion goes beyond the scope of this document.

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