Part 7 of 13. Inverse-square orbits conserve a vector that no rotation or translation explains. The symmetry behind it, a four-dimensional rotation group, also explains the degeneracies in the hydrogen spectrum.
The Kepler problem, a particle pulled toward a fixed center by an inverse-square force, conserves one more quantity than the symmetries of Part 6 account for: the Laplace-Runge-Lenz vector. In January 1926 Pauli used it to derive the hydrogen spectrum from matrix mechanics, submitting ten days before Schrödinger submitted his wave-equation solution.
In the Kepler problem,
time translation and rotations give energy and angular momentum. The combination
also has on solutions. It points from the focus to perihelion with length , where is the eccentricity. A closed ellipse is what a fixed looks like.
Almost any change to the force breaks it. Figure 1 adds to the potential, the form of the leading general-relativistic correction. The force stays central and time-independent, so energy and angular momentum survive.
Figure 1. An orbit with from , , integrated by RK4 with the potential . The violet arrow is ; faint spokes mark each perihelion passed. The strips track the full energy , , and the angle of . Green holds and red drifts. At all three hold and the planet retraces one ellipse.
First-order perturbation theory predicts a turn of per orbit (), which depends only on the exactly conserved . With it becomes Einstein's for Mercury.
Figure 2. Measured perihelion advance per orbit (dots) against , for the launch of Figure 1 at the chosen speed. Each dot averages three perihelion-to-perihelion turns, located where changes sign. The line is the first-order prediction . The dots leave the line as grows, a second-order effect; the readout fits to the dots with and compares with the prediction.
No transformation of positions alone, as in Part 5, produces . Its symmetry moves positions and momenta together, depending on the velocity, which Noether's 1918 paper allows.
Any phase-space function generates a flow, , . Momentum generates translations, angular momentum rotations, and a component of moves a planet onto a different orbit of the same energy.
Figure 3. Sixty planets equally spaced in time on the circular orbit of radius 1 (dashed), each pushed through phase space by the flow of for a distance ; faint curves trace the positions of every fifth one along the way. They land on an ellipse with the same energy and major axis, eccentricity and angular momentum , bunched toward aphelion because they are still equally spaced in time.
On bound orbits, rescale . The six conserved quantities and then have Poisson brackets
the Lie algebra of , the rotations of four-dimensional space: adds three rotations that mix the spatial directions with a fourth. Fock showed in 1935 that bound states of one energy map onto the 3-sphere , where these act as rotations.
Figure 4. All thirty-six brackets among , computed by finite differences at a seeded random state in three dimensions. Each cell shows the number and, beneath it, the generator it should equal. Orange cells equal a component of , violet cells a component of . For unbound states, with , the lower-right block changes sign and the algebra becomes that of the Lorentz group .
Rotations alone group the states of a central potential into multiplets of , with different energies for different . In hydrogen the energy depends on alone, so level holds states, the dimension of an SO(4) representation. Splitting gives two commuting rotation algebras, and Pauli read the spectrum off their Casimirs without solving a differential equation.
Figure 5. Bound levels of the radial Schrödinger equation (atomic units) for , found numerically on a 1,200-point grid for to 3, plotted as the effective quantum number . At the lines of each meet at an integer; any breaks the LRL symmetry, keeps rotations, and fans them apart. The term stands in for Figure 1's , which is too singular for the quantum problem, and has the exact solution the readout checks against.
In real hydrogen, relativistic fine structure and the Lamb shift split levels of the same slightly, the quantum counterpart of Figure 1's precession.
The vector is old: Jakob Hermann and Johann Bernoulli found it around 1710, Laplace used it in 1799, Hamilton rediscovered it in the 1840s. The symmetry stayed hidden because it does not act on positions alone.
Part 8 turns to a symmetry that moves nothing in space at all: rotating the phase of a quantum wave function.