EM Lagrangian density

The electromagnetic Lagrangian density is the Lorentz-invariant scalar L = −¼ F_{μν} F^{μν} − A_μ J^μ from which all of classical electromagnetism follows by the principle of least action. The first term is the kinetic contribution, ½ ε₀ (|E|² − c²|B|²) up to constants and sign conventions, encoding the field's own dynamics; the factor of −¼ (rather than −½) is set so that F^{μν} is antisymmetric and pairs of indices are summed without double-counting. The second term is the source coupling, which in 3-space components reads ρφ − A·J — the standard interaction energy of charge with potential and current with vector potential. Both terms are Lorentz scalars, so the integrated action S = ∫ L d⁴x is a Lorentz invariant.

Apply the Euler-Lagrange equation to A_ν as the dynamical field: the variation δS/δA_ν = 0 gives the inhomogeneous Maxwell equation ∂_μ F^{μν} = μ₀ J^ν, recovering Gauss's law (ν = 0) and Ampère–Maxwell (ν = 1, 2, 3). The other two Maxwell equations (Faraday's law and no-monopoles) follow automatically from F^{μν} = ∂^μ A^ν − ∂^ν A^μ via the Bianchi identity. The Lagrangian is invariant under the gauge transformation A_μ → A_μ + ∂_μ Λ; integrating the source-coupling term by parts and dropping a boundary term shows this invariance requires ∂_μ J^μ = 0, which is local charge conservation. This is Noether's theorem (1918) applied to the gauge symmetry: the symmetry of the Lagrangian generates the conservation law of the four-current. The EM Lagrangian is therefore the entire theory of classical electromagnetism in two terms — the kinetic part −¼ F_{μν} F^{μν} (built from the field tensor) and the source part −A_μ J^μ (linear in the four-potential and four-current). The deeper connection to Yang-Mills theory and the Standard Model lives in the same structure: replace the abelian U(1) gauge group of electromagnetism with a non-abelian SU(N) and the resulting Lagrangian becomes the kinetic term of the strong or weak nuclear force.