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The mathematician Emmy Noether, who made groundbreaking advances in abstract algebra and whose eponymous theorems articulated the deep connection between symmetries and conserved quantities in physics, was born #OTD in 1882.

Image: Public domain (photographer unknown)

Image: Public domain (photographer unknown)

Emmy Noether began university at a time when women studying mathematics were only allowed to sit in on lectures. A professor’s permission was usually required.

She spent her first two years at Erlangen, then a year at Göttingen where she attended lectures by Hilbert, Klein, Minkowski, and Schwarzschild. But she wasn’t allowed to be an “official” student, so she returned to Erlangen.

In 1904 the rules were changed, and she began her doctorate at Erlangen. Three years later she was done, summa cum laude. But women still weren’t allowed to assume academic posts, so she moved home. She continued to work, though, on problems in abstract algebra.

Noether became known as a talented and creative mathematician. In 1915, Hilbert and Klein asked her to return to Göttingen to help with their program in mathematical physics. At the time, Hilbert was racing against Einstein to articulate the ideas underlying General Relativity.

Faculty from other departments objected to a woman taking an academic post at the university. Klein dismissed them. “After all,” he said, "we are a university, not a bathhouse."

It was during this period at Göttingen that Noether proved the two theorems that bear her name. It is the first theorem, often just called “Noether’s Theorem,” that associates conserved quantities with symmetries of physical systems.

cwp.library.ucla.edu/articles/noeth…

cwp.library.ucla.edu/articles/noeth…

A symmetry is something you can do to a system that doesn’t change its physics. For example, all other things being equal, a collision between two masses works out the same way in your office and in the lab down the hall. A translation down the hall is a symmetry of the physics.

Likewise, if you collide the two masses in exactly the same way, you’ll get the same result in a minute or an hour or a year that you got earlier today. So a translation in time can be a symmetry. You might also rotate the experiment and find that the physics is unchanged.

Noether’s Theorem states that for every symmetry of a physical system there is a conserved quantity — a property of the system that does not change over time. For translations in space it's momentum, for translating in time it’s energy, and for rotations it's angular momentum.

For a complicated system, the symmetries may be more involved than the examples I gave above. In that case, Noether’s theorem shows us how, starting with the basic setup for describing the physics of the system, to determine the conserved quantity associated with each symmetry.

This is really useful information! The physics of a collision might be messy and hard to follow, but knowing that there are quantities that remain the same before and after can help us pin down the end result of that process.

For a quantum mechanical system there is often some redundancy in how we describe the physics. Exploiting that redundancy to alter the description in a way that leaves the physics unchanged is called a gauge symmetry. Noether’s Theorem gives conserved quantities for those, too.

(Classical theories can also have gauge symmetries -- electromagnetism is an example.)

The conservation laws of particle physics emerge this way, so Noether’s Theorem is at the heart of each principle that guided the development of the Standard Model.

In all these cases, Noether’s Theorem relies on what is called a “variational formulation” of the physics. Almost every modern physical theory can be formulated in this way, so Noether’s Theorem applies to just about everything.

Noether’s results were important in the early days of general relativity, helping to untangle some thorny issues with energy and energy transport in the theory.

But many prominent physicists of that generation and the next questioned the primacy of variational principles, and so the importance of Noether’s work wasn’t fully appreciated.

In the second half of the 20th century variational principles reemerged as the language in which most theories were formulated, and Noether’s work became much more prominent. Today, it’s considered foundational.

Hilbert eventually secured a position for Noether at Göttingen. She remained there until the early 1930s, when the Nazi government forced Jewish academics out of academic positions.

In 1933 Emmy Noether moved to the US for a visiting professor position at Bryn Mawr. She remained there for two years, and frequently lectured at Princeton. She passed away in 1935, quite suddenly, while recovering from a procedure to treat an ovarian cyst.

Anyway, happy 136th birthday to Emmy Noether, and happy 100th birthday later this year to the "Invariante Variationsprobleme" paper where she published the two Noether's Theorems. Here's an English translation on the arXiv:

arxiv.org/abs/physics/05…

arxiv.org/abs/physics/05…

And here's a (retina) desktop-sized version of that image I posted earlier in the thread.

dropbox.com/s/a6ad0jky8i73…

dropbox.com/s/a6ad0jky8i73…

☝️A thread from yesterday about Emmy Noether.

A symmetry:

[rotates object]

◾ → ◆ → ◾

A gauge symmetry:

[waves hands, says “ta-da!”]

◾ → ◾

[rotates object]

◾ → ◆ → ◾

A gauge symmetry:

[waves hands, says “ta-da!”]

◾ → ◾

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