Chapter 5: Lagrangian Formulation
The Lagrangian formulation provides the deepest understanding of electromagnetism, connecting to symmetries via Noether's theorem and paving the way to quantum field theory.
Electromagnetic Lagrangian
\( \mathcal{L} = -\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu} - J^\mu A_\mu \)
Field kinetic term + interaction term
The Euler-Lagrange equations for Aμ give Maxwell's inhomogeneous equations. The action S = ∫ℒ d⁴x is Lorentz invariant.
Energy-Momentum Tensor
The stress-energy tensor Tμν encodes energy density, momentum density, and stress:
\( T^{\mu\nu} = \frac{1}{\mu_0}\left( F^{\mu\alpha}F^\nu_{\;\alpha} - \frac{1}{4}\eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta} \right) \)
Conservation: \( \partial_\mu T^{\mu\nu} = -F^{\nu\alpha}J_\alpha \) (energy-momentum transfer to charges).
Symmetries and Conservation Laws
Gauge Symmetry → Charge Conservation
Invariance under Aμ → Aμ + ∂μχ implies ∂μJμ = 0.
Translation Symmetry → Energy-Momentum Conservation
Spacetime translation invariance gives ∂μTμν = 0 (in absence of sources).