Ada Lovelace Day: Emmy Noether and Symmetry, Revisited

Today (October 7) is Ada Lovelace Day, in which we celebrate women in technology, science, and mathematics. Ada Lovelace (1815-1852) was daughter of George Gordon, the poet better known as Lord Byron, but her greater claim to fame lies in writing the first computer programs…before any computer existed to run it. She wrote the program for Charles Babbage’s Analytical Engine, so she had a design to build for, but Babbage was never to build the computer: it was too ambitious, too expensive, and frankly nothing like it existed in the 19th century. (Babbage’s own lack of focus no doubt helped sink the project as well.) Nevertheless, Lovelace’s programs had all the basic features computer languages possess today, including flow control (“if-then-else” statements, etc.) and the manipulation of symbols. The concepts she wrote about in 1842 are still taught in modern numerical programming courses today, including the one I taught last fall. Ada died of cancer at age 36.

Although I’ve done a lot of computer programming, I’m not a computer scientist, so I’ll leave others to talk about Ada and her colleagues. As a theoretical physicist with a highly mathematical focus, I will instead highlight a mathematician who is famous among physicists but not well known among the general public: Emmy Noether. (This post ran in a slightly different version on March 8 of this year.)

Noether (1882-1935) has two major accomplishments to her name: she was a leader in the development of what is known today as abstract algebra (one of the major branches of mathematics), and she proved a theorem that bears her name that connects symmetries of nature to physical conservation laws. Algebra isn’t something I can talk intelligently about, so I’ll let others eulogize her for her contributions to that area. Her other major contribution is one that not only shows up in my research, but one I think can be understood by non-scientists.

Noether’s theorem, which she proved in 1915, says that there is a direct correspondence between a symmetry of nature and the conservation of a quantity. The most famous conservation law in physics is probably conservation of energy: you cannot create or destroy energy, simply transform it from one type to another. Noether’s theorem shows that the conservation of energy is related to time: if you run the clock backwards on an experiment and everything still looks “normal”, then energy is conserved. Here’s a simple experiment: take a ball and roll it up a ramp. It will roll up a certain distance, stop, then roll back down. If you run the clock in reverse, the motion of the ball looks the same! Noether’s theorem says this is no accident, but a statement about a symmetry in time. (Nitpickers will point out that this is a gross oversimplification, and they’re right. I hope this captures the spirit of the theorem well enough to satisfy most of you, though.) However, not all types of energy are “useful”: some are known as dissipative. The prime example of dissipative energy is friction—you can’t use energy converted into friction to (for example) lift an object. It’s lost to any useful purpose. Friction breaks the symmetry of time: you lose the same amount of energy to friction whether the clock is running forward or backwards, so a system with friction will not look “right” if you run the clock in reverse.

If you rotate a featureless sphere by any angle around any axis running through the sphere's center, it will look exactly the same. This is known as spherical (or SO(3) for the math fans) symmetry.

Other types of symmetry are more familiar: a sphere spinning around an axis has rotational symmetry, and if the sphere is featureless like a pool cue ball, you can pick any axis for it to rotate around. In ordinary classical physics, symmetry of this type leads to conservation of angular momentum, which figure skaters exploit by pulling in their arms to spin faster; in particle physics, symmetry of the same type (known as SU(2) symmetry for those physicists or mathematicians reading along at home) is associated with the nuclear forces. If you restrict the rotation to one axis (known as U(1) symmetry), the associated force is the electromagnetic force. (Supersymmetry goes a few steps further, and predicts a whole zoo of as-yet undiscovered particles—but it’s all still closely tied in to Noether’s theorem.) There’s also reflection symmetry of various sorts, rotations by fixed angles (think of a starfish with its five arms), translations by fixed distances (as with crystal lattices) and fairly abstract symmetries that are hard to describe in everyday language; each of these are associated with conservation laws if they are exhibited within a physical model. Even physicists who don’t know specifically what the theorem says and don’t know how to use it in a formal way still apply it, whether they know it or not.

A featureless cylinder has a special axis of rotation running down the middle; rotation by any angle around this axis will be indistinguishable. However, if you leave the label on the soup can, the symmetry is broken. By Noether's theorem, this means the conservation law associated with U(1) symmetry will no longer hold.

Obviously a physical and mathematical principle this powerful is extremely noteworthy. I find it fitting to honor Emmy Noether on the day we honor her fellow mathematician Ada Lovelace, not least because there are still significant gender imbalances in the world of science and math. Noether had to teach without official recognition on several occasions in her life; as a woman in Germany in the early 20th century, she needed to get special permission even to study her subject. (Even the support of famous mathematicians and scientists like David Hilbert and Albert Einstein failed to win her remunerative academic positions.) Thankfully, though we have a long way to go still, we have made progress in many countries, yet there are still nations where women are denied basic educational opportunities. How many Emmy Noethers and Ada Lovelaces might there be, who are lacking the opportunity to excel in a field to which they are denied access by restrictive cultural norms?

Here and there a cygnet is reared uneasily among the ducklings in the brown pond, and never finds the living stream in fellowship with its own oary-footed kind. Here and there is born a Saint Theresa, foundress of nothing, whose loving heart-beats and sobs after an unattained goodness tremble off and are dispersed among hindrances, instead of centring in some long-recognizable deed. — George Eliot, “Middlemarch”

15 Responses to “Ada Lovelace Day: Emmy Noether and Symmetry, Revisited”

  1. 1 glenmartin March 2, 2012 at 00:19

    I should point out that, strictly speaking, Noether’s Theorem does not apply to discrete symmetries, such as those associated with a crystalline lattice or reflection symmetries. The various proofs of Noether’s Theorem only hold for continuous symmetries.

    As an example of a conservation law no longer holding due to symmetry-breaking, the Law of Conservation of Energy breaks down in scenarios where time-translation symmetry is broken, which is frequently the case in General Relativity due to the dynamic nature of space-time. In General Relativity, that conservation law is replaced by conservation of the energy-momentum 4-vector.

    • 2 Matthew R. Francis March 2, 2012 at 08:15

      Yes, it’s true that discrete symmetries don’t have a Noether theorem conservation law. That is a mistake on my part: I was listing symmetries, and failed to distinguish between continuous and non-continuous types. Lack of reflection symmetry is coupled with lack of translational symmetry in many cases, so I use that example to show (e.g.) why momentum is not conserved when a ball bounces off a wall.

      Energy is a slippery concept in general relativity, since there isn’t an expression for the energy involved with gravitation. That quickly gets into a technical discussion though, and I’d probably end up quoting from my PhD thesis by the time I’m done, so I’ll keep the comment brief. ;)

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