Noether's theorem

Noether's theorem, proved by Emmy Noether in 1918, is the most beautiful theorem in classical physics. It says: if the Lagrangian (or, more generally, the action) of a physical system is invariant under a continuous one-parameter group of transformations, then there is a conserved quantity — a Noether charge — that can be read directly off the generator of the transformation.

The three classical instances are: time-translation symmetry → energy conservation; space-translation symmetry → momentum conservation; rotational symmetry → angular-momentum conservation. The theorem explains why exactly these three conservation laws hold: the laws of classical physics are invariant under translations in time, translations in space, and rotations of space, and so — by Noether — they admit exactly these three conserved charges and no others.

The theorem extends beyond mechanics. In quantum field theory, conservation of electric charge comes from a global phase symmetry of wave functions; conservation of colour charge from SU(3) gauge symmetry; conservation of lepton and baryon numbers from further global symmetries. Noether herself stated a second theorem, less famous but equally deep, that handles gauge (local) symmetries differently — this is the theorem Einstein consulted her about while completing general relativity. The forward and backward directions of the theorem turn physics into a search for symmetries: find a symmetry, find a conserved quantity; lose a conserved quantity, blame a broken symmetry.