Proof — Noether's theorem (symmetries and conservation laws)
Statement
For a physical system described by an action functional invariant under a continuous symmetry group (Lie group of transformations of fields and spacetime), Noether’s theorem produces conserved currents: each generator of the symmetry yields a continuity equation, hence conserved charges (energy from time translation, momentum from spatial translation, angular momentum from rotation, etc.). The result applies to classical field theory and underpins gauge symmetries and conserved quantities in modern physics; formalizable in analysis on manifolds (ZFC). Noether also founded the theory of Noetherian rings and modules in commutative algebra—a separate pillar of modern mathematics.
Formula
δS = 0, symmetry ⇒ ∂_μ j^μ = 0
Equation
Noether current j^μ from infinitesimal symmetry of L
Proof sketch
Noether, "Invariante Variationsprobleme," Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse (1918). Standard derivation varies the action under a local symmetry parameter; boundary terms yield the conservation law via the Euler–Lagrange equations.
Source
Noether, E. Invariante Variationsprobleme. Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1918, 235–257.
Domains
mathematics, physics
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- Theorem: Noether's theorem (symmetries and conservation laws)
- Proof: Noether's theorem (symmetries and conservation laws)