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*by Roberto Lalli***Eugene Paul (Jenő Pál) Wigner ****Nobel Prize in Physics 1963 ****"for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles".**

Jenő Pál Wigner was a Hungarian-born theoretical physicist whose groundbreaking work has deeply affected several branches of physics. He pioneered the application of group theory to quantum mechanics as well as the application of quantum mechanics to quantum field theory, solid-state physics, physical chemistry, and nuclear physics. Born in 1902 in Budapest from a bourgeois Jewish family, Wigner followed a life path very similar to that of other Hungarian Jewish physicists of his generation who had a far-reaching impact on the development of nuclear energy and nuclear weapons in the United States during the Second World War: the mathematician John von Neumann, and the physicists Edward Teller and Leó Szilárd. The four scientists all showed an exceptional mathematical talent and all built the early stage of their scientific and academic career in Germany. All of them left Germany between 1930 and 1933 due to the increasing atmosphere of anti-Semitism and to the political successes of Adolf Hitler and National Socialism. All of them eventually played key, albeit different, roles in the launch and developments of the Manhattan Project - the American program for the construction of the first atomic bombs - as well as in the American nuclear arms race during the Cold War era and the related debates on nuclear proliferation. **On the Relevance of Symmetry in Quantum Mechanics**

Before dramatic political, economic and social transformations forced Wigner to leave Germany permanently in 1930, he had already set the building blocks of his most fundamental contributions to theoretical physics: By applying group-theoretical methods to quantum mechanics, Wigner demonstrated that symmetry principles could be employed to describe the behaviour of many-body quantum systems. Although he was not alone in this endeavour, he was the first physicist to use group-theoretical arguments and symmetry properties to derive physical laws. At the time, group theory was a well-developed field in abstract mathematics, but physicists had only a superficial knowledge of its methods. Beginning in 1927, Wigner systematically employed group-theoretic techniques to solve problems concerning the application of the newly born quantum mechanics to poly-electronic atoms. Three related elements were crucial in shaping his singular approach to quantum mechanics. First, from 1921 to 1925 he had studied chemical engineering at the *Technische Universität Berlin*, where he had the possibility to work at the *Kaiser-Wilhelm-Institut für Physikalische Chemie und Elektrochemie *under the supervision of the Hungarian chemist and philosopher of science Michael Polanyi. During this experience, Wigner began working on the symmetry properties of crystals from a theoretical perspective. Second, soon after having received his doctorate, Wigner became research assistant of Karl Weissenberg, mathematician and expert of X-ray crystallography, at the *Berliner Universität *(today *Humboldt-Universität zu Berlin*) from 1926 to 1927. Weissenberg assigned Wigner the task to obtain a general and elegant demonstration of the symmetry properties of crystals by applying group theory. The third element was Wigner’s friendship with von Neumann, who introduced Wigner to the more recent advancements of group theory and with whom Wigner co-authored three fundamental papers of applied group theory in 1928.

In mid-1926, Werner Heisenberg had already showed that the solutions of the Hamiltonian of the helium atom were symmetric with respect to the exchange of the positions of the two electrons. This was the first step in the quantum mechanical treatment of many-body systems through the study of the symmetries of the wave functions with respect to permutations of the individual systems. A few months later, Dirac proved the connection between the symmetric properties of the wave function and the statistics of n identical particles: If the wave function is symmetric the related particles follow the Bose-Einstein statistics, while if the wave function is anti-symmetric the Pauli exclusion principle holds and the particles follow what was later called the Fermi-Dirac statistics. Starting from the papers of Heisenberg and Dirac, Wigner’s earliest great contribution was to base the study of the n-electron atoms and their spectra on the more recent advancement of group theory. Thanks to the suggestions and the active collaboration of von Neumann, Wigner became acquainted with the representation theory of groups developed between the end of the 19th century and the early 20th century. This theory allows the study of the features of an abstract group through the analysis of the representation of its elements as linear transformations of vector spaces. Between 1927 and 1931, Wigner used the representation of a group onto a matrix group to derive several properties of the atomic spectra. Wigner’s procedure started by classifying the groups of transformations that left the Hamiltonian invariant (the symmetry groups), which included permutation of electrons of the atom, coordinate rotations in 3-dimensional space around the nucleus if no external field is present (or rotations around the axis of the field if a constant external field is present) and spatial reflections. The irreducible representations of these symmetry groups allowed Wigner to find the invariant subspaces and, consequently, label the eigenstates characterizing the atomic states. Among the main achievements of Wigner’s program were the derivations of several properties of atomic spectra, including the intensities and selection rules, the displacements of the Stark effect, the Landé g factor and the Laporte selection rule.

Apart from these specific applications, Wigner’s main target was the rigorous formalization of the group-theoretical methodology for quantum theory. Many mathematicians and mathematically minded physicists and chemists soon recognized the validity of this approach and made their own contributions. In particular, the mathematician Hermann Weyl published in 1928 the first book dealing with the applications of group theory to quantum mechanics, and in the same year chemists Walter Heitler and Fritz London began employing group-theoretical considerations to the analysis of the covalent bond and the stability of molecules. In spite of the successes of group-theoretical arguments, the abstraction of the theory made it incomprehensible to many physicists. Within the scientific community, many did not appreciate the abstract mathematical turn physics was taking and saw group theory as an unphysical theory. Some physicists even stigmatized the spread of this approach as “gruppenpest” (group plague). Since the late 1920s, alternative and simplified methods for explaining the behaviour of many-body systems and their spectra were proposed, which found the favour of a great number of physicists who were reluctant to employ group theory in their calculations. For this reason, in 1931 Wigner published his book *Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren*, which incorporated all the results of his previous papers on mono-atomic systems and which was more accessible to experimental physicists than other accounts of applied group theory, including the more mathematically rigorous approach published by Weyl. Although Wigner’s book was instrumental in spreading and promoting the group theoretical tools among a broader physics community, most physicists continued to avoid employing the group-theoretical points of view. This situation changed only after the Second World War, when group theory gradually became the fundamental mathematical language of theoretical nuclear physics and elementary particle physics. Starting from the late 1940s, symmetry considerations began to have a central role in the advancement of theories. Clear indications that the group theoretical approach gained momentum within the physics community only after the late 1940s are that Wigner’s 1931 book was translated into English only in 1959, and that he was not nominated for the Nobel Prize before 1950. In fact, his group theoretical works were not much quoted in the physics literature until the Italian-Israeli physicist Giulio Racah made group theory a fundamental tool of theoretical physics by demonstrating in 1949 that group theory greatly simplified the analysis of complex spectra. The clearest sign of this transformation is the 1957 Nobel Prize in Physics awarded to the Chinese-born American physicists Chen Ning Yang and Tsung-Dao Lee “for their penetrating investigation of the so-called parity laws which has led to important discoveries regarding the elementary particles.” In his Nobel lecture, Yang recognized Wigner’s groundbreaking work in the demonstration that the Laporte rule is a consequence of the reflection invariance (parity symmetry) of the electromagnetic forces in the atom. Beginning from the late 1950s, Wigner began receiving several important recognitions for his studies of symmetrical structures of physical systems, and in 1963 he was awarded one of the Nobel Prizes in Physics “for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles.”

**Spreading the Group-theoretic Tools in Physics Before the Second World War**

After having framed his program for the application of irreducible representations of symmetry groups to quantum mechanics, Wigner made fundamental contributions to several branches of theoretical physics, including quantum field theory, nuclear physics, solid-state physics, and physical chemistry. Wigner became a founding father of all these fields by playing a pivotal role between the late 1920s and the late 1930s. In particular, he regarded symmetry principles as fundamental instruments in order to derive physical laws – a conviction that continued to shape his approach to scientific problems.

In 1928, Wigner co-authored with Pascual Jordan a paper that had a deep influence on the development of quantum field theory. In 1927, Dirac had quantized the electromagnetic field and obtained the commutation rules governing the creation and annihilation of photons – a work Jordan and Oskar Klein had generalized to many-boson systems by taking into account Coulomb interactions between charged bosons. Jordan then asked Wigner for help in constructing the formalism for the quantization of fields whose quanta were fermions (namely, particles whose statistics is governed by the Pauli exclusion principle). By applying group-theoretical arguments to the quantization of the non-relativistic Schrödinger field of fermions, Jordan and Wigner derived the two operators that increase and decrease the number of particles in a certain quantum state (the creation and annihilation operators) and demonstrated that these operators anti-commute in the case of fermions, while those of bosons commute. This paper, along with other works by Jordan in the same period, laid the foundation of the quantum field theory, which, in turn, came to be the theoretical framework of the unified theories of particle physics and condensed matter physics formalized from the 1950s till the late 1970s.

The field in which Wigner made more contributions in this period was theoretical nuclear physics. The discovery of the neutron in 1932 led several theoretical physicists to develop models concerning the neutron-proton (n-p) interactions in the nucleus. The first physicist to elaborate a quantum theoretical model of nuclear forces was Werner Heisenberg, the very same year of the discovery of the neutron. In Heisenberg’s nuclear model, the mechanism underlying the attraction between nucleons resembled the mechanism already developed for explaining the forces binding atoms in a molecule - namely, exchange forces depending on the sharing of electrons. More specifically, Heisenberg assumed three different forces: The n-p attractive force depended on the sharing of one electron; the weaker attraction between two neutrons (n-n) came from the exchange of two electrons; lastly, the force between two protons (p-p) was simply the Coulomb repulsive force. Heisenberg’s model had several limits, but it represented the first non-relativistic quantum mechanical model of the nucleus and set the stage for further developments. Wigner soon joined the field by investigating the n-p scattering, and the measurements of mass and spin of the newly discovered deuteron. During this early stage, Wigner proposed a model for the forces between nucleons that came to be known as Wigner forces. Wigner’s model represents one of the four different descriptions of nuclear forces elaborated by physicists in the early 1930s (the others being Heisenberg’s model, and the nuclear potentials developed by Ettore Majorana in 1932 and James H. Bartlett in 1936). Although Wigner’s model was soon discarded, the German-born nuclear physicist Hans Bethe considered Wigner’s early papers as fundamental in the development of the field because they showed that nuclear forces had to be strong and short-ranged. In the mid-1930s, along with Russian-born American physicist Gregory Breit, Wigner also published a number of papers on the mechanisms of nuclear reactions. In these papers, the authors developed a tool of calculation that is still standard in nuclear physics, called the Breit-Wigner distribution. In their model of neutron-nucleus scattering, Breit and Wigner put forward a model of the interaction that had many similarities with the compound nucleus proposed by Bohr in the same period and which became the predominant model for several years. Perhaps, Wigner’s most fundamental contributions to theoretical nuclear physics concerned the applications of group theory and symmetry principles to nuclear states. To describe the n-p exchange forces, Heisenberg had introduced a third quantum number w whose value depended on the charge of the nucleon: It was +1/2 for protons (positive charge) and -1/2 for neutrons (null charge). Around the mid-1930s, experiments had showed that the forces between nucleons were independent of the types of nucleons; namely, p-p interactions were of the same type and had the same strength as n-p and n-n interactions, contrary to what Heisenberg had assumed. Beginning in 1937, Wigner published a series of papers in which he used the invariance of the nuclear Hamiltonian with respect to the exchange of charge and spin between nucleons to derive fundamental properties of the nuclear wave equations. In his paper “On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei,” Wigner demonstrated that the quantum number w proposed by Heisenberg (which Wigner christened “isotopic spin”) was fundamental in order to formulate the group theoretical treatment of nuclear forces as well as to study their energy levels. Wigner’s treatment made it clear that protons and neutrons might be understood as two different states of the same particle with respect to the nuclear force (now strong force) and that their different charge depended on a different value of the isotopic spin. Since then, isospin (which is how isotopic spin is currently called) has become a central aspect in the development of the quantum field theories of fundamental forces. This concept is at the base of what is now called flavour symmetry on which the quark model developed in 1968 is based.

In 1933, Wigner began contributing to the collective efforts to establish the building blocks of the quantum theory of solids, whose main target was to explain microscopically well-known macroscopic phenomena by the new theoretical tools of quantum mechanics and quantum statistics. Principally based on the theory of conduction in metals elaborated by Felix Bloch in 1928, the theory described conduction electrons in terms of the occupation of series of energy bands. As promising as this model was in explaining various properties of metals and the phenomenological distinctions between conductors and insulators, until 1933 no simple procedure to calculate the band energies of real substances had been elaborated. Because of the regularities of crystals, solid-state-physics was a suitable field for applying the symmetry principles. Starting from this approach, Wigner and his first doctoral student, Frederick Seitz, developed a simplified computational model that allowed the calculation of the band structure of sodium. The agreement between their calculations and the observed properties of sodium was surprising and the authors proposed a program to further elaborate their method. Their two-part paper “On the constitution of real sodium” paved the way for the study of the quantum properties of real materials and had an extraordinary impact on the studies of the electronic structure of metals.

In the same decades, Wigner occupied himself with the theory of chemical reactions. His interest in this field began very early, when he was working on his doctoral dissertation with Polanyi in 1923. In the early 1930s, Wigner made important contributions to the theory of the rates of elementary chemical reactions called the Transition State Theory (TST), developed in the same period by Polanyi and Henry Eyring. In spite of some limitations, the theory is still employed for understanding in a qualitative way how reactions occur. The TST allows for a calculation of the reaction rates in case the quantum states of the electrons do not change during the reaction (adiabatic case). Using classical statistics Wigner correctly drew the approximate shape of the potential of the reaction. He introduced a saddle point that described the value of the activation energy of the reaction as function of the distance between the nuclei forming the molecules. The importance of Wigner’s papers on chemical reactions was not limited to the formulation of the potential curve. Wigner divided the problem of chemical reaction in a convenient way into sections that concerned classical statistics and parts that should be addressed through quantum considerations. In this way, his papers provided a useful approach that directed further researches. In a more indirect way, Wigner’s studies of chemical reactions gave a framework for developing models for nuclear reactions in the late 1930s.

Wigner’s program of formulating a general group theoretical approach to physical problems made an ultimate step forward in 1939 when he first considered the invariance of the Hamiltonian with respect to Lorentz transformations. From a group-theoretic perspective the problem was to find the unitary representations of the inhomogeneous Lorentz group, also known as the Poincaré group. This group includes time and spatial translations, 3-dimensional space rotations and pure Lorentz transformations (Lorentz boosts). This problem was not new. Ettore Majorana and, after him, Paul Dirac had already tried to formulate a quantum-relativistic theory of particles. Wigner’s approach in his epoch-making paper “The Unitary Representations of the Inhomogenous Lorentz Group” was more general and became a standard reference after the Second World War, when a new generation of theoretical physicists reconsidered group theory in order to formulate quantum-relativistic theories of particle interactions. In this paper, Wigner put forward a classification of all the irreducible representations of the Poincaré group and a description of their infinitesimal generators. The major outcome of the paper was to demonstrate that an elementary system has no other invariant characteristics besides mass and intrinsic spin, if its mass is non-null, and helicity (projection of the spin along the direction of the momentum) if its mass is null. Wigner’s approach was very modern, but, as it happened with Wigner’s previous works on group theory, his paper had a delayed impact. Wigner had to publish it as a mathematical paper in the journal *Annals of Mathematics *because of its high mathematical content. Around forty years later, it became the standard reference for theoretical physics and contemporary textbooks still rely on it, although the notation is, of course, changed.**Protagonist of the Nuclear Era**

Word War II caused radical and dramatic transformations in several aspects of human life and in the relationships between nations. The word of physics did not go untouched either. Apart from the theoretical evolutions in nuclear physics, and the construction of new modern facilities for experimentation, physics practitioners saw a radical change in their organization. The target-focused and team-organized work that physicists made during the war for the race to the first atomic weapon shaped the way in which physicists did their activities. Los Alamos became an example of organization that accelerated the passage to the big-science way of doing research in large communities of researchers with different expertise connected by a strong organizational system and a common target. Wigner himself was one of the protagonists of the American nuclear program during and after the war and he, like others, modified his activities in relation with the social transformation of the physics world. During the war, Wigner was already recognized as one of the leaders in nuclear physics and he was asked to direct the theoretical team of the Metallurgical Laboratory of the University of Chicago. This group collaborated with the experimental group headed by Fermi. The two groups succeeded in initiating and controlling the first self-sustaining nuclear chain reaction on December 2, 1942. Wigner had to resolve specific problems in the endeavour to build the first atomic pile. More fundamentally, Wigner became a group leader and an expert of engineering problems related to nuclear reactors. His group designed the first nuclear reactor that was built by the American chemical company DuPont at Hanford, which produced the plutonium for the second atomic weapon dropped on Nagasaki on August 9, 1945. After the war, Wigner continued to hold important directive positions in nuclear research facilities becoming the first research director of the Oak Ridge National Laboratory. In 1947, he returned to Princeton University as a full professor, where he continued to make important contributions in several fields of physics and trained a large number of students.

After the war, Wigner contributed to the change in perspective of theoretical physicists who came to rely on group theory to advance the knowledge of fundamental forces. His broad views of scientific problems led Wigner to produce several philosophical works in which he tried to deepen the understating of the role of symmetry principles in physics. Wigner’s major philosophical works on symmetries were later collected in the book Symmetries and Reflections, which constitutes an invaluable example of the way in which philosophical considerations might be a guide in the evolution of theoretical methods. **Bibliography **

Bonolis, L. (2005). From the rise of the group concept to the stormy onset of group theory in the new quantum mechanics: A saga of the invariant characterization of physical objects, events and theories. Rivista del Nuovo Cimento 27: 39-109.

Borrelli, A. (2009). The emergence of selection rules and their encounter with group theory, 1913–1927. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 40: 327-337.

Chayut, M. (2001). From the Periphery: the genesis of Eugene P. Wigner's application of group theory to quantum mechanics. Foundations of Chemistry 3: 55-78.

Seitz, F., Vogt, E., and Weinberg. A. M. (2000) Eugene Paul Wigner. 17 November 1902-1 January 1995. Biographical Memoirs of Fellows of the Royal Society 46: 579-592.

Gross, D. J. (1995). Symmetry in Physics: Wigner's Legacy. Physics Today, 48, 46-50.

Westfall, C. (2008). Wigner, Eugene Paul. New Dictionary of Scientific Biography 25: 293-297.

Wigner, E. P. (1992). The Recollections of Eugene P. Wigner: As Told to Andrew Szanton. New York: Basic Books.

Wightman, A. (1993-1997). The Collected Works of Eugene Paul Wigner: The Scientific Papers, Vols. 1-4. Berlin, Springer.

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