# Linking symmetry and conservation laws, she changed physics forever

July 1, 2018 12:08 PM Subscribe

"It’s not easy to find quotes of Noether reflecting on the significance of her work. Once she made a discovery, she seemed to move on to the next thing. She referred to her own Ph.D. thesis as 'crap,' or 'Mist' in her native German. But Noether recognized that she changed mathematics: 'My methods are really methods of working and thinking; this is why they have crept in everywhere anonymously,' she wrote to a colleague in 1931."

Emmy Noether was a brilliant mathematician in a time when brilliance in female mathematicians was not encouraged. Despite the support of renowned male colleagues, including David Hilbert, and the general acknowledgement of her superior mathematical abilities, she was for many years treated as a second-class academic, allowed to teach at the University of Göttingen only as a nominal assistant to men, without pay. Nevertheless, she persisted.

When, in 1915, Hilbert and Felix Klein were trying to understand the implications of Einstein's new General Theory of Relativity, the two mathematicians, each outstanding in his own right, found they were unable to solve certain problems posed by the theory. Stumped, they turned to Noether as one of the best minds to solve it. This led her to formulate what became known as Noether's Theorem, the mathematical basis for much of the subsequent physics of the 20th century, and continuing into the 21st.

Briefly stated, Noether's Theorem shows that for every symmetry in the laws of physics, there is a corresponding conservation law, and vice versa. The importance of conservation laws had long been understood in classical physics, deriving directly from Newtonian mechanics. Symmetry was also understood to be important, with the invariance of physical laws with respect to changes of reference frame dating at least to Galilean relativity. Einstein had elevated symmetry in the understanding of physics first with special and then with general relativity. But Noether was the first to show that these two methods of understanding physical laws were in fact the same.

The laws of physics do not depend on where you are in space; this symmetry corresponds to the conservation of momentum. They also do not depend on when you are in time; this corresponds to the conservation of energy. They also do not depend on how you are oriented in space; this corresponds to the conservation of angular momentum. Noether showed physicists that these tools could be used to connect known symmetries and conservation laws, but also to discover new symmetries from known conservation laws or vice versa. For example, the conservation of electric charge corresponds to a gauge symmetry in the electromagnetic potential. Ultimately, the entire Standard Model of particle physics stands on Noether's shoulders.

Noether's influence was not just in mathematical physics. She has been called "the mother of modern algebra" (academic paywall, 2 page preview available) for her work in abstract algebra, and her name is attached to a wide range of algebraic objects (e.g., the Noetherian ring). Her wide-ranging intellect transformed both the study of the mathematical problems that captured her attention and the minds and careers of the students who learned from her, known in Göttingen as "the Noether boys".

As a Jewish woman, Emmy Noether was forced to leave Göttingen when the Nazis took power. She spent the last years of her life teaching at Bryn Mawr college in Pennsylvania. In 1935, at the age of 53, she died from complications following surgery to remove an ovarian cyst. Her ashes are buried in a courtyard of the M. Carey Thomas library on the Bryn Mawr campus.

Emmy Noether has been discussed previously on MetaFilter.

Emmy Noether was a brilliant mathematician in a time when brilliance in female mathematicians was not encouraged. Despite the support of renowned male colleagues, including David Hilbert, and the general acknowledgement of her superior mathematical abilities, she was for many years treated as a second-class academic, allowed to teach at the University of Göttingen only as a nominal assistant to men, without pay. Nevertheless, she persisted.

When, in 1915, Hilbert and Felix Klein were trying to understand the implications of Einstein's new General Theory of Relativity, the two mathematicians, each outstanding in his own right, found they were unable to solve certain problems posed by the theory. Stumped, they turned to Noether as one of the best minds to solve it. This led her to formulate what became known as Noether's Theorem, the mathematical basis for much of the subsequent physics of the 20th century, and continuing into the 21st.

Briefly stated, Noether's Theorem shows that for every symmetry in the laws of physics, there is a corresponding conservation law, and vice versa. The importance of conservation laws had long been understood in classical physics, deriving directly from Newtonian mechanics. Symmetry was also understood to be important, with the invariance of physical laws with respect to changes of reference frame dating at least to Galilean relativity. Einstein had elevated symmetry in the understanding of physics first with special and then with general relativity. But Noether was the first to show that these two methods of understanding physical laws were in fact the same.

The laws of physics do not depend on where you are in space; this symmetry corresponds to the conservation of momentum. They also do not depend on when you are in time; this corresponds to the conservation of energy. They also do not depend on how you are oriented in space; this corresponds to the conservation of angular momentum. Noether showed physicists that these tools could be used to connect known symmetries and conservation laws, but also to discover new symmetries from known conservation laws or vice versa. For example, the conservation of electric charge corresponds to a gauge symmetry in the electromagnetic potential. Ultimately, the entire Standard Model of particle physics stands on Noether's shoulders.

Noether's influence was not just in mathematical physics. She has been called "the mother of modern algebra" (academic paywall, 2 page preview available) for her work in abstract algebra, and her name is attached to a wide range of algebraic objects (e.g., the Noetherian ring). Her wide-ranging intellect transformed both the study of the mathematical problems that captured her attention and the minds and careers of the students who learned from her, known in Göttingen as "the Noether boys".

As a Jewish woman, Emmy Noether was forced to leave Göttingen when the Nazis took power. She spent the last years of her life teaching at Bryn Mawr college in Pennsylvania. In 1935, at the age of 53, she died from complications following surgery to remove an ovarian cyst. Her ashes are buried in a courtyard of the M. Carey Thomas library on the Bryn Mawr campus.

Emmy Noether has been discussed previously on MetaFilter.

Warning: derail.

The late 19th and early 20th centuries were an amazing time for math and physics in Germany; so many exceptional individuals like Noether and Arnold Sommerfeld who were not only first-class thinkers but who had a gift for teaching and collaboration that vastly multiplied what they could ever have done individually. And then, of course, the Nazis -- expelling or killing Jews, demoting non-Jews with Jewish spouses and denigrating "Jewish physics". The United States became the new leader in physics and sci/tech in general, and German academia has never really recovered. For decades, the United States has maintained this leadership fed with a constant stream of the best and brightest from around the world. And now it becomes increasingly harder to immigrate into the US, while fields like climate science are routinely denigrated by American reactionaries purely on the basis of politics.

posted by Slothrup at 2:33 PM on July 1, 2018 [13 favorites]

The late 19th and early 20th centuries were an amazing time for math and physics in Germany; so many exceptional individuals like Noether and Arnold Sommerfeld who were not only first-class thinkers but who had a gift for teaching and collaboration that vastly multiplied what they could ever have done individually. And then, of course, the Nazis -- expelling or killing Jews, demoting non-Jews with Jewish spouses and denigrating "Jewish physics". The United States became the new leader in physics and sci/tech in general, and German academia has never really recovered. For decades, the United States has maintained this leadership fed with a constant stream of the best and brightest from around the world. And now it becomes increasingly harder to immigrate into the US, while fields like climate science are routinely denigrated by American reactionaries purely on the basis of politics.

posted by Slothrup at 2:33 PM on July 1, 2018 [13 favorites]

*Briefly stated, Noether's Theorem shows that for every symmetry in the laws of physics, there is a corresponding conservation law, and vice versa.*

This is one of the ones that gives me chills. It’s just this beautiful, shimmering

*thing*that you want to turn over and over in your mind until you can finally grok all of it. Like the world’s best cat toy, but for nerds of a certain stripe.

posted by schadenfrau at 2:37 PM on July 1, 2018 [22 favorites]

Yeah. The way I was first introduced to it in my undergrad E&M class was something like "Energy and momentum are conserved

posted by biogeo at 3:55 PM on July 1, 2018 [2 favorites]

*because*spacetime is isotropic." I think that's a pretty specific interpretation of Noether, but it was definitely one of those moments in my physics education that blew my mind a bit.posted by biogeo at 3:55 PM on July 1, 2018 [2 favorites]

There's a book called Mathematicians Fleeing From Nazi Germany that's worth taking a look at if you have access to a university library. Many positions were found or created to get people visas (because "impending genocide" apparently isn't good enough), but inevitably not enough. Hausdorff, a name that should be known to every math major*, committed suicide to avoid deportation to a concentration camp.

*I guess it's

posted by hoyland at 4:02 PM on July 1, 2018 [4 favorites]

*I guess it's

*possible*to avoid his name if your real analysis class is taught only in R^n. You*should*know Noether's name, but it's plausible to me that some undergrad algebra classes don't make it to rings at all.posted by hoyland at 4:02 PM on July 1, 2018 [4 favorites]

Also, the women in math group at Berkeley is called the Noetherian Ring.

posted by hoyland at 4:04 PM on July 1, 2018 [7 favorites]

posted by hoyland at 4:04 PM on July 1, 2018 [7 favorites]

schadenfrau: Yes. When I stumbled across her theorem I was awestruck. And the effect lasted for days. It still comes back and smacks me between the eyes from time to time. It had bothered me in my Undergraduate education about where the traditional Conservation laws came from. I was always told they were just what had been historically observed to always be true. None of my teachers knew about this (or bothered to tell me). Was it possible they weren't true? What would that even mean? And then this... [shakes head]

posted by aleph at 7:00 PM on July 1, 2018 [6 favorites]

posted by aleph at 7:00 PM on July 1, 2018 [6 favorites]

Yes, I also remember encountering Noether's theorem for the first time. I believe it was Feynman in The Character of Physical Law. To me, it was just astounding and the sort of thing that you wish you had learned earlier because it suddenly seems like it is scratching at something fundamental.

I thought it seemed like the kind of magical leap that Feynman himself would take but then learned that it was Noether that did all of this. Noether who? As others have reported, this is the first time you hear of her and it seems inexplicable and you look at her work and you realize we had a great towering genius in our midst and, unlike Einstein, she remains unknown simply because she was a woman.

posted by vacapinta at 2:14 AM on July 2, 2018 [7 favorites]

I thought it seemed like the kind of magical leap that Feynman himself would take but then learned that it was Noether that did all of this. Noether who? As others have reported, this is the first time you hear of her and it seems inexplicable and you look at her work and you realize we had a great towering genius in our midst and, unlike Einstein, she remains unknown simply because she was a woman.

posted by vacapinta at 2:14 AM on July 2, 2018 [7 favorites]

biogeo: Is spacetime isotropic only in a gravity free region? Doesn't seem to be true of spacetime around a Black hole or any region with gravity giving it a non-zero curvature.

posted by aleph at 9:18 AM on July 2, 2018 [1 favorite]

posted by aleph at 9:18 AM on July 2, 2018 [1 favorite]

Yeah, things don't always work as intended in GR. Conservation of energy is also tricky at best in it due to the dynamical nature of spacetime.

posted by edd at 9:20 AM on July 2, 2018 [1 favorite]

posted by edd at 9:20 AM on July 2, 2018 [1 favorite]

That sounds like a question for someone whose physics training is more complete than mine, but I think the argument would be that it's at least locally isotropic

posted by biogeo at 11:34 AM on July 2, 2018

*within its curved manifold*. I never took GR as an undergrad and didn't pursue physics past my B.S., so I know enough to appreciate and enjoy it but not enough to explain it well or understand it as a true physicist would.**edd**, is this one of the many cases where mass leads to symmetry-breaking?posted by biogeo at 11:34 AM on July 2, 2018

Yup here's another, I can still remember the first time I encountered Noether's Theorem. My mind was blown then, and I admit (many decades later) it is still blown.

(Another woman treated shabbily by world in general and the Nazis in particular was Lise Meitner.)

posted by phliar at 12:55 PM on July 2, 2018

(Another woman treated shabbily by world in general and the Nazis in particular was Lise Meitner.)

posted by phliar at 12:55 PM on July 2, 2018

biogeo: I think you're thinking of symmetry breaking in the Higgs mechanism leading to mass? In which case no, that's something different.

posted by edd at 1:51 PM on July 2, 2018 [1 favorite]

posted by edd at 1:51 PM on July 2, 2018 [1 favorite]

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Chalk drop.

posted by clew at 12:23 PM on July 1, 2018 [35 favorites]