Noether’s Theorem: Why Symmetry Means Conservation

Physics students learn conservation laws as a list of facts: energy is conserved, momentum is conserved, angular momentum is conserved. Emmy Noether showed in 1918 that they are not separate facts at all. Each one is the shadow of a symmetry in the laws of physics. Our Emmy Noether sticker says it in four words: “Symmetry In. Conservation Out.” This post extends our explainer Who Really Discovered It? Physics Credit Myths, because Noether is a prime example of a scientist whose name deserves to be far better known.

Who was Emmy Noether?

The MacTutor History of Mathematics archive gives the essentials (MacTutor^[1]). Emmy Noether was born on 23 March 1882 in Erlangen, Germany, and died on 14 April 1935 in Bryn Mawr, Pennsylvania. In 1915 David Hilbert and Felix Klein invited her to Göttingen because Hilbert was working on ideas in the theory of relativity close to Einstein’s and needed an expert in invariant theory. The university would not at first give a woman the right to teach, so Hilbert advertised her courses under his own name; only in 1919, after a long battle, was she allowed to complete her habilitation. MacTutor notes she is best known to mathematicians for her work in abstract algebra.

She left Germany in 1933, after being dismissed by the Nazi government, and took a visiting professorship at Bryn Mawr College. She is one of the faces on our Physicist Portraits Vol. 1 sticker sheet and our Women in Physics sticker sheet, alongside other physicists whose names deserve to be better known.

What does Noether’s theorem say?

Her 1918 paper, “Invariant Variation Problems”, is available in English translation (Noether, transl. Tavel 1971^[2]; arXiv version^[3]). Its abstract states that the problems it treats admit a continuous group in Lie’s sense, which means symmetries that can be applied in arbitrarily small steps, like a shift or a rotation.

The Stanford Encyclopedia of Philosophy describes the paper as containing two theorems: the first leads to a connection between global symmetries and conservation laws, and the second to results about local symmetries (Stanford Encyclopedia of Philosophy^[4]). When physicists say “Noether’s theorem” they usually mean the first. In plain words:

If the laws of a system do not change under a continuous symmetry, then some quantity is conserved.

• Laws the same at every place (symmetry under shifts in space) → momentum is conserved.
• Laws the same at every moment (symmetry under shifts in time) → energy is conserved.
• Laws the same in every direction (symmetry under rotations) → angular momentum is conserved.

Energy conservation, which OpenStax treats as one of the foundations of mechanics (OpenStax University Physics, §8.3^[5]), turns out to rest on the fact that the laws of physics did not change between yesterday and today.

How does the theorem work? A worked example

Noether worked in the language of the Lagrangian, L, a single function that sums up a system. For a simple mechanical system, L = kinetic energy − potential energy, measured in joules. The motion obeys the Euler–Lagrange equation:

d/dt (∂L/∂q̇) = ∂L/∂q

where q is a coordinate (such as a position in metres), q̇ = dq/dt is its rate of change, t is time in seconds, and ∂ means a partial derivative, how L changes when only one variable changes.

Now suppose L does not depend on q at all, so shifting q changes nothing. That is a symmetry. Then the right-hand side is zero, and

∂L/∂q = 0 ⇒ p = ∂L/∂q̇ = constant

where p is called the conserved momentum belonging to q.

Our example: a puck of mass m = 0.2 kg sliding on a level, frictionless table, with x the horizontal position. Its Lagrangian is L = ½ m ẋ² (no potential energy changes as it slides sideways). L does not depend on x, so p = ∂L/∂ẋ = m ẋ stays constant. If the puck moves at 3 m/s, its momentum is 0.2 × 3 = 0.6 kg·m/s, and it stays 0.6 kg·m/s forever. The symmetry “the table is the same everywhere” has produced conservation of momentum.

The time version works the same way. If L does not depend explicitly on time, then the quantity E = q̇ (∂L/∂q̇) − L is conserved. For our puck, E = ½ m ẋ², the kinetic energy: 0.5 × 0.2 × 3² = 0.9 J.

What does Noether’s theorem not cover?

The first theorem needs continuous symmetries. Discrete symmetries, which come in one jump, are outside it. The most famous is parity, the mirror symmetry that swaps left and right. In 1957 Chien-Shiung Wu and her colleagues tested it in the beta decay of cobalt-60 nuclei in their paper “Experimental Test of Parity Conservation in Beta Decay” (Wu et al., 1957^[6]), and found that the weak force does not respect mirror symmetry. That year’s Nobel Prize in Physics went to Tsung-Dao Lee and Chen Ning Yang “for their penetrating investigation of the so-called parity laws” (Nobel Prize 1957^[7]). Wu, who did the experiment, was not among the laureates, another entry for our credit myths list. Our Chien-Shiung Wu sticker puts it as “She Broke Symmetry. And Records.” For the physics of beta decay itself, see Beta Decay Unveiled.

The second theorem, about local symmetries, is subtler. The Stanford Encyclopedia notes that it shows conservation laws can have a different status when a global symmetry is part of a larger local symmetry. That is one reason energy conservation in general relativity is still discussed carefully by physicists and philosophers.

Why does Noether’s theorem matter?

MacTutor’s biography sums up the physics result: to every infinitesimal transformation of the Lorentz group there corresponds a conservation theorem, a result often called Noether’s theorem. Our Noether’s Ledger poster shows one case: for a body in a central potential the Lagrangian does not depend on the angle, so angular momentum is conserved. Modern physics often runs the argument in reverse: physicists decide which symmetries a theory should have, and the conservation laws follow. In that sense, her 1918 paper sits underneath most of the physics on this blog.

Quick FAQ

What is Noether’s theorem in simple terms?

Every continuous symmetry of the laws of physics comes with a conserved quantity.

Which symmetry gives energy conservation?

The laws being the same at all times: symmetry under shifts in time.

When did Emmy Noether prove it?

In 1918, while working at Göttingen.

Does it apply to parity?

No. Parity is a discrete mirror symmetry, so the first theorem does not apply. Wu’s 1957 experiment showed the weak force breaks it.

References

1. MacTutor, Emmy Noether biography. https://mathshistory.st-andrews.ac.uk/Biographies/Noether_Emmy/
2. E. Noether (1918), “Invariant variation problems,” English transl. M. A. Tavel, Transport Theory and Statistical Physics 1, 186–207 (1971) (DOI 10.1080/00411457108231446; metadata verified via Crossref). https://doi.org/10.1080/00411457108231446
3. E. Noether (1918), “Invariant Variation Problems,” transl. M. A. Tavel, arXiv:physics/0503066. https://arxiv.org/abs/physics/0503066
4. K. Brading, E. Castellani & N. Teh, “Symmetry and Symmetry Breaking,” Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/symmetry-breaking/
5. W. Moebs, S. J. Ling & J. Sanny, University Physics Volume 1 (OpenStax, Rice University), §8.3 “Conservation of Energy”. https://openstax.org/books/university-physics-volume-1/pages/8-3-conservation-of-energy
6. C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes & R. P. Hudson (1957), “Experimental Test of Parity Conservation in Beta Decay,” Phys. Rev. 105, 1413–1415 (DOI 10.1103/PhysRev.105.1413; metadata verified via Crossref). https://doi.org/10.1103/PhysRev.105.1413
7. NobelPrize.org, The Nobel Prize in Physics 1957 (Lee, Yang). https://www.nobelprize.org/prizes/physics/1957/summary/

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.