Editing Eugene Wigner (section)

==Middle years==
Wigner returned to Budapest, where he went to work at his father's tannery, but in 1926, he accepted an offer from [[Karl Weissenberg]] at the [[Max Planck Institute for Physics|Kaiser Wilhelm Institute]] in Berlin. Weissenberg wanted someone to assist him with his work on [[X-ray]] [[crystallography]], and Polanyi had recommended Wigner. After six months as Weissenberg's assistant, Wigner went to work for [[Richard Becker (physicist)|Richard Becker]] for two semesters. Wigner explored [[quantum mechanics]], studying the work of [[Erwin Schrödinger]]. He also delved into the [[group theory]] of [[Ferdinand Georg Frobenius|Ferdinand Frobenius]] and  [[Eduard Ritter von Weber]].{{sfn|Szanton|1992|pp=101–106}}

Wigner received a request from [[Arnold Sommerfeld]] to work at the [[University of Göttingen]] as an assistant to the great mathematician [[David Hilbert]]. This proved a disappointment, as the aged Hilbert's abilities were failing, and his interests had shifted to logic. Wigner nonetheless studied independently.{{sfn|Szanton|1992|pp=109–112}} He laid the foundation for the theory of symmetries in quantum mechanics and in 1927 introduced what is now known as the [[Wigner D-matrix]].<ref>{{cite journal |last=Wigner |first=E. |year=1927 |title=Einige Folgerungen aus der Schrödingerschen Theorie für die Termstrukturen  |language=de|journal=[[Zeitschrift für Physik]] |volume=43 |issue=9–10 |pages=624–652 |doi=10.1007/BF01397327 |bibcode = 1927ZPhy...43..624W |s2cid=124334051 }}</ref> Wigner and [[Hermann Weyl]] were responsible for introducing group theory into quantum mechanics. The latter had written a standard text, ''Group Theory and Quantum Mechanics'' (1928), but it was not easy to understand, especially for younger physicists. Wigner's ''Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra'' (1931) made group theory accessible to a wider audience.{{sfn|Szanton|1992|pp=116–119}}
[[File:Jucys diagram for Wigner 6-j symbol.svg|thumb|[[Jucys diagram]] for the Wigner [[6-j symbol]].  The plus sign on the nodes indicates an anticlockwise reading of its surrounding lines.  Due to its symmetries, there are many ways in which the diagram can be drawn.  An equivalent configuration can be created by taking its mirror image and thus changing the pluses to minuses.]]

In these works, Wigner laid the foundation for the theory of [[symmetry in physics|symmetries]] in [[quantum mechanics]].<ref>{{cite journal |last=Wightman |first=A.S. |year=1995 |url=https://www.ams.org/notices/199507/wigner.pdf |title=Eugene Paul Wigner 1902–1995   |journal=[[Notices of the American Mathematical Society]] |volume=42 |issue=7 |pages=769–771}}</ref> [[Wigner's theorem]], proven by him in 1931, is a cornerstone of the [[mathematical formulation of quantum mechanics]]. The theorem specifies how physical [[symmetries]] such as rotations, translations, and [[CPT symmetry]] are represented on the [[Hilbert space]] of [[Quantum state|states]]. According to the theorem, any symmetry transformation is represented by a [[unitary transformation|linear and unitary]] or [[antiunitary operator|antilinear and antiunitary]] transformation of Hilbert space. The representation of a symmetry group on a Hilbert space is either an ordinary [[representation (group theory)|representation]] or a [[projective representation]].{{sfn|Wigner|1931|pp=251–254}}{{sfn|Wigner|1959|pp=233–236}}

In the late 1930s, Wigner extended his research into atomic nuclei. By 1929, his papers were drawing notice in the world of physics. In 1930, [[Princeton University]] recruited Wigner for a one-year lectureship, at 7 times the salary that he had been drawing in Europe. Princeton recruited von Neumann at the same time. Jenő Pál Wigner and János von Neumann had collaborated on three papers together in 1928 and two in 1929. They anglicized their first names to "Eugene" and "John", respectively.{{sfn|Szanton|1992|pp=127–132}} When their year was up, Princeton offered a five-year contract as visiting professors for half the year. The Technische Hochschule responded with a teaching assignment for the other half of the year. This was very timely, since the [[Nazi]]s soon rose to power in Germany.{{sfn|Szanton|1992|pp=136, 153–155}} At Princeton in 1934, Wigner introduced his sister Margit "Manci" Wigner to the physicist [[Paul Dirac]], with whom she remarried.{{sfn|Szanton|1992|pp=163–166}}

Princeton did not rehire Wigner when his contract ran out in 1936.{{sfn|Szanton|1992|pp=171–172}} Through [[Gregory Breit]], Wigner found new employment at the [[University of Wisconsin–Madison|University of Wisconsin]]. There, he met his first wife, Amelia Frank, who was a physics student there. However, she died unexpectedly in 1937, leaving Wigner distraught. He therefore accepted an offer in 1938 from Princeton to return there.{{sfn|Szanton|1992|pp=173–178}} Wigner became a [[naturalized citizen]] of the United States on January 8, 1937, and he brought his parents to the United States.{{sfn|Szanton|1992|pp=184–185}}