wigner.qxp 4/27/98 10:06 AM Page 769 Eugene Paul Wigner 1902–1995 A. S. Wightman Eugene Wigner died in Princeton, NJ, on Janu- as a physicist, assistant to the physicist Richard ary 1, 1995, at the age of 92. He had been one Becker. of the last survivors of the generation that wit- During the next decade and a half Wigner nessed the creation of quantum mechanics and continued his study of the theory of chemical re- participated in the exciting initial years of its de- actions but used the then new quantum me- velopment. He spent most of six active decades chanics. He did related work with Victor Weis- on the faculty of Princeton University. Although skopf on the theory of line breadth in atomic he was best known for his physical and mathe- spectra as well as a study of nuclear reaction matical analyses of symmetry in quantum me- rates with Gregory Breit. However, the main chanics, he also made important contributions focus of his effort was in the application of to solid-state physics, physical chemistry, nuclear group theory to the study of the symmetry prop- engineering, and epistemology. In his later years, erties of stationary states of atoms, molecules, he found himself in the unusual position of atomic nuclei, and crystals. being highly esteemed by physicists, mathe- It was also during this period that Wigner maticians, chemists, engineers, and philoso- made a transition from Germany to the United phers. States. From 1930 to 1933 Wigner and von Neu- Eugene Wigner was born Jenö Pál Wigner in mann had a common arrangement: they spent Budapest, Hungary, on November 17, 1902. Since one term each year at their jobs in Berlin and one he was a somewhat sickly child, his parents at Princeton University. In the spring of 1933 the arranged for his early education to occur at National Socialists came to power in Germany, home. However, later on he spent four years at and the Berlin positions of von Neumann and the famous Lutheran gymnasium (high school) Wigner vanished. Von Neumann then joined the of Budapest, where he had the good fortune to faculty of the new Institute for Advanced Study. have as friend and classmate (one class behind Wigner spent three years full-time in Princeton him) Jancsi (=Johann=John) von Neumann. and then went to Wisconsin for two years. In the Wigner was attracted by mathematics and fall of 1938 he was back in Princeton in an en- physics, but, following his father’s wish that he dowed professorship, just in time to hear the study something that could be useful in the news of the discovery of nuclear fission, a phe- leather tannery where his father was a foreman, nomenon whose consequences dominated the Wigner got a degree in chemical engineering next decade of his life. from the Technische Hochschule in Berlin. His Wigner and his friend Leo Szilard foresaw as thesis (1925), written under the supervision of clearly as anyone the disastrous consequences Michael Polanyi, was on the theory of chemical of the Third Reich’s acquiring nuclear weapons reactions. Wigner’s acumen so impressed Polanyi before the Allies. They persuaded Albert Einstein that he recommended him for his first position to write a letter alerting President Roosevelt. The result was the Manhattan Project, a large- 235 238 A. S. Wightman is professor emeritus of mathematical scale effort to separate U from U in ura- physics at Princeton University, Princeton, NJ. nium ore and to create nuclear reactors to pro- JULY 1995 NOTICES OF THE AMS 769 wigner.qxp 4/27/98 10:06 AM Page 770 duce plutonium as well as to design bombs However, he found the case of n 4 electrons which used these products as explosives. Wigner too complicated to do by hand. On≥ the advice of was heavily involved in design studies for the nu- von Neumann, he studied the pre-World War I clear reactors. Most of the work took place in the papers of Frobenius, Schur, and Burnside on the mathematics department of the University of representation theory of finite groups, as well Chicago, code-named the Metallurgical Labora- as the later papers of Weyl and of Schur on con- tory. It was here that Wigner acquired his repu- tinuous groups. The latter enabled him to enlarge tation as a formidable nuts-and-bolts engineer. his study to the consideration of the action on When his design work for the plutonium pro- eigenfunctions, of rotations R of the coordi- duction reactors was done, Wigner turned to nates of n electrons ~x1, ..., ~xn : the design of power reactors. This continued ~x1, ...~xn R~x1, ...R~xn. after the war; he spent 1946–47 as director of → research at Oak Ridge on leave from Princeton. He recognized that if the Hamiltonian commutes Wigner and his coworker Alvin Weinberg col- with the action on wave functions of permuta- lected their knowledge and tions of coordinates or with experience in the definitive the action on wave func- treatise, The Physical The- tions of rotations of coor- ory of Neutron Chain Re- dinates, then the linear actors (1958). subspace spanned by the He returned to academic Although he was eigenfunctions of a fixed life in 1947 but, over the best known for his eigenvalue is left invariant next three decades, often by these actions and, in the served as a consultant to physical and subspace, yields a unitary the federal government in representation of the per- nuclear matters. Both his mathematical mutation and the rotation scientific achievements and group. Such a representa- his service were richly rec- analyses of tion is a direct sum of ir- ognized with prizes and symmetry in reducibles, so the dimen- awards, of which only a few sion of the linear space will be mentioned: Medal quantum spanned by the eigenfunc- for Merit, 1946; Fermi tions must be a sum with Award, 1958; Max Planck mechanics, he also possible multiplicities of Medal, 1961; Nobel Prize made important the dimensions of the irre- for Physics, 1963; National ducible representations of Medal of Science, 1969. contributions to the groups. This is the ele- After his retirement in mentary group theoretical 1971, he remained active solid-state physics, explanation for the ubiq- until the late 1980s. uitous appearance in In the remainder of this physical chemistry, atomic physics of degen- obituary I have singled out nuclear erate multiplets of multi- for discussion a few points plicity 2j +1 where j is a from Wigner’s work on the engineering, and positive integer or half-odd application of group the- integer. Later on, in the ory to the study of sym- epistemology. context of nuclear physics, metry in quantum me- this argument led to a the- chanics. More details and ory of super-multiplets in more complete coverage which the group SU(2) is are to be found in Volume I replaced by the group of Wigner’s Collected Works in the magistral an- SU(4). notations of Brian Judd, Part II “Applied Group Up to this point, Wigner had treated the un- Theory 1926–1935”, and George W. Mackey, Part physical case of spinless electrons. However, in III “Mathematical Papers”. the very same issue of the Zeitschrift für Physik Wigner began his study of symmetry in quan- in which he had published these considerations, tum mechanics with the problem of classifying there appeared Pauli’s paper on nonrelativistic the behavior of eigenfunctions of the Schrödinger electrons with spin 1/2. Within a few months von Hamiltonians for atoms under permutations of Neumann and Wigner published the first of three the electrons. Inspired by a paper of Heisenberg papers generalizing everything that Wigner had on helium (two electrons), he first treated the done to the case of spin 1/2 Pauli electrons. case of three electrons, without recourse to the These papers were not easy to read, and it seems representation theory of the permutation group. plausible that they, together with Hermann 770 NOTICES OF THE AMS VOLUME 42, NUMBER 7 wigner.qxp 4/27/98 10:06 AM Page 771 Weyl’s classic book Gruppentheorie und Quan- tenmechanik (1928) were the origin of the use by physicists of the phrase “die Gruppenpest” to describe this approach to spectroscopy. (The German word is often translated “group pest”, but the alternative “group plague” is probably better, if you take into account some of the pas- sionate animadversions on group theory by physicists in those days.) In any case, it is in- structive to compare Weyl’s book with Wigner’s Gruppentheorie und Ihre Anwendung auf die Quantenmechanik der Atomspektren (1931). Both books have introductory chapters on linear transformations, groups, and quantum mechanics. Here Weyl puts more emphasis on vector spaces; Wigner on calculations with ma- trices. Wigner confines his attention to the per- mutation group and rotation group or its cov- ering group SU(2). However, he goes into much more detail on the representation theory of these groups. For example, he derives the Wigner Eckart formula for the matrix elements of ten- sor operators. This permits him to derive the in- tensity relations for spectral lines that follow from rotation invariance. He also gives a quan- titative and general analysis of the splitting of spectral lines in the presence of external sym- metry-breaking interactions. Weyl, on the other leaving a vector of mass 2 = m2 fixed. When hand, discusses the Lorentz group, its covering m2 > 0, the little group is isomorphic to SO(3) group SL(2,C), and the relation of their finite- or SU(2) and so is determined by a positive in- dimensional representations to quantum field teger or half-odd integer, the spin.