Figure 2: Dirac (left) and Eliezer (center). Unidentified person is not me (regrettably). | Figure 3: This is me (2nd from right) and Eliezer (3rd from right). |
English physicist and Cambridge University professor Paul Dirac was an avid mountain climber and occasionally ascended such well-known peaks as Mount Elbruz in the Caucasus. In preparation for such excursions, Dirac would often climb trees in the hills just outside Cambridge - wearing the same black suit in which he was invariably seen around the university campus.
"It would be intriguing to explore whether this is about a miracle or it is the group-theoretical approach which leads to this formula."In 1995, Weinberg declared it to be:
"Absolutely chance coincidence."Even as recently as 2004, we find [14]:
"In truth, when adding the orbital and spin momenta, the values of j run from (l_{min} + 1/2) = 1/2 … (l_{max} + 1/2) = n − 1/2, so that 1 ≤ (j + 1/2) ≤ n. Because this spectrum coincides with that of the values of n_{φ}—both varying from 1 to n—the numerical results of both theories are the same."For my part, I'm less persuaded by either Eliezer's original suggestion, or Heisenberg's 1968 comment, that the paradox might be resolved by an analysis of dynamical symmetries [15]. For one thing, the SO(4) symmetry of the Pauli hydrogen atom [5] is broken by relativistic modification of the Coulomb potential, thus lifting the (2l + 1)^{−2} spectral degeneracy. But that also leaves only the geometrical symmetry, i.e, the usual angular momentum states of SO(3). It's not clear how that smaller symmetry can support a group-theoretic derivation of the more complicated formula for the relativistic energy eigenvalues. It now seems clear to me that the original Bohr-Sommerfeld quantization rules, based on classical-particle (planetary) orbits of the electron [13], were naive in that they did not (and could not, in 1916) include the de Broglie "matter wave" effects of the (scalar) electron that ultimately appeared more clearly in the WKB approximation—a decade later.