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The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century

Yvette Kosmann-Schwarzbach
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Sources and Studies in the History of Mathematics and Physical Sciences
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Michael Berg
, on

This book is positively brimming with the kind of arcana that appeals irresistibly to both historians of mathematics (and theoretical physics) and mathematicians proper who have a parochial interest in the lives and adventures of titans of the past. I used to claim that the measure of the complement of the latter set in the set of all mathematicians has measure zero, but I’m not so certain now. I guess that testifies to my growing status as a curmudgeon, but I think there is plenty of anecdotal evidence that mathematical people nowadays are not all that interested in anecdotes about their forebears — or even scholarly treatises on the historical moments and movements with which these mathematicians of the past were involved. Today’s young mathematicians appear far too busy with (inter)networking and committee quasi-work to sit down leisurely with a book like Constance Reid’s Hilbert and map themselves back in space-time to the Göttingen of the early twentieth century to spy on these grand adventures that shaped today’s mathematical world.

A book like Yvette Kosmann-Schwarzbach’s The Noether Theorems is a very welcome potential antidote to this myopia. It concerns itself with Emmy Noether, one of the most important players in early modern mathematics, with her work with (and for) David Hilbert and Felix Klein, the reigning mandarins of the whole discipline at the time, and mathematical Göttingen’s schizoid involvement with the physics of Einstein and (later) Heisenberg.

Here is the background: Physicists and mathematicians who are involved with relativity (for example) understand something very different by “Noether’s Theorems” than the assertions that are now more often called the first and second (and even third, if you recall Herstein’s Topics in Algebra) isomorphism theorem for such algebraic structures as groups, rings, or vector spaces. In physics, Emmy Noether’s contribution under this title has to do with her proof that there is a correspondence between conservation laws, invariants of a certain sorts, and symmetries.

David Hilbert brought Noether to Göttingen to establish this fact as sound mathematics, to be used in his and his fellows’ attempt to understand the burgeoning physics flowing from the fountain pens of Einstein and the quantum mechanics Wunderkinder (in the Institute next door) at a dizzying rate. (My apologies for playing fast and loose with historical events: Heisenberg’s matrix mechanics appeared on the scene when he was 22, i.e. in 1923; Noether came to Göttingen in 1915. However, Heisenberg was Max Born’s student, and Born’s thesis advisor was none other than Hilbert’s colleague Runge, and Born’s doctoral thesis concerned a theme of great interest to Klein. And earlier Born had served as one of Hilbert’s assistants … [Isn’t Wikipedia a wonderful thing?])

In any case, the grand objective of properly mathematizing hypermodern physics harmonized with a long-established dream of Felix Klein and (later) Richard Courant. The wanted to follow the tradition started by Göttingen’s scientific progenitor, Carl Friedrich Gauß, who, after all, revolutionized, e.g., number theory and geometry (of various flavors) while running Göttingen’s astronomical observatory: mathematics and physics are two lungs of the same body, as someone (who? Google and I failed to find out) has said. Hilbert certainly subscribed to this view with gusto, even going so far as to proclaim the marching orders that “physics is far too difficult for physicists” — see Reid’s book for the details.

Kosmann-Schwarzbach presents her objective in a cogent Preface starting off with the following tantalizing quotation from Emmy Noether (1918): “What follows thus depends upon a combination of the methods of the formal calculus of variations and Lie’s theory of groups.” Kosmann-Schwarzbach then states that “[t]his book is about a fundamental text containing two theorems and their converses which established the relation between symmetries and conservation laws for variational problems.” So we learn right off what machinery Noether brought into the game and that the framing of physical theories à la Hilbert and his school solidly presupposes the formalisms (e.g. Lagrange’s or Hamilton’s) for physical laws that lend themselves to the calculus of variations. This is a particularly telling point in that this is certainly how, for instance, quantum mechanics is presented, largely as a consequence of how it was crafted in Europe in the latter days of Hilbert.

The other, and earlier, physics revolution on the historical scene was Einstein’s work on general relativity, done largely at Berlin’s Kaiser Wilhelm Institut. As Kosmann-Schwarzbach conveys in the book, Einstein and the Göttingen mathematicians were in very close contact with each other, with Einstein coming to the Georg-August Universität to give colloquia on a number of occasions. The Noether text the book is concerned with is “Invariante Variationsprobleme” (Göttinger Nachrichten, 1918), whose very date of publication indicates that Hilbert’s focus at the time was not quantum mechanics (yet) but relativity: certainly his contributions to relativity are far more explicit and substantial than what he did in quantum mechanics, discounting the fact, of course, that quantum mechanics dramatically requires Hilbert space.

Kosmann-Schwarzbach points out in the first chapter of part II of her book that the Noether theorems fit into a matrix (so to speak) of work on general relativity by Felix Klein, a past master of Riemannian geometry, and Hilbert, and also their work on conservation of energy. Thus this future giant of abstract algebra (or Moderne Algebra, as van der Waerden’s seminal book has it) is revealed in the present context as a master not only of invariant theory (in the older sense: after all, she did her doctoral work with Gordan) but of Hilbert’s beloved calculus of variations and of Klein’s Lie theory, then of relatively recent vintage. It was evidently on the basis of these strengths that she was called to Göttingen.

Noether soon produced her 1918 paper, mentioned above. It is the all-but-exclusive focus of Kosmann-Schwarzbach’s book, which starts off with a translation of the article, “followed … by a detailed analysis of its inception, as well as an account of its reception in the scientific community.” The paper is placed in a proper historical and mathematical context by the author’s discussion of, e.g., “some developments in the theory of invariants in the nineteenth century which culminate in the definition and study of differential invariants” and “several works in mechanics dating from the beginning of the twentieth century in which Sophus Lie’s … methods … began to be applied …” And then: “We … summarize the contents of Noether’s article in modern language.”

Kosmann-Schwarzbach proceeds to review how Noether’s work was received by her contemporaries, specifically “the mathematicians Felix Klein, David Hilbert and Hermann Weyl, and the physicists Einstein and Wolfgang Pauli.” She then goes on to convey how the two main theorems of her paper, the first theorem on conservation laws, and the second one on differential identities, received “quite different diffusion,” and she closes this fine book with an “outline [of] the genuine generalizations of Noether’s results that began to appear after 1970, in the field of calculus of variations and in the theory of integrable systems.”

I have recently had occasion to look into these Noether Theorems in connection with my own work and welcome the present book enthusiastically: it will be of great service to me. More importantly, however, Kosmann-Schwarzbach’s The Noether Theorems is a wonderful scholarly contribution to the history of modern mathematics as it interacts with modern physics, and a very evocative account of the roles played by a number of major figures in early twentieth century German science. The present book is the 2010 translation of Kosmann-Schwarzbach’s 2004 French original: it’s a wonderful thing that her fine scholarship is now available to a larger audience in the new lingua franca of science.



Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Preface.-Acknowledgements.- I. "Invariant Variational Problems". II.- Invariance and Conservation Laws in the Twentieth Century. The Inception and Reception of the Noether Theorems. - 1. The Inception of the Noether Theorems. -2. The Noether Theorems.-3. The Noether Theorems as Seen by Contemporaries and by Historians of Science.-4. From Bessel-Hagen to Hill, 1921–1951.-5. The Reception of Noether's First Theorem after 1950.-6. The Reception of Noether's Second Theorem after 1950.-7. After 1970—Genuine Generalizations.-Conclusion.-Appendix I. Postcard from Noether to Klein.-Appendix II. Letter from Noether to Klein. -Appendix III. Letter from Klein to Pauli.-Appendix IV. Letter from Noether to Einstein.-Lectures at the Mathematical Society of Göttingen.-References.-Index.