INFERENCE / Vol. 5, No. 3 Reflections on a Revolution John Iliopoulos, reply by Sheldon Lee Glashow In response to “The Yang–Mills Model” (Vol. 5, No. 2). Internal Symmetries As Glashow points out, particle physicists distinguish To the editors: between space-time and internal symmetry transforma- tions. The first change the point of space and time, leaving Gauge theories brought about a profound revolution in the the fundamental equations unchanged. The second do not way physicists think about the fundamental forces. It is this affect the space-time point but transform the dynamic vari- revolution that is the subject of Sheldon Glashow’s essay. ables among themselves. This fundamentally new concept Gauge theories, such as the Yang–Mills model, use two was introduced by Werner Heisenberg in 1932, the year mathematical concepts: group theory, which is the natural the neutron was discovered, but the real history is more language to describe the physical property of symmetry, complicated.3 Heisenberg’s 1932 papers are an incredible and differential geometry, which connects in a subtle way mixture of the old and the new. For many people at that symmetry and dynamics. time, the neutron was a new bound state of a proton and Although there exist several books, and many more an electron, like a small hydrogen atom. Heisenberg does articles, relating historical aspects of these theories,1 a not reject this idea. Although for his work he considers real history has not yet been written. It may be too early. the neutron as a spin one-half Dirac fermion, something When a future historian undertakes this task, Glashow’s incompatible with a proton–electron bound state, he notes precise, documented, and authoritative essay will prove that “under suitable circumstances [the neutron] can invaluable. A large part is written in the first person, since break up into a proton and an electron in which case the the author is one of the main actors in this fantastic play. conservation laws of energy and momentum probably do I cannot think of any nontrivial comments, let alone any not apply.”4 In the β-decay controversy that opposed Wolf- criticism, so I will add my views on the subject trying to gang Pauli, who postulated the existence of a neutrino, to avoid duplication. Niels Bohr, who advocated non-conservation of energy Most revolutions in physics occur when an unexpected and momentum, Heisenberg does not take any clear experimental result contradicts current theoretical beliefs. stand, but he sides more with his master Bohr than with But the revolution that brought geometry into physics had his friend Pauli: “The admittedly hypothetical validity of a theoretical, and I would say aesthetic, motivation. It Fermi statistics for neutrons as well as the failure of the came from a few theorists who tried to go beyond a simple energy law in β-decay proves the inapplicability of pres- phenomenological model. ent quantum mechanics to the structure of the neutron.” Most of the time, these theorists played the leading In fact, Heisenberg’s fundamental contribution should be role. Glashow is one of them. As he explains, an import- appreciated not despite these shortcomings, but precisely ant motivation came from the study of what are called because of them. It should be remembered that in 1932, weak interactions. It may sound strange that a revo- experimental data on nuclear forces were almost entirely lution in particle physics was initiated by the weakest absent. Heisenberg had to guess the values of the nuclear among the forces,2 but this happened often before: weak attractive forces between nucleon pairs by using a strange interactions triggered many such revolutions. Maybe we analogy with molecular forces. He postulated a p – n and should meditate on the fundamental significance of tiny an n – n nuclear force, but not a p – p one, so his theory effects. was not really isospin invariant. Nevertheless, he made the In this letter, I touch on several aspects of this revolu- conceptual step to describe protons and neutrons together tion and bring some additional elements of history. These and to introduce the idea of rotations in an abstract space include internal symmetries, gauge theories, and the that would interchange a proton and a neutron.5 In the theory of weak interactions. following years, three important developments allowed 1 / 6 LETTERS TO THE EDITORS Heisenberg’s initial suggestion to become a complete iso- of the wave function in the Schrödinger theory implies spin invariant theory. the introduction of an electromagnetic field.8 Naturally, The first, and probably the most important, was the one would expect non-Abelian gauge theories to be con- progress in experimental techniques, which brought more structed following the same principle immediately after detailed and more precise data. They showed the need for isospin symmetry was established in the 1930s. But here the introduction of a p – p force and confirmed the charge history took an unexpected route. independence of all nuclear forces. The development of the general theory of relativity The second was the 1933 formulation by Enrico Fermi offered a new paradigm for a gauge theory. In the next of a theoretical model for the amplitude of neutron β-de- decades, it became the starting point for all studies on the- cay. The influence of this work by far exceeds this initial ories invariant under local transformations trying to unify problem and covers subjects even beyond the theory of gravity and electromagnetism, the only forces known at elementary particles. Since that time, quantum field theory the time. Theodor Kaluza’s attempt, completed by Oskar has become the universal language and one of the corner- Klein,9 is today often used in supergravity and superstring stones of modern theoretical physics. To every particle theories. corresponds a quantum field that describes the excitations Particle physicists date the birth of non-Abelian gauge that physicists call particles. theories to 1954, with the publication of the fundamental The third was Hideki Yukawa’s introduction of the paper by Chen Ning Yang and Robert Mills.10 The impact meson as an intermediary for the nuclear forces. In 1938, of this work on high-energy physics has often been empha- Nicholas Kemmer had the idea to incorporate Yukawa’s sized, but here I want to mention some earlier and little meson into Heisenberg’s isospin formalism and write the known attempts which, according to present views, have first fully isospin invariant pion–nucleon interaction. He followed a very strange route. assumed the existence of three pions with charges +1, –1, The first is due to Klein. In an obscure conference in and 0, which he grouped together in an isospin triplet. In 1938, he presented a paper with the title “On the Theory 1938, there was evidence for the existence of a charged of Charged Fields,” in which he attempts to construct an Yukawa meson, although it was the wrong one, but there SU(2) gauge theory for the nuclear forces.11 This paper was no evidence for a neutral one. Kemmer understood follows an incredibly circuitous road: Klein considers that isospin symmetry required such a neutral meson. In general relativity in a five-dimensional space, he compac- 0 1950, π was discovered at the Berkeley electron synchro- tifies à la Kaluza–Klein, but he takes the g4μ components of tron by Jack Steinberger, Wolfgang Panofsky, and Jack the metric tensor to be 2 × 2 matrices. In spite of several Steller. Thus π0 is the first particle whose existence was technical problems, he finds the correct expression for the predicted as a requirement for an internal symmetry and field strength tensor of SU(2). He was adding mass terms the first to be discovered in an accelerator.6 by hand, and it is not clear whether he worried about the It took six years, as well as the work of many physicists, resulting breaking of gauge invariance. It is not known for Heisenberg’s original suggestion of 1932 to become the whether this paper has inspired anybody else’s work, and full isospin symmetry of hadronic physics we know today. Klein himself mentioned it only once in a 1955 conference It is a remarkably short time, given the revolutionary nature in Bern.12 of the idea. The conceptual change has been very import- The second work in the same spirit is due to Pauli, who ant: for the first time, nontrivial internal symmetries have in 1953, in a letter to Abraham Pais,13 developed precisely been considered in physics. Heisenberg’s isospin space this approach: the construction of the SU(2) gauge theory was three-dimensional, and the transformations look like as the flat space limit of a compactified higher dimensional familiar rotations. The concept was subsequently enlarged theory of general relativity. He was closer to the approach as new particles were discovered and larger internal sym- followed today because he considered a six-dimensional metry groups were brought into evidence. The space for theory with the compact space forming an S2. He never elementary particle physics became a multi-dimensional published this work, and we do not know whether he was manifold, with complicated geometrical and topological aware of Klein’s 1938 paper. He had realized that a mass properties, and only a subspace of it, the four-dimensional term for the gauge bosons breaks the invariance, and he Minkowski space, is directly accessible to our senses. had an animated argument during a seminar by Yang at the Institute for Advanced Studies in Princeton in 1954.14 Gauge Theories and Geometry What is certainly surprising is that both Klein and Pauli, fifteen years apart one from the other, decided to con- The origins of gauge theory can be traced back to classical struct the SU(2) gauge theory for strong interactions and electrodynamics,7 but the importance it has acquired today both chose to follow this totally counterintuitive method.