Noether's Theorem for Fields

Core Idea

Noether's theorem for fields states that every continuous symmetry of the Lagrangian density yields a conserved current j^mu with partial_mu j^mu = 0. Spacetime translations give energy-momentum conservation; internal symmetries give conserved charges like electric charge.

Explainer

You already know Noether's theorem from classical mechanics: if the Lagrangian is invariant under time translations, energy is conserved; under spatial translations, momentum is conserved; under rotations, angular momentum is conserved. The field-theory version promotes these conserved quantities from global scalars to conserved currents. A conserved current j^mu satisfies the continuity equation partial_mu j^mu = 0, which says that the charge density j⁰ can only change at a point if there is a flux of current through its boundary. The total charge Q = integral j⁰ d^3x is constant in time, provided the current vanishes at spatial infinity.

For spacetime translations, Noether's theorem produces the energy-momentum tensor T^{mu nu}. The component T⁰⁰ is the energy density, T⁰ⁱ is the momentum density, and the conservation law partial_mu T^{mu nu} = 0 encodes conservation of both energy and momentum. For internal symmetries -- transformations that act on the field values rather than on spacetime coordinates -- the theorem gives conserved currents associated with the symmetry group. The most important example is the global U(1) symmetry phi -> e^{i alpha} phi of a complex field, which yields a conserved current whose charge is electric charge (or more generally, particle number minus antiparticle number).

The derivation follows the same logic as in particle mechanics but with the field-theoretic Euler-Lagrange equation. If a continuous transformation phi -> phi + epsilon delta phi leaves the Lagrangian density invariant (or changes it by a total divergence), then the current j^mu = (partial L / partial (partial_mu phi)) delta phi is conserved on-shell (when the equations of motion are satisfied). The energy-momentum tensor arises from the special case where the transformation is a spacetime translation: delta phi = partial_nu phi, and the resulting T^{mu nu} is a rank-2 tensor rather than a four-vector.

What makes Noether's theorem indispensable in quantum field theory is that it links the symmetries you impose on the Lagrangian to the conservation laws that constrain scattering processes. Every Feynman diagram must conserve all Noether charges at every vertex. Furthermore, the theorem survives quantization in most cases, but with a crucial caveat: some classical symmetries are anomalous, meaning they are broken by quantum effects. The study of anomalies -- which classical symmetries survive quantization and which do not -- is one of the most important topics in modern quantum field theory.