Emmy NoetherSymmetry In. Conservation Out.

The sticker says Symmetry In. Conservation Out. That is Emmy Noether's theorem, drawn as a vending machine.

In 1918, in Göttingen, Noether proved that every continuous symmetry of a system's laws comes with a conserved quantity. If the laws don't change when you shift the clock, energy is conserved. If they don't change when you slide everything sideways, momentum is conserved. Turn the whole set-up through an angle and angular momentum is conserved. The three cans in the machine are exactly those three trades.

Before Noether, physicists found conservation laws one at a time and trusted them because experiments kept agreeing. After her, you go looking for the symmetry, and the conservation law falls out of the mathematics. The formulas above are the mechanical version: if nudging the coordinates leaves the Lagrangian unchanged, the quantity Q stays fixed as the system moves.

Her 1918 paper is far more general than the table on the machine. It covers fields as well as particles, and a second theorem deals with symmetries that can vary from place to place, the kind found in general relativity and in modern gauge theories. Charge conservation is now traced to a symmetry of the quantum phase. That is a later application of her result, not something she wrote down.

She did this work while the University of Göttingen would not let a woman habilitate, the qualification needed to lecture in her own name. For years her courses were advertised under David Hilbert's name. She was finally allowed to habilitate in 1919.

Press θ. Collect L.