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Book Review: Emmy Noether's Wonderful Theorem

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Noether’s theorem is one of the towering achievements of early twentieth century physics, in some ways even more illuminating than special relativity or the uncertainty principle. In shorthand it sounds so simple: every symmetry produces a conserved quantity, is one way to put it. The math behind the theorem is anything but simple, and it is those details which this book explores in elaborate and almost loving attention.

I’d like to claim that I understood a full 5% of this book, and the theorem which is its subject, but let’s be honest: it’s much, much less than that. Still I was able to get some basic understanding for the basic building blocks of Noether’s theorem, and to appreciate just how elegant it is. For that reason I’m glad I read this book.

It’s styled as a textbook aimed at introductory-level physics students - I imagine someone fresh out of a semester or two of 100-level classes. I’ve never been one, so I can only speculate, but it seems like the text is operating at a level quite a bit more elevated with that. It probes deeply into at least three or four major branches of modern physics, without much of a pause as it hops back and forth. Perhaps that is more an artifact of the broad reach of Noether’s theorem than anything else.

Relatedly, there is something of a perverse fascination with the algebraic manipulations that lead one into this or that application of Noether’s theorem. There is rather less focus on what you might call the poetry of the theorem - although there is the occasional flourish or elaboration here or there. I suspect that the author appreciates this side of the theorem but is simply too eager to solve one more differential equation. Admittedly - it may not be the place of this work to take up the subject of poetry in physics. In any case I tuned out the bulk of the algebra, trying to get a handle on the larger picture instead. It was difficult going indeed, but of course, no one has ever confused me for a talented student of physics.

I will say that the questions and exercises which accompany each chapter are a delight unto themselves; it’s possible to learn just as much in these sections as from the main body of the chapter. I would strongly recommend them to any reader, even (especially) those who have no intention of completing the exercises.

Any theorem which has done as much to crystalize our understanding of the physical world as Emmy Noether’s deserves a great deal of study and appreciation. For all that it is a challenging, difficult book, I really applaud the degree to which this work has really tried to plumb the depths of this theorem.