Notes on Fearful Symmetry#
A. Zee’s Fearful Symmetry is a tour of how 20th-century physics came to organize itself around symmetry rather than around forces or facts. It’s from the 1980s, so some of its specific bets have aged unevenly, but the methodological story still holds up. These are my study notes written with Claude’s help as I try to learn more about physics, symmetry and group theory.
Symmetry in classical physics#
Physics and Science in the 19th century generally demanded that evidence should precede theory. Faraday played with wires and frog legs. Maxwell wrote down equations that summarized the resulting body of facts. The he Lorentz invariance of those equations was something nobody noticed until much later, and once noticed, it forced a rewrite of mechanics and the nature of time, with E=mc² as a consequence.
Einstein went the other way with his ground-breaking general theory of relativity. He started from a symmetry (the equivalence principle, which says a uniform gravitational field cannot be locally distinguished from constant acceleration) and derived the consequences. General covariance, which demands that physics keep its form under arbitrary coordinate transformations, then forces the warping of spacetime and the rest of general relativity. This inversion of theory before evidence is the through-line of Zee’s book.
Emmy Noether’s theorem continues this path and connects continuous symmetries to conservation laws with uncomfortable precision. Invariance of the action under translation in time gives energy conservation. Invariance under translation in space gives momentum conservation. Energy is conserved, in this picture, because the laws of physics do not care what time it is. There is something almost embarrassing about this once you internalize it.
Reformulating physics through action principles instead of forces also turned out to be the right frame for quantum mechanics, where Dirac and Feynman’s path integral lets a particle take every possible path weighted by its action.
Parity, and the experiment that broke it#
Noether covers the continuous symmetries. There are discrete ones too, and not all of them hold. Parity is the symmetry between a process and its mirror image. If you film a physical event and flip it left-to-right, parity conservation says the mirrored film should still show valid physics. Gravity respects this. So does electromagnetism. Until 1956, every physicist treated parity as one of nature’s exact symmetries, on roughly the same footing as energy conservation.
T.D. Lee and C.N. Yang noticed that nobody had actually checked. The strong and electromagnetic interactions had been tested and conserved parity, but no experiment had specifically tested it for the weak force. They wrote a paper in 1956 proposing experiments that could decide the question. The reaction from the rest of the field ranged from indifferent to amused. Pauli wrote to Weisskopf saying he refused to believe God was a “weak left-hander.” Feynman bet $50 against parity violation. Felix Bloch told colleagues at Stanford he would bet his hat on it; he later told Lee he was lucky he didn’t own one.
Chien-Shiung Wu took the proposal seriously. Wu was a beta-decay specialist at Columbia, and after a conversation with Lee in the spring of 1956 she set about designing the experiment. She picked cobalt-60 as the source. Cobalt-60 nuclei have spin and undergo beta decay, and they can be polarized using adiabatic demagnetization, a low-temperature technique developed at Oxford a few years earlier. Wu contacted Ernest Ambler at the National Bureau of Standards, whose lab had the cryogenic equipment to do the polarization properly.
The setup is simple to state. Cool a cobalt-60 sample to within hundredths of a degree of absolute zero so thermal motion doesn’t randomize the spins. Apply a magnetic field to align all the nuclear spins. Then count the beta-decay electrons coming off in the direction of the spin and compare against the count coming off opposite to the spin. If parity holds, the two counts have to be equal: the mirror image of a spinning nucleus emitting electrons one way is a spinning nucleus emitting them the other way, and a parity-symmetric law has no preference between the two.
The two counts were not equal. Electrons came out preferentially opposite to the nuclear spin direction. As the sample warmed up and the polarization was lost, the asymmetry vanished, which ruled out instrumental effects. Wu had her data in late December 1956. The result was published in February 1957, alongside a confirming experiment by Garwin and Lederman on muon decay.
What this meant was that the weak force has a handedness built into it. The neutrinos produced in beta decay are left-handed; their antineutrinos are right-handed. The mirror image of a weak-interaction process is, in general, not a valid physical process at all. This was the first symmetry physicists had assumed was exact and turned out not to be, and it set the precedent for the later discovery of CP violation, which is part of why the universe contains more matter than antimatter.
Lee and Yang received the Nobel Prize in 1957, less than nine months after Wu’s results were published. Wu did not. Pauli was outraged on her behalf, and Jack Steinberger later called the omission the biggest mistake in the committee’s history. Wu was awarded the inaugural Wolf Prize in 1978. She did not publicly complain, though in a letter to Steinberger she wrote that her work being overlooked still hurt.
The particle zoo, organized by group theory#
By the 1960s experimenters had piled up dozens of “elementary” particles. Group theory rescued them from chaos. Gell-Mann’s eightfold way sorted hadrons into geometric patterns the way the periodic table once sorted chemistry, even predicting a missing particle that experimenters then went and found.
The deeper move was making the symmetry local. The intuition is roughly this: instead of requiring physics to look the same when you rotate every quark in the universe by the same amount, you require it to still look the same when you rotate them by different amounts at different points in spacetime. The only way to satisfy this stricter requirement is by introducing new force-carrying particles, called gauge bosons, that travel between points to compensate for the mismatch. The forces that fall out of this construction look exactly like the ones we see in nature. The simplest case gives electromagnetism, with the photon as its gauge boson. Yang and Mills worked out the math in 1954 for the harder cases involving more elaborate symmetries, and that framework is what the strong and weak forces are built on, with gluons and the W and Z bosons playing the role of the photon.
The strong force has a counter-intuitive feature called asymptotic freedom: quarks pull harder on each other the further apart they get, and barely interact when close. This is why a lone quark has never been observed. They’re permanently confined inside hadrons, and the gluons holding them together stay hidden too.
Symmetries that aren’t quite there#
Not every symmetry physicists assumed held actually does. Parity, the symmetry between a process and its mirror image, was for a long time treated as obviously true, and the weak force violated it. Wu’s experiment in 1956 showed that cobalt-60 nuclei prefer to emit beta-decay electrons in one direction relative to their spin, which is something a mirror-symmetric law has no way to do.
Spontaneous symmetry breaking covers a different case: the equations stay symmetric, but the lowest-energy state of the system doesn’t. This is what lets the unified electroweak symmetry SU(2) × U(1) present itself, at the energies we live at, as a massless photon alongside the heavy W and Z bosons. The symmetry is exact in the equations; the vacuum just doesn’t share it.
Toward unification#
Spontaneous symmetry breaking is another way a symmetry can fail to show up at the energies we live at. The equations stay symmetric, but the lowest-energy state of the system doesn’t. This is what lets the weak and electromagnetic forces, which appear as a single unified force in the equations, show up in everyday physics as the very different things they seem to be: a massless photon for electromagnetism, and the heavy W and Z bosons for the weak force. The symmetry is exact in the equations; the vacuum just doesn’t share it.
Grand unification asks the obvious next question. Can all three forces of the Standard Model be derived from a single underlying symmetry, with the photon, the W, the Z, and the eight gluons all playing the role of gauge bosons of one larger group? Georgi and Glashow proposed a candidate in 1974. Among other things, it explains why the proton and the electron carry equal and opposite charges, a fact the Standard Model otherwise just stipulates.
The price is that the Georgi-Glashow model predicts proton decay at a half-life of around 10³⁰ years. We’ve built large detectors and waited. So far, no decay.