Emmy Noether's Wonderful Theorem

by Dwight E. Neuenschwander

In the judgment of the most competent living mathematicians, Fr�ulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.--Albert Einstein

The year was 1915, and the young mathematician Emmy Noether had just settled into G�ttingen University when Albert Einstein visited to lecture on his nearly finished general theory of relativity. Two leading mathematicians of the day, David Hilbert and Felix Klein, dug into the new theory with gusto, but had difficulty reconciling it with what was known about the conservation of energy. Knowing of her expertise in invariance theory, they requested Noether's help. To solve the problem, she developed a novel theorem, applicable across all of physics, which relates conservation laws to continuous symmetries--one of the most important pieces of mathematical reasoning ever developed.

Noether's first and second theorem was published in 1918. The first theorem relates symmetries under global spacetime transformations to the conservation of energy and momentum, and symmetry under global gauge transformations to charge conservation. In continuum mechanics and field theories, these conservation laws are expressed as equations of continuity. The second theorem, an extension of the first, allows transformations with local gauge invariance, and the equations of continuity acquire the covariant derivative characteristic of coupled matter-field systems. General relativity, it turns out, exhibits local gauge invariance. Noether's theorem also laid the foundation for later generations to apply local gauge invariance to theories of elementary particle interactions.

In Dwight E. Neuenschwander's new edition of Emmy Noether's Wonderful Theorem, readers will encounter an updated explanation of Noether's first theorem. The discussion of local gauge invariance has been expanded into a detailed presentation of the motivation, proof, and applications of the second theorem, including Noether's resolution of concerns about general relativity. Other refinements in the new edition include an enlarged biography of Emmy Noether's life and work, parallels drawn between the present approach and Noether's original 1918 paper, and a summary of the logic behind Noether's theorem.

Genres: Physics, Mathematics, Science, Nonfiction, History, Biography, Stem

344 pages, Paperback

First published September 12, 2010

Reviews

[Manny · Rating 4 out of 5]

- Agent Krzorxxx, you may now approach the throne.

- Your Majesty, I am overwhelmed. Allow me to express my humblest--

- Pish and tush, young fellow. They've given me half an hour with you, let's not waste it on empty phrases. Tell me about your latest mission.

- Sire, I doubt that the life of a minor contact agent will be of any--

- Krzorxxx, no false modesty. The Section Head is very pleased with you. He tells me you have received four commendations during the last hemi-cycle alone. He is particularly impressed with your performance on, hm, "Earth". I asked for a direct briefing. Pray begin.

- Ah, yes your Majesty. I am as always yours to command. The mission was a delicate one, the "Earth" civilization being near the Cusp but as yet by no means sure of attaining it--

- How do PsychEval estimate their chances?

- Currently about 21.3%, your Majesty. I am sorry I cannot report a better figure--

- Nonsense, Krzorxxx, you know as well as I do that it was under 10% when you left. Excellent work. Proceed.

- Thank you, your Majesty. I incarnated as a minor cleric named Emmy Noether--

- You again wished to improve the Earthlings' understanding of ethical matters?

- No, your Majesty. On this mission, we had determined that philosophy was more important. As your Majesty is aware, achievement of the Cusp depends on both--

- Indeed, indeed. Please go on.

- Yes, your Majesty. I obtained the confidence of the local priesthood using standard methods and was inducted into the fraternity as a lay member. Fortunately, I had direct access to some of the more powerful officials of the Temple of Göttingen--

- Fortunately, my foot. Good planning. Proceed.

- Yes Sire. I commenced the usual campaign of covert education following schedule 28b. Abstract algebra, algebraic topology, the use of symmetry in penetrating the veil of reality--

- This all went well?

- Very well, Sire. I believe, at any rate.

- Please tell me more about the symmetries. I have always had a weakness for that subject.

- Sire, I could naturally only sketch out the most elementary aspects. But I was able to suggest the quantitative relationship between symmetry and constancy in a way that excited the attention of a few priests. These Earthlings are not as lacking in intelligence as some people make out. I remember two in particular, "Weyl" and "Einstein"--

- Krzorxxx, your sympathy for these primitive savages does you credit, but you must keep your affections firmly under control. We cannot afford to lose any more field operatives. That is an order.

- Yes Sire.

- So, symmetries and conservation. On my home planet, as you are perhaps aware, we have a rather pleasant poem which expresses the fundamental theorem. I remember learning it when I was still a larval form.

- I know it well, Sire.

- But no doubt it was hard to translate into the Earthling tongue?

- I did my best, Sire. They grasped the ideas quite quickly. They have already advanced to an incomplete understanding of the inertial field.

- Have they indeed?

- Yes Sire. A certain "Higgs"--

- Your Section Head was not exaggerating, Krzorxxx. You have done well.

- Thank you, Sire.

- In fact, I cannot help wondering if you may not have done too well. You have observed the Prime Directive at all times?

- Scrupulously, Sire.

- There is no chance that any of the Earthlings could have suspected your true nature?

- None, Sire.

- And how can you be so sure?

- It is a little complicated to explain, Sire. I incarnated as a woman--

- A "woman"?

- Sire, the Earthlings are sexually differentiated. In the reproductive process, the "woman" occupies a position regarded as inferior due to--

- Spare me the details, Krzorxxx. If I understand you correctly, you are saying this biological curiosity is enough to protect you? There is no chance the Earthlings will notice that all their key philosophical insights over the last centicycle come from one source?

- No Sire. I know it sounds far-fetched, but PsychEval tested it thoroughly. Over 99%. They ran extensive simulations. The only Earthlings who have even a theoretical possibility of noticing are a few mathematicians.

- "Mathematicians"?

- An obscure mystical cult, your Majesty. There are no more than a handful of them. If necessary, they can easily be eliminated.

- You no doubt understand the necessity for these questions, Krzorxxx?

- I do, your Majesty.

- Your answers are satisfactory. I confirm your immediate promotion.

- Sire, I am at a loss for words. I most abjectly--

- Save your breath, young fellow. And keep up the good work.

- Thank you, Sire.

- No need, Senior Agent Krzorxxx. Dismissed.

[Laura · Rating 4 out of 5]

I picked up this book because the first time I heard of Emmy Noether I was already a tenured physics professor, and this seemed a glaring omission that I wanted to understand. The truth turns out to be that it wasn't that I'd never seen applications of her theorem connecting symmetry and conservation, but the person behind it had been left out of all explanations of it. Did I miss hearing about Noether because I was an observational astronomer in grad school and really didn't pay any more attention to mathematical physics than I had to, or because the world still finds it easy to belittle and ignore the contributions of women to science? After reading this book I'm convinced that Noether's name should be there next to to Dirac, Einstein and the other men of this era of the birth of modern physics given how fundamental and broadly applicable her theorem is.

This book is not a history of science text for the casual reader. It is a text on Noether's Theorem suitable for early graduate students, with questions and problems and requires a familiarity with a mathematical representation of physics usually not seen before the most advanced undergraduate courses. I bogged down reading it at first as I found myself reading as if I had to lecture on it: working out all the details from one step to the next, thinking of how I'd explain it to my seniors and solving all the problems. This book is quite beyond the level of anything I teach however, so eventually I opted for simply enjoying seeing the applications and insight into fundamental connections in ways I had never considered. The book works well for either style of reading.

I highly recommend this book for all physics graduate students within their first year. I wish I had read it then!

[Erik]

This is amazing, well written and argued, with thoughtful questions, and I'm just talking about the stuff explaining classical mechanics. Incidentally, the converse of Noether's theorem is also true: physical conservation laws imply symmetries pp 75 to 79. I've always been curious why we don't read it that way since all of our information about spacetime comes through physical experiments in spacetime. Another surprise, Noether's theorem unites energy and momentum conservation already in pre-relativity physics, p 79.

[Sean · Rating 5 out of 5]

At the onset I treated this book as a hybrid between a popular physics book and a text book, perhaps something like Feynman's QED. I may have been right for the first few chapters but that perspective made reading the second half of this book a real challenge! Having gotten through it I would more accurately describe this as being an enthralling, elegant and readable textbook. Suitable for the physics graduate student and beyond, Emmy Noether's Wonderful Theorem begins by reviewing variational calculus and Lagrangian mechanics then dives into some of the fundamental ideas behind the most modern and sometimes complicated physical theories. Chapters 6 and 7 are kind of a slog, but 8 and 9 are much smoother as most grad students in physics have a pretty sound background in quantum mechanics. Each chapter is accompanied by conceptual questions first and then more quantitative questions. Definitely an awesome read which highlights many beautiful aspects of one of the most underrated theorems of the last hundred years.

[Mark Johnson · Rating 4 out of 5]

I got this monograph because I one of my fictional characters becomes obsessed with Emmy Noether (he is a math geek who gets dumped by his wife). While I cannot recommend this book to anyone who isn't a math geek, I would urge everyone who has never heard of Emmy Noether to check out her Wikipedia page at http://en.wikipedia.org/wiki/Emmy_Noe.... This puts the lie to the notion that 'woman aren't hard-wired to understand mathematics'. In addition to Noether's theorem, which has been described as one of the most important equations in the history of theoretical physics, on a par with Pythagoras' theorem (!), Noether was a pioneer in the newborn field of abstract algebra and provided major contributions to Lie group theory, ring theory, algebraic topology, Galois group theory, hypercomplex numbers, representation theory, and noncommutative algebras.

To enjoy this book, one should be familiar with calculus, at a minimum; for full appreciation, at least a passing acquaintance with the calculus of variations or--what amounts to the same thing--Lagrangian and Hamiltonian mechanics would be extremely helpful. Although these topics are introduced in the text, the treatment is extremely abbreviated and is unlikely to provide insight to a reader encountering these concepts for the first time.

After a biography of Emmy Noether, the book develops Noether's theorem, beginning with a consideration of symmetry in physical systems, equating symmetry to invariance under a coordinate transformation. The topic of conserved quantities in physical systems is then addressed, in the course of which the Euler-Lagrange equation--a fundamental result in the calculus of variations--is demonstrated and illustrated with real-life examples. Noether's theorem is then formally proven, preceded by a proof of the Rund-Trautman identity.

Noether's theorem, in a nutshell, states that a conserved quantity is associated with any physical system that is invariant under a coordinate transformation. Thus, a spinning bicycle wheel, which is invariant with rotation of the coordinate axes, displays conservation of angular momentum. The initial motivation for the theorem was an attempt to solve a perceived flaw in Einstein's general theory of relativity. David Hilbert noticed that Einstein's theory of gravitation seemed to violate the principle of conservation of energy because gravitational energy can, itself, gravitate. Noether's theorem not only determined that energy was conserved in Einstein's differentiably symmetrical system, but that the principle holds for any such system.

To appreciate the rest of the book requires some knowledge of the theory of gauge invariance and electromagnetic field theory as well as the theory of quantum fields; very few general readers will be able to make it through these sections. There are 'questions for reflection' and problem sets at the end of each chapter.

This monograph is written clearly and concisely and I highly recommend it to anyone with the requisite background in math and physics to appreciate it.

[Blaise Pascal · Rating 4 out of 5]

This book is an excellent book to learn about the structure and implications of Emmy Noether's theorems tying together symmetry and conservation laws. It covers the formulations of the theorems (and related theorems) in both Lagrangian and Hamiltonian approaches, and gives plenty of examples including classical mechanics, special relativity, and shows how it carries over to quantum mechanics and quantum field theories.

This is a textbook, which I didn't realize when I bought it. Each chapter ends with questions and exercises. My guess is that it is a graduate-level textbook. I would probably have gotten more out of it if I hadn't read straight through and actually took the time to work the exercises. Even so, it gave me a better understanding of Lagrangian and Hamiltonian approaches to mechanics, as well as more insight into how that quantum stuff works.

My main problem with the book came from the Kindle formatting of it. It read well on my computer screen, but on my tablet there were serious problems with the rendering of the mathematics portions of the book (which was continuous throughout the book). The equations rendered too small compared to the text, and were virtually unreadable.

[Shai Sachs · Rating 3 out of 5]

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.

[Ashvika Amalan · Rating 5 out of 5]

"Emmy Noether's Wonderful Theorem" leans towards a textbook, but don't treat it as one. It's a valuable book to anyone interested in math and physics. The book includes both Lagrangian and Hamiltonian formulations of the theorems, and several examples of classical mechanics and special relativity, whilst connecting quantum mechanics and quantum field theories.

Neuenschwander has done an excellent job demonstrating many of the mathematical equations with real-life examples, especially the Euler-Lagrange equation (chap. 6.2, pg. 96).

It's DEFINITELY not the easiest to understand, but any math 'geek' would love this book!!

[Alex Lee · Rating 4 out of 5]

Conceptually it makes great sense that someone who was interested in group theory would be able to find a way to chart consistency between multiple frames of reference. Beyond the first half of the book though I kept getting lost in the math. Still, it's quite worth reading. Neuschwander makes much effort to explain much of historic and mathematical asides to her theorem, and for that I appreciate it.

[Rhonald Lua · Rating 4 out of 5]

I was browsing the Rice University Library, looking for another book, when I saw this lovely book about Emmy Noether. I'm fascinated with books, especially those that tell a story, a biography or history of science and math, while at the same time "meaty" in math and physics . I enjoy and remember the material better this way.

I have a physics degree, but from this book, I learned and understood for the first time a complete statement of Noether's theorems, and the difference between the first and second theorems. I also understood Parametric Invariance, that was a pleasant experience. I also liked reading about how Einstein arrived at his Field Equations, such as how he arrived at the coefficients on the "Geometry" side of the Field Equations by inferring them using a special coordinate transformation without knowing the Bianchi Identity and Noether's second theorem.

The book is accessible, but there are many typos and a few mistakes. The short proof of Parametric Invariance given that the Lagrangian is homogeneous of order 1 under velocity rescalings, it looks incorrect and can be written better. I reworked the proof in my notes so that it makes more sense.

[John Micallef · Rating 5 out of 5]

Very interesting exploration of one of the pillars of modern physics.

[Jeff Stevens · Rating 4 out of 5]

Good book on a topic I have been wanting to understand. I can't say I fully understand it yet, but I understand considerably more than before reading this book.

[Ken Rideout · Rating 4 out of 5]

What a great book this would have been for me back as a young grad student (or older undergrad?). The math was just a little beyond me now-a-days, but I recognize most of it as things I supposedly could do back in the day. This book puts it all in a nice package and motivates it all with central themes and over-arching aesthetic sensibilities. Finding common ground in classical conservation laws, quantum mechanics, and principles of least action. I actually spent time looking at every page even though I could not follow all the derivations. Why couldn't I have a had a professor the likes of Dr. Neuenschwander? (although, who knows, a younger me may not have been on board for the ride!). First time in a LONG time that I regret not following through on the mathematically complicated (partials and tensors in the same equation and everything seemed too generic) physics that perplexed me in school.
If I could follow all the math, it might have gotten 5 stars... As it is, it validates my initial forays into sharing a little of Noether's theorem with my high school students (symmetries give you conservation laws is how I put it).

[Morgan · Rating 4 out of 5]

Emmy Noether's Wonderful Theorem derives the eponymous theorem and then explores applications in various areas of physics. It's well written and engaging, with thought provoking questions and useful exercises.

Noether's theorem relates symmetries in physics with conservation laws, and does so using tools that were later used in the development of quantum mechanics and field theory: variation calculus and Lagrangians.

One chapter of the book gives a short biography of Emmy Noether, who was truly an amazing mathematician. The rest of the book introduces variational calculus and lagrangians, and then goes through derivations and explanations of Noether's theorem. It's not an easy book to read, and I had to read a few chapters twice to understand what was going on. This is a great book for people that are really into mathematical physics, but it definitely isn't written for someone just beginning in physics.

[Jeffrey Perren · Rating 3 out of 5]

Superb, but highly technical textbook of Noether's work on the relation of mathematical symmetry to physical laws.