Bessel-Hagen and the first uses

Three years after Noether's paper, a young Göttingen mathematician used it, and in using it added something she had not written. The addition is now attributed to her more often than to him.

Bessel-Hagen

Erich Bessel-Hagen was born in 1898 and died in 1946. He came to Göttingen and fell under Klein's influence to an unusual degree: he lived in Klein's house and served as his private assistant from 1921 to 1923, and later helped prepare Klein's lectures on nineteenth-century mathematics for publication.

In September 1921 the Mathematische Annalen published his Über die Erhaltungssätze der Elektrodynamik at pages 258 to 276 of volume 84. It is the first substantial application of Noether's theorems, and his citations to her are precise — to particular pages and particular sections of her paper, which he abbreviates simply by her name.

What the paper does

Bessel-Hagen applies both theorems to Maxwell's equations and asks what the full symmetry group of electrodynamics yields.

The answer is larger than the Lorentz group. Electrodynamics is invariant under the fifteen-parameter conformal group, a fact established by Bateman and Cunningham a decade earlier and known then as the group of spherical wave transformations. Feeding that group into the first theorem produces conserved quantities that had not been written down, beyond the familiar energy, momentum and angular momentum.

The addition: invariance up to a divergence

Noether's theorems as she stated them require the action integral to be invariant. Bessel-Hagen observed that this is stronger than necessary. It is enough that the integrand change by a divergence, since a divergence integrates to a boundary term and leaves the variational problem alone.

This is now called divergence invariance, or quasi-invariance, and the symmetries it admits are called divergence symmetries. Bessel-Hagen also identified why the weaker condition has to be defined carefully. Strict invariance under an infinitesimal transformation carries over automatically to the one-parameter group that transformation generates. Invariance up to a divergence does not. The definition therefore has to be attached to the infinitesimal transformations, not to the finite group — a point that is routinely skipped in modern statements.

The standard illustration is the Galilean boost. The Lagrangian of a free particle is not invariant under a boost; it changes by a total time derivative. Under strict invariance the boost gives nothing. Under divergence invariance it gives a conserved quantity, and that quantity is the uniform motion of the centre of mass.

This is a correction to the usual account, and this volume marks it everywhere it arises. Divergence symmetries are commonly presented as part of Noether's theorem. They are not in her paper. She works with invariance of the integral, relation (1) of her first section. She does observe that an integral differing from another by a divergence admits the same group with the same variation, which is adjacent to Bessel-Hagen's point and is not the same as it. The generalisation is his.

The first uses, and their scale

There were few. A study of Bessel-Hagen's paper reports that it drew its first proper citation in 1934 and only a handful more over the following two decades.

That figure describes his paper. It is not evidence about the reception of Noether's, and this volume does not treat it as such. What the reception of Noether's paper actually was over the same period is unresolved.

Provenance: Bessel-Hagen's paper was not reached in full text. It is indexed and digitised, but no copy was retrievable in the course of preparing this volume, and the only English translation is under copyright. Nothing above is quoted. The account of the paper's contents and of its citation history is taken from the secondary literature. Closing this gap requires the Mathematische Annalen volume 84 scan. See Sources.