Sorry for a naive question.(adsbygoogle = window.adsbygoogle || []).push({});

In EM textbook and QM path integral textbook, the action and Lagrangian in electromagnetic interaction are

S = L dt = e(\phi – A v) dt ---equ.(1)

But in QFT textbook, the action and Lagrangian density are

S = L d^4x = A J d^4x ---equ.(2)

As I understand, in equ.(2), J = \rho U = \rho \gamma V

In which \rho is density, U is the 4-velocity=dx/d\tau, and V is the common velocity=dx/dt, \gamma is \sqrt (1-v^2/c^2).

So equ.(2) will have a factor of \gamma, but equ.(1) does not have.

So where is my mistake?

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# Question about Lagrangian in electromagnetic interaction

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