Ask Ethan: How do symmetries lead to conservation laws? | by Ethan Siegel | Starts With A Bang! | Apr, 2024


Emmy Noether, the person who proved Noether’s theorem, which connects symmetries and invariances of a theory with an associated conserved quantity. At right, the illustration of conservation of linear momentum, a consequence of spatial-translation invariance, is shown. (Credit: Mathematical Association of America (L)/R Nave (R))

First derived by Emmy Noether, for every symmetry a theory possesses, there’s an associated conserved quantity. Here’s the profound link.

In this Universe, there are certain physical quantities that are — quite importantly — always conserved. Laws like the “conservation of energy,” “conservation of momentum,” and “conservation of angular momentum” are cornerstones of theories ranging from Newtonian mechanics all the way to quantum electrodynamics and beyond, and apply equally to planets in the Solar System as they do to collisions taking place inside the world’s most powerful particle accelerators. But these aren’t just empirical laws that we observe to be true; they’re an inevitable consequence of certain symmetries that these theories exhibit: an unavoidable connection required by Noether’s Theorem.

What’s the connection? That’s the topic of this week’s question from Kaleberg, who enjoyed this prior article on symmetries in physics and inquired with this follow-up:

“Excellent. Now do an article on Noether’s Theorem and show how symmetry leads to conservation laws. Doesn’t spatial invariance imply conservation of momentum, and doesn’t time invariance imply conservation of energy?”

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