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These notes have been posted at feynmanlectures.info with the gracious permission of the estate of Richard P. Feynman and The California Institute of Technology. The originals can be found in Box 76, Folder 2 in The Feynman Papers at the Caltech Archives.
In 1983 Rockefeller University sponsored Shelter Island II, a two day conference that highlighted aspects of the developments in “fundamental” physics made by the high energy physics community after the establishment of the standard model.^{1} One session of the first day of the conference was devoted to recalling Shelter Island I, and everyone who had attended Shelter Island I and was alive—Bethe, Bohm, Feynman, Lamb, Marshak, Pais, Pauling, Rabi, Schwinger, Weisskopf, Wheeler—was invited.^{2}
Shelter Island I was a watershed. Held in June 1947, it was the conference at which Lamb presented the most recent data of his experiment with Robert Retherford that measured the energy difference between the 2s_{1/2} and 2p_{1/2} levels in hydrogen. His robust data established that contrary to the solutions of the Dirac equation for the energy levels of an electron in a Coulomb field, the 2s and 2p levels of the hydrogen atom were not degenerate, but that the 2s level was “shifted up by about 0.037cm^{–1} (1100 Mc).” Lamb's talk was followed by Rabi's, who presented the data John Nafe, Edward Nelson, and he had obtained on the hyperfine structure of hydrogen and deuterium.^{3} The discussions that was generated by these findings and the lecture the following morning by Hendrik Kramers—in which he presented his work on mass renormalization in simple models of extended charges—were the point of departure for Bethe's seminal calculation of the Lamb shift in non-relativistic quantum electrodynamics and Feynman's and Schwinger's formulation of quantum electrodynamics.
Also on the first day of Shelter Island I Bruno Rossi reported on the experiment that had been carried out in Rome by Conversi, Pancini, and Piccioni on the absorption of mesons in the atmosphere. The ensuing discussion gave rise to the paper by Marshak and Bethe in which the Marshak two-meson hypothesis was formulated.^{4} Marshak suggested that there were two kinds of mesons. The heavier ones were produced at the top of the atmosphere by cosmic rays in nuclear reactions. He identified these with the mesons Yukawa had postulated. The π mesons then decay into the lighter μ mesons, which do not interact strongly with nuclei. This became the point of departure for understanding and representing the weak and strong forces, culminating with the formulation of the electroweak theory and quantum chromodynamics.
Shelter Island II highlighted the fact that the theoretical physics community in 1983
Regarding d.: In the 1960s and 1970s, mathematicians and physicists witnessed a confluence of concepts and ideas. On one side were the abelian and non-abelian gauge theories describing the electroweak and strong interactions, as well as important results regarding supersymmetry in alleviating the divergence difficulties in relativistic quantum field theories. On the other, was an internally motivated extension of Riemannian geometry involving mathematical structures known as fiber bundles. When studying the topology and differential geometry of surfaces in an arbitrary number of dimensions, mathematicians found it useful to introduce auxiliary spaces, such as the space consisting of the tangent planes to the surface being studied or the space of lines normal to that surface. Such spaces are called fiber bundles, the fibers being the auxiliary spaces and the bundle the totality of fibers that are fitted together. The classification of bundles and their invariants had been studied in the 1930s and 1940s by Hassler Whitney, Norman Steenrod, Lev Pontrjagin, Shiing-Shen Chern, and others. They obtained integral formulas for the invariants of bundles that generalize the Gauss–Bonnet formula. The latter states that the integral of the Gaussian curvature over an entire surface is a topological invariant—that is, the value of the integral is not changed by local deformation of the surface—and is a multiple of 2π. These results were generalized in important works by Atiyah and Singer which culminated in 1963 with their index theorem. Roughly speaking, the theorem concerns the analytic index of the solutions, f, on a space X of a system of differential equations P(f) = 0. The analytical index of the system is an integer closely related to the dimension of the space of solutions. The Atiyah–Singer index theorem relates the analytical index to the topology of the space X. This theorem thus provides an answer whether such solutions exist or not without having to exhibit them analytically.
In his lecture at Shelter Island II Roman Jackiw gave an intuitive presentation of why the Atiyah–Singer index theorem is of relevance in non-abelian gauge fields: such theories divide themselves into topologically distinct disconnected classes determined by an integer k, related to the analytical index.^{5}
The mathematician Isadore Singer, who with Michael Atiyah had formulated the Atiyah–Singer index theorem, made a presentation at Shelter Island II on the content of that theorem. A year earlier he had written an article in Physics Today which gave a very informative and accessible overview of the connection between gauge theories, differential geometry, and topology.^{6}
Singer's lecture at Sheleter Island II was not included in the proceedings. However, Feynman took some notes of the lectures.^{7} The pages below are taken from Box 76, Folder 2 in Feynman's papers at the Caltech Archives, labeled “Singer–Index Theorem.”^{8}
Feynman's notes indicate that Singer was presenting the main result of the Atiyah–Singer index theorem in a way that makes connection to the language of supersymmetry. He was summarizing the “heat kernel proof of the index theorem” in the language of supersymmetry. The connection between supersymmetry and the Atiyah–Singer index theory was “in the air” at Shelter Island II. The connection was made explicit in a paper entitled “Supersymmetric Derivation of the Atiyah–Singer Index Theorem and the Chiral Anomaly” by Friedan and Windey (1984). The acknowledgment of that paper begins with the statement: “We would like to thank I. M. Singer whose report of a discussion with E. Witten initiated this work.”^{9}
S.S. Schweber
Brandeis University and Harvard University