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. […] Ada Lovelace Day: Emmy Noether and Symmetry, Revisited « Galileo’s Pendulum mathematician Emmy Noether (1882-1935) […]

  2. […] two directions to go: clockwise or counter-clockwise. With three dimensions to work with, you have a lot of choices of directions for rotation, so order means a lot, as anyone who has tried carrying a sofa-bed up a narrow staircase can […]

  3. […] weeks ago, many people wrote stirring praise for the great women in science, math, and technology. My own choice was Emmy Noether, a mathematician whose work is important to my research, so it seemed fitting to create a shirt with her visage. It’s available for women, girls, and […]

  4. […] Symmetry in nature tells us a lot about the forces that shape an object: stars and planets are mostly spherical because of gravity, for example, but slightly flattened at the poles because of rotation. Crystal lattices in solids form repeating patterns due to the forces between atoms or molecules comprising them. Even biological symmetries—two arms, legs, eyes, lungs, and so forth in humans, five arms in starfish, etc.—are due to genetic expression; while these aren’t “forces” in the usual physics sense, there is a definite mathematical kinship. Chirality in human hands: even with both left and right hands oriented with fingers the same way, they aren't identical. The palm of the left hand is facing outward, while the knuckles of the right hand face outward. Or is it the other way around? I won't tell. I even removed my ring. […]

  5. […] names by that time) have mass, while the photon does not. (This is a case of symmetry breaking; my post on Emmy Noether has a little more information.) However, the mass of the Higgs itself is not part of the Standard […]

  6. […] of light, which is always perpendicular to the direction the photon is traveling.) My post on Emmy Noether has more information about rotational […]

  7. […] about Albert Einstein (the founder of my primary area of study, general relativity), or about Emmy Noether (whose famous theorem marks reveals deep mysteries of nature). I’ve highlighted Charles […]

  8. 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.

    1. 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. ;)

  9. […] century mathematician, Emmy Noether, for the New York Times. (Too bad she didn’t link back to my tribute, but then again, Angier doesn’t sell Emmy Noether t-shirts either, I […]

  10. […] The W and Z bosons form a triplet analogous to the three possible axes of rotation in a sphere: the math is the same. This type of symmetry is known as SU(2) symmetry, which is related to the math of quaternions. […]

  11. […] Conservation laws are some of the deepest principles in physics. At their most straightforward (as with our murder weapon), they simply tell how mechanical systems work, but at their most profound, they lead to results such as the Higgs mechanism, by which particles acquire mass. We don’t need to understand particle physics for Busman’s Honeymoon (“Bunter, I suspect a new particle is carrying momentum away from this collision!”): introductory physics is enough. The murder weapon in Busman’s Honeymoon. Using a weighted cactus pot as a pendulum, the murderer set a booby trap. The speed the pot has at the bottom of its swing is the same as the speed it would have if it were dropped vertically from the same height. […]

  12. […] who wrote the first computer algorithm (long before there was a computer to run it!). To quote my post from last year: Ada Lovelace (1815-1852) was daughter of George Gordon, the poet better known as Lord Byron, but […]

  13. […] two directions to go: clockwise or counter-clockwise. With three dimensions to work with, you have a lot of choices of directions for rotation, so order means a lot, as anyone who has tried carrying a sofa-bed up a narrow staircase can […]

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