Noether's theorems

Noether's theorems: map
Reading the volume How to read this volume Notation Glossary Index of symmetries Figure gallery
Reading by level Start here: the plain-language primer The first theorem, stated and proved The second theorem, stated and proved The two theorems side by side The mathematics: Lie groups and Lie algebras The applications: General relativity and the energy question The history: Göttingen and the energy problem
Sources Sources Bibliography Chronology Further reading
Related Hamilton's principle On Physical Lines of Force

The symmetry-to-conservation map

Emmy Noether read her results to the Göttingen Mathematical Society on 23 July 1918. Three days later Felix Klein presented the paper, Invariante Variationsprobleme, to the Göttingen Scientific Society, and it appeared in that Society's Nachrichten for the year. Noether held no post at Göttingen when she wrote it.

The paper contains two theorems. They are not two statements of one result, and they do not do the same work. Most accounts of "Noether's theorem" give the first and pass over the second. The paper was written to prove the second.

Both theorems start from the same object. Physics writes down an integral, called the action, built from a function of some variables and their derivatives. Both theorems ask what follows when a continuous group of transformations leaves that integral unchanged. They differ in one thing only: what kind of group is permitted. The difference in what comes back is large enough that treating the two as one result loses the paper.

The first theorem

What it takes. A continuous group of transformations that leaves the action unchanged and depends on a fixed, finite number of constant parameters. Rotation of space is such a group, with three parameters. Shift in time is another, with one.

What it returns. One relation for each parameter. Each relation says that a particular combination of the expressions on the left of the equations of motion is a divergence — the mathematical form of something flowing without accumulating anywhere.

Where the result holds. The relations themselves are identities. They are true of the action whether or not anything obeys the equations of motion. Impose those equations and each relation becomes a conservation law: a quantity that does not change as the system evolves. The identity is unconditional; the conservation law is not.

The theorem runs backwards as well. Noether proved the converse: the relations imply the invariance they came from. That direction is what makes an observed conservation law evidence of a symmetry, rather than a coincidence that happens to be consistent with one.

The second theorem

What it takes. A continuous group that depends on arbitrary functions of the independent variables rather than on constants. The transformation is free to differ from place to place, and its derivatives up to some order may enter. The coordinate freedom of general relativity is such a group. So is the freedom in the potentials of electromagnetism.

What it returns. One identity for each arbitrary function, holding among the expressions on the left of the field equations and their derivatives.

Where the result holds. Everywhere, and unconditionally. Nothing has to obey a field equation for these to be true. They are not conservation laws and they do not become conservation laws. They are relations the field equations satisfy by construction, which is a stronger and stranger thing to have.

This theorem also has a converse, and Noether proved it.

The distinction

Set the two sets of slots beside each other and the difference reads off them.

The first theorem is fed constants and returns quantities that are conserved once the physics holds. The second is fed functions and returns relations that hold before any physics is imposed. The first tells you what a system keeps. The second tells you that some of its equations were never independent.

The consequence is blunt. A theory with a symmetry that varies from place to place has fewer independent field equations than it has fields. You cannot specify all of them freely, because the theory itself has already fixed some in terms of the rest.

What the theorems are about

Both theorems are statements about the action. Neither is a statement about the equations of motion.

This distinction is the most common error in accounts of the subject. A transformation that carries solutions to solutions need not leave the action unchanged, and if it does not, no conservation law follows from it. Symmetry of the equations and symmetry of the action are different properties, and only the second feeds either theorem.

Noether also worked at a generality that later treatments usually drop. Her statement allows several independent variables, derivatives of any order, and transformations that mix the dependent and independent variables together. Most textbook versions state a special case and leave the reader to suppose it is the theorem.

Why it matters

The first theorem's results are famous, and citing them proves little: energy, momentum and angular momentum were known to be conserved long before 1918. What the theorem added was the reason.

The second theorem's results were not known, and without it they remain unexplained.

The final section of the paper settles a claim of Hilbert's. Energy conservation in general relativity does not take the form it takes in every other field theory. Attempts to write a local energy density for the gravitational field produce quantities that depend on the choice of coordinates, and can be made to vanish at any chosen point. This was read for years as a defect of the formalism, something a better construction might repair.

Noether showed it is a prediction. A theory invariant under a group of transformations that vary from place to place, and containing the shifts among that group, has energy relations of exactly this degenerate kind — and has them if and only if that condition is met. The property is group-theoretic. Gravity is one instance of it, not the cause of it. No better construction is coming, because there is nothing to repair.

The same theorem accounts for the constraint structure of every gauge theory built since, and for the identity that Einstein's equations satisfy automatically. These are not applications discovered later and traced back. They are the theorem.

Knowing the boundary matters too. Not every conserved quantity comes from a symmetry of an action, and this volume marks the cases that do not.

How to read this volume

The volume has eight tracks. Every page carrying mathematics names a plain-language counterpart at the top, and each counterpart links back, so a reader can enter at either level and cross over at any point.

• Start with the plain-language primer if you want the argument with no mathematics at all.
• Göttingen and the energy problem gives the question the paper was written to settle, and the history track follows it to what the name means today.
• The first theorem is proved in full, with worked derivations of the standard conserved quantities.
• The second theorem is proved separately and at the same length, and the two are set side by side on one page.
• The mathematics track carries the machinery the theorems need, from Lie groups to the quantum statement and its failure.
• The applications track works the physics case by case, including one conserved quantity the theorems do not explain.
• Notation, glossary, chronology, an index of symmetries against their conserved quantities and a bibliography weighted toward primary sources make up the reference apparatus.

Provenance: this page carries no equations and no quotation. Noether's prose is paraphrased throughout and no English wording of hers is transcribed anywhere in this volume; her equations are reproduced in her own notation, from the German original. See Sources.