Particle physics and representation theory

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https://en.wikipedia.org/wiki/Particle_physics_and_representation_theory 

 

There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner. It links the properties of elementary particles to the structure of Lie groups and Lie algebras. According to this connection, the different quantum states of an elementary particle give rise to an irreducible representation of the Poincaré group. Moreover, the properties of the various particles, including their spectra, can be related to representations of Lie algebras, corresponding to "approximate symmetries" of the universe.

General picture

Symmetries of a quantum system

In quantum mechanics, any particular one-particle state is represented as a vector in a Hilbert space ${\displaystyle {\mathcal {H}}}$. To help understand what types of particles can exist, it is important to classify the possibilities for ${\displaystyle {\mathcal {H}}}$ allowed by symmetries, and their properties. Let ${\displaystyle {\mathcal {H}}}$ be a Hilbert space describing a particular quantum system and let ${\displaystyle G}$ be a group of symmetries of the quantum system. In a relativistic quantum system, for example, ${\displaystyle G}$ might be the Poincaré group, while for the hydrogen atom, ${\displaystyle G}$ might be the rotation group SO(3). The particle state is more precisely characterized by the associated projective Hilbert space ${\displaystyle \mathrm {P} {\mathcal {H}}}$, also called ray space, since two vectors that differ by a nonzero scalar factor correspond to the same physical quantum state represented by a ray in Hilbert space, which is an equivalence class in ${\displaystyle {\mathcal {H}}}$ and, under the natural projection map ${\displaystyle {\mathcal {H}}\rightarrow \mathrm {P} {\mathcal {H}}}$, an element of ${\displaystyle \mathrm {P} {\mathcal {H}}}$. 

By definition of a symmetry of a quantum system, there is a group action on ${\displaystyle \mathrm {P} {\mathcal {H}}}$. For each ${\displaystyle g\in G}$, there is a corresponding transformation ${\displaystyle V(g)}$ of ${\displaystyle \mathrm {P} {\mathcal {H}}}$. More specifically, if ${\displaystyle g}$ is some symmetry of the system (say, rotation about the x-axis by 12°), then the corresponding transformation ${\displaystyle V(g)}$ of ${\displaystyle \mathrm {P} {\mathcal {H}}}$ is a map on ray space. For example, when rotating a stationary (zero momentum) spin-5 particle about its center, ${\displaystyle g}$ is a rotation in 3D space (an element of ${\displaystyle \mathrm {SO(3)} }$), while ${\displaystyle V(g)}$ is an operator whose domain and range are each the space of possible quantum states of this particle, in this example the projective space ${\displaystyle \mathrm {P} {\mathcal {H}}}$ associated with an 11-dimensional complex Hilbert space ${\displaystyle {\mathcal {H}}}$. 

Each map ${\displaystyle V(g)}$ preserves, by definition of symmetry, the ray product on ${\displaystyle \mathrm {P} {\mathcal {H}}}$ induced by the inner product on ${\displaystyle {\mathcal {H}}}$; according to Wigner's theorem, this transformation of ${\displaystyle \mathrm {P} {\mathcal {H}}}$ comes from a unitary or anti-unitary transformation ${\displaystyle U(g)}$ of ${\displaystyle {\mathcal {H}}}$. Note, however, that the ${\displaystyle U(g)}$ associated to a given ${\displaystyle V(g)}$ is not unique, but only unique up to a phase factor. The composition of the operators ${\displaystyle U(g)}$ should, therefore, reflect the composition law in ${\displaystyle G}$, but only up to a phase factor:

${\displaystyle U(gh)=e^{i\theta }U(g)U(h)}$,

where ${\displaystyle \theta }$ will depend on ${\displaystyle g}$ and ${\displaystyle h}$. Thus, the map sending ${\displaystyle g}$ to ${\displaystyle U(g)}$ is a projective unitary representation of ${\displaystyle G}$, or possibly a mixture of unitary and anti-unitary, if ${\displaystyle G}$ is disconnected. In practice, anti-unitary operators are always associated with time-reversal symmetry.

Ordinary versus projective representations

It is important physically that in general ${\displaystyle U(\cdot )}$ does not have to be an ordinary representation of ${\displaystyle G}$; it may not be possible to choose the phase factors in the definition of ${\displaystyle U(g)}$ to eliminate the phase factors in their composition law. An electron, for example, is a spin-one-half particle; its Hilbert space consists of wave functions on ${\displaystyle \mathbb {R} ^{3}}$ with values in a two-dimensional spinor space. The action of ${\displaystyle \mathrm {SO(3)} }$ on the spinor space is only projective: It does not come from an ordinary representation of ${\displaystyle \mathrm {SO(3)} }$. There is, however, an associated ordinary representation of the universal cover ${\displaystyle \mathrm {SU(2)} }$ of ${\displaystyle \mathrm {SO(3)} }$ on spinor space. 

For many interesting classes of groups ${\displaystyle G}$, Bargmann's theorem tells us that every projective unitary representation of ${\displaystyle G}$ comes from an ordinary representation of the universal cover ${\displaystyle {\tilde {G}}}$ of ${\displaystyle G}$. Actually, if ${\displaystyle {\mathcal {H}}}$ is finite dimensional, then regardless of the group ${\displaystyle G}$, every projective unitary representation of ${\displaystyle G}$ comes from an ordinary unitary representation of ${\displaystyle {\tilde {G}}}$. If ${\displaystyle {\mathcal {H}}}$ is infinite dimensional, then to obtain the desired conclusion, some algebraic assumptions must be made on ${\displaystyle G}$ (see below). In this setting the result is a theorem of Bargmann. Fortunately, in the crucial case of the Poincaré group, Bargmann's theorem applies.

The requirement referred to above is that the Lie algebra ${\displaystyle {\mathfrak {g}}}$ does not admit a nontrivial one-dimensional central extension. This is the case if and only if the second cohomology group of ${\displaystyle {\mathfrak {g}}}$ is trivial. In this case, it may still be true that the group admits a central extension by a discrete group. But extensions of ${\displaystyle G}$ by discrete groups are covers of ${\displaystyle G}$. For instance, the universal cover ${\displaystyle {\tilde {G}}}$ is related to ${\displaystyle G}$ through the quotient ${\displaystyle G\approx {\tilde {G}}/\Gamma }$ with the central subgroup ${\displaystyle \Gamma }$ being the center of ${\displaystyle {\tilde {G}}}$ itself, isomorphic to the fundamental group of the covered group.

Thus, in favorable cases, the quantum system will carry a unitary representation of the universal cover ${\displaystyle {\tilde {G}}}$ of the symmetry group ${\displaystyle G}$. This is desirable because ${\displaystyle {\mathcal {H}}}$ is much easier to work with than the non-vector space ${\displaystyle \mathrm {P} {\mathcal {H}}}$. If the representations of ${\displaystyle {\tilde {G}}}$ can be classified, much more information about the possibilities and properties of ${\displaystyle {\mathcal {H}}}$ are available.

The Heisenberg case

An example in which Bargmann's theorem does not apply comes from a quantum particle moving in ${\displaystyle \mathbb {R} ^{n}}$. The group of translational symmetries of the associated phase space, ${\displaystyle \mathbb {R} ^{2n}}$, is the commutative group ${\displaystyle \mathbb {R} ^{2n}}$. In the usual quantum mechanical picture, the ${\displaystyle \mathbb {R} ^{2n}}$ symmetry is not implement by a unitary representation of ${\displaystyle \mathbb {R} ^{2n}}$. After all, in the quantum setting, translations in position space and translations in momentum space do not commute. This failure to commute reflects the failure of the position and momentum operators—which are the infinitesimal generators of translations in momentum space and position space, respectively—to commute. Nevertheless, translations in position space and translations in momentum space do commute up to a phase factor. Thus, we have a well-defined projective representation of ${\displaystyle \mathbb {R} ^{2n}}$, but it does not come from an ordinary representation of ${\displaystyle \mathbb {R} ^{2n}}$, even though ${\displaystyle \mathbb {R} ^{2n}}$ is simply connected. 

In this case, to obtain an ordinary representation, one has to pass to the Heisenberg group, which is a nontrivial one-dimensional central extension of ${\displaystyle \mathbb {R} ^{2n}}$.

Poincaré group

The group of translations and Lorentz transformations form the Poincaré group, and this group should be a symmetry of a relativistic quantum system (neglecting general relativity effects, or in other words, in flat space). Representations of the Poincaré group are in many cases characterized by a nonnegative mass and a half-integer spin (see Wigner's classification); this can be thought of as the reason that particles have quantized spin. (Note that there are in fact other possible representations, such as tachyons, infraparticles, etc., which in some cases do not have quantized spin or fixed mass.)

Other symmetries

The pattern of weak isospins, weak hypercharges, and color charges (weights) of all known elementary particles in the Standard Model, rotated by the weak mixing angle to show electric charge roughly along the vertical.

While the spacetime symmetries in the Poincaré group are particularly easy to visualize and believe, there are also other types of symmetries, called internal symmetries. One example is color SU(3), an exact symmetry corresponding to the continuous interchange of the three quark colors.

Lie algebras versus Lie groups

Many (but not all) symmetries or approximate symmetries form Lie groups. Rather than study the representation theory of these Lie groups, it is often preferable to study the closely related representation theory of the corresponding Lie algebras, which are usually simpler to compute.

Now, representations of the Lie algebra correspond to representations of the universal cover of the original group. In the finite-dimensional case—and the infinite-dimensional case, provided that Bargmann's theorem applies—irreducible projective representations of the original group correspond to ordinary unitary representations of the universal cover. In those cases, computing at the Lie algebra level is appropriate. This is the case, notably, for studying the irreducible projective representations of the rotation group SO(3). These are in one-to-one correspondence with the ordinary representations of the universal cover SU(2) of SO(3). The representations of the SU(2) are then in one-to-one correspondence with the representations of its Lie algebra su(2), which is isomorphic to the Lie algebra so(3) of SO(3).

Thus, to summarize, the irreducible projective representations of SO(3) are in one-to-one correspondence with the irreducible ordinary representations of its Lie algebra so(3). The two-dimensional "spin 1/2" representation of the Lie algebra so(3), for example, does not correspond to an ordinary (single-valued) representation of the group SO(3). (This fact is the origin of statements to the effect that "if you rotate the wave function of an electron by 360 degrees, you get the negative of the original wave function.") Nevertheless, the spin 1/2 representation does give rise to a well-defined projective representation of SO(3), which is all that is required physically.

Approximate symmetries

Although the above symmetries are believed to be exact, other symmetries are only approximate.

Hypothetical example

As an example of what an approximate symmetry means, suppose an experimentalist lived inside an infinite ferromagnet, with magnetization in some particular direction. The experimentalist in this situation would find not one but two distinct types of electrons: one with spin along the direction of the magnetization, with a slightly lower energy (and consequently, a lower mass), and one with spin anti-aligned, with a higher mass. Our usual SO(3) rotational symmetry, which ordinarily connects the spin-up electron with the spin-down electron, has in this hypothetical case become only an approximate symmetry, relating different types of particles to each other.

General definition

In general, an approximate symmetry arises when there are very strong interactions that obey that symmetry, along with weaker interactions that do not. In the electron example above, the two "types" of electrons behave identically under the strong and weak forces, but differently under the electromagnetic force.

Example: isospin symmetry

An example from the real world is isospin symmetry, an SU(2) group corresponding to the similarity between up quarks and down quarks. This is an approximate symmetry: While up and down quarks are identical in how they interact under the strong force, they have different masses and different electroweak interactions. Mathematically, there is an abstract two-dimensional vector space

${\displaystyle {\text{up quark}}\rightarrow {\begin{pmatrix}1\\0\end{pmatrix}},\qquad {\text{down quark}}\rightarrow {\begin{pmatrix}0\\1\end{pmatrix}}}$

and the laws of physics are approximately invariant under applying a determinant-1 unitary transformation to this space:

${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}\mapsto A{\begin{pmatrix}x\\y\end{pmatrix}},\quad {\text{where }}A{\text{ is in }}SU(2)}$

For example, ${\displaystyle A={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}$ would turn all up quarks in the universe into down quarks and vice versa. Some examples help clarify the possible effects of these transformations:

• When these unitary transformations are applied to a proton, it can be transformed into a neutron, or into a superposition of a proton and neutron, but not into any other particles. Therefore, the transformations move the proton around a two-dimensional space of quantum states. The proton and neutron are called an "isospin doublet", mathematically analogous to how a spin-½ particle behaves under ordinary rotation.
• When these unitary transformations are applied to any of the three pions (
  π⁰,
  π⁺, and
  π⁻), it can change any of the pions into any other, but not into any non-pion particle. Therefore, the transformations move the pions around a three-dimensional space of quantum states. The pions are called an "isospin triplet", mathematically analogous to how a spin-1 particle behaves under ordinary rotation.
• These transformations have no effect at all on an electron, because it contains neither up nor down quarks. The electron is called an isospin singlet, mathematically analogous to how a spin-0 particle behaves under ordinary rotation.

In general, particles form isospin multiplets, which correspond to irreducible representations of the Lie algebra SU(2). Particles in an isospin multiplet have very similar but not identical masses, because the up and down quarks are very similar but not identical.

Example: flavour symmetry

Isospin symmetry can be generalized to flavour symmetry, an SU(3) group corresponding to the similarity between up quarks, down quarks, and strange quarks. This is, again, an approximate symmetry, violated by quark mass differences and electroweak interactions—in fact, it is a poorer approximation than isospin, because of the strange quark's noticeably higher mass.

Nevertheless, particles can indeed be neatly divided into groups that form irreducible representations of the Lie algebra SU(3), as first noted by Murray Gell-Mann and independently by Yuval Ne'eman.

Einstein–Szilárd letter

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https://en.wikipedia.org/wiki/Einstein%E2%80%93Szil%C3%A1rd_letter 

 

A copy of the letter

The Einstein–Szilárd letter was a letter written by Leó Szilárd and signed by Albert Einstein that was sent to the United States President Franklin D. Roosevelt on August 2, 1939. Written by Szilárd in consultation with fellow Hungarian physicists Edward Teller and Eugene Wigner, the letter warned that Germany might develop atomic bombs and suggested that the United States should start its own nuclear program. It prompted action by Roosevelt, which eventually resulted in the Manhattan Project developing the first atomic bombs.

Origin

Leó Szilárd

 

Albert Einstein

The letter was conceived and written by Szilárd, and signed by Einstein

Otto Hahn and Fritz Strassmann reported the discovery of uranium fission in the January 6, 1939 issue of Die Naturwissenschaften, and Lise Meitner identified it as nuclear fission in the February 11, 1939 issue of Nature. This generated intense interest among physicists. Danish physicist Niels Bohr brought the news to the United States, and the U.S. opened the Fifth Washington Conference on Theoretical Physics with Enrico Fermi on January 26, 1939. The results were quickly corroborated by experimental physicists, most notably Fermi and John R. Dunning at Columbia University.

Hungarian physicist Leó Szilárd was living in the United States at the time and realized that the neutron-driven fission of heavy atoms could be used to create a nuclear chain reaction which could yield vast amounts of energy for electric power generation or atomic bombs. He had first formulated such an idea in 1933 upon reading Ernest Rutherford's disparaging remarks about generating power from his team's 1932 experiment using protons to split lithium. However, Szilárd had not been able to achieve a neutron-driven chain reaction with neutron-rich light atoms. In theory, if the number of secondary neutrons produced in a neutron-driven chain reaction was greater than one, then each such reaction could trigger multiple additional reactions, producing an exponentially increasing number of reactions.

Szilárd teamed up with Fermi to build a nuclear reactor from natural uranium at Columbia University, where George B. Pegram headed the physics department. There was disagreement about whether fission was produced by uranium-235, which made up less than one percent of natural uranium, or the more abundant uranium-238 isotope, as Fermi maintained. Fermi and Szilárd conducted a series of experiments and concluded that a chain reaction in natural uranium could be possible if they could find a suitable neutron moderator. They found that the hydrogen atoms in water slowed neutrons but tended to capture them. Szilárd then suggested using carbon as a moderator. They then needed large quantities of carbon and uranium to create a reactor. Szilárd was convinced that they would succeed if they could get the materials.

Szilárd was concerned that German scientists might also attempt this experiment. German nuclear physicist Siegfried Flügge published two influential articles on the exploitation of nuclear energy in 1939. After discussing this prospect with fellow Hungarian physicist Eugene Wigner, they decided that they should warn the Belgians, as the Belgian Congo was the best source of uranium ore. Wigner suggested that Albert Einstein might be a suitable person to do this, as he knew the Belgian Royal Family.

The letter

On July 12, 1939, Szilárd and Wigner drove in Wigner's car to Cutchogue on New York's Long Island, where Einstein was staying. When they explained about the possibility of atomic bombs, Einstein replied: Daran habe ich gar nicht gedacht (I did not even think about that). Szilárd dictated a letter in German to the Belgian Ambassador to the United States. Wigner wrote it down, and Einstein signed it. At Wigner's suggestion, they also prepared a letter for the State Department explaining what they were doing and why, giving it two weeks to respond if it had any objections.

This still left the problem of getting government support for uranium research. Another friend of Szilárd's, the Austrian economist Gustav Stolper, suggested approaching Alexander Sachs, who had access to President Franklin D. Roosevelt. Sachs told Szilárd that he had already spoken to the President about uranium, but that Fermi and Pegram had reported that the prospects for building an atomic bomb were remote. He told Szilárd that he would deliver the letter, but suggested that it come from someone more prestigious. For Szilárd, Einstein was again the obvious choice. Sachs and Szilárd drafted a letter riddled with spelling errors and mailed it to Einstein.

Szilárd also set out himself for Long Island again on August 2. Wigner was unavailable, so this time Szilárd co-opted another Hungarian physicist, Edward Teller, to do the driving. After receiving the draft, Einstein dictated the letter first in German. On returning to Columbia University, Szilárd dictated the letter in English to a young departmental stenographer, Janet Coatesworth. She later recalled that when Szilárd mentioned extremely powerful bombs, she "was sure she was working for a nut". Ending the letter with "Yours truly, Albert Einstein" did nothing to alter this impression. Both the English letter and a longer explanatory letter were then posted to Einstein for him to sign.

The letter dated 2 August and addressed to President Roosevelt warned that:

"In the course of the last four months it has been made probable — through the work of Joliot in France as well as Fermi and Szilárd in America — that it may become possible to set up a nuclear chain reaction in a large mass of uranium, by which vast amounts of power and large quantities of new radium-like elements would be generated. Now it appears almost certain that this could be achieved in the immediate future. This new phenomenon would also lead to the construction of bombs, and it is conceivable — though much less certain — that extremely powerful bombs of a new type may thus be constructed. A single bomb of this type, carried by boat and exploded in a port, might very well destroy the whole port together with some of the surrounding territory. However, such bombs might very well prove to be too heavy for transportation by air."

It also specifically warned about Germany:

"I understand that Germany has actually stopped the sale of uranium from the Czechoslovakian mines which she has taken over. That she should have taken such early action might perhaps be understood on the ground that the son of the German Under-Secretary of State, von Weizsäcker, is attached to the Kaiser-Wilhelm-Institut in Berlin where some of the American work on uranium is now being repeated."

At the time of the letter, the estimated material necessary for a fission chain reaction was several tons. Seven months later a breakthrough in Britain would estimate the necessary critical mass to be less than 10 kilograms, making delivery of a bomb by air a possibility.

Delivery

Roosevelt's reply

The Einstein–Szilárd letter was signed by Einstein and posted back to Szilárd, who received it on August 9. Szilárd gave both the short and long letters, along with a letter of his own, to Sachs on August 15. Sachs asked the White House staff for an appointment to see President Roosevelt, but before one could be set up, the administration became embroiled in a crisis due to Germany's invasion of Poland, which started World War II. Sachs delayed his appointment until October so that the President would give the letter due attention, securing an appointment on October 11. On that date he met with the President, the President's secretary, Brigadier General Edwin "Pa" Watson, and two ordnance experts, Army Lieutenant Colonel Keith F. Adamson and Navy Commander Gilbert C. Hoover. Roosevelt summed up the conversation as: "Alex, what you are after is to see that the Nazis don't blow us up."

Roosevelt sent a reply thanking Einstein, and informing him that:

"I found this data of such import that I have convened a Board consisting of the head of the Bureau of Standards and a chosen representative of the Army and Navy to thoroughly investigate the possibilities of your suggestion regarding the element of uranium."

Einstein sent two more letters to Roosevelt, on March 7, 1940, and April 25, 1940, calling for action on nuclear research. Szilárd drafted a fourth letter for Einstein's signature that urged the President to meet with Szilárd to discuss policy on nuclear energy. Dated March 25, 1945, it did not reach Roosevelt before his death on April 12, 1945.

Results

Roosevelt decided that the letter required action, and authorized the creation of the Advisory Committee on Uranium. The committee was chaired by Lyman James Briggs, the Director of the Bureau of Standards (currently the National Institute of Standards and Technology), with Adamson and Hoover as its other members. It convened for the first time on October 21. The meeting was also attended by Fred L. Mohler from the Bureau of Standards, Richard B. Roberts of the Carnegie Institution of Washington, and Szilárd, Teller and Wigner. Adamson was skeptical about the prospect of building an atomic bomb, but was willing to authorize $6,000 ($100,000 in current USD) for the purchase of uranium and graphite for Szilárd and Fermi's experiment.

The Advisory Committee on Uranium was the beginning of the US government's effort to develop an atomic bomb, but it did not vigorously pursue the development of a weapon. It was superseded by the National Defense Research Committee in 1940, and then the Office of Scientific Research and Development in 1941. The Frisch–Peierls memorandum and the British Maud Reports eventually prompted Roosevelt to authorize a full-scale development effort in January 1942. The work of fission research was taken over by the United States Army Corps of Engineers's Manhattan District in June 1942, which directed an all-out bomb development program known as the Manhattan Project.

Einstein did not work on the Manhattan Project. The Army and Vannevar Bush denied him the work clearance needed in July 1940, saying his pacifist leanings and celebrity made him a security risk. At least one source states that Einstein did clandestinely contribute some equations to the Manhattan Project. Einstein was allowed to work as a consultant to the United States Navy's Bureau of Ordnance. He had no knowledge of the atomic bomb's development, and no influence on the decision for the bomb to be dropped. According to Linus Pauling, Einstein later regretted signing the letter because it led to the development and use of the atomic bomb in combat, adding that Einstein had justified his decision because of the greater danger that Nazi Germany would develop the bomb first. In 1947 Einstein told Newsweek magazine that "had I known that the Germans would not succeed in developing an atomic bomb, I would have done nothing."

Eugene Wigner

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Eugene_Wigner 

Eugene Wigner
Born	Wigner Jenő Pál November 17, 1902 Budapest, Austria-Hungary
Died	January 1, 1995 (aged 92) Princeton, New Jersey, US
Citizenship	American (post-1937) Hungarian (pre-1937)
Alma mater	Technical University of Berlin
Known for	Bargmann–Wigner equations Law of conservation of parity Wigner D-matrix Wigner–Eckart theorem Wigner's friend Wigner semicircle distribution Wigner's classification Wigner distribution function Wigner quasiprobability distribution Wigner crystal Wigner effect Wigner energy Relativistic Breit–Wigner distribution Modified Wigner distribution function Wigner–d'Espagnat inequality Gabor–Wigner transform Wigner's theorem Jordan–Wigner transformation Newton–Wigner localization Wigner–Inonu contraction Wigner–Seitz cell Wigner–Seitz radius Thomas–Wigner rotation Wigner–Weyl transform Wigner–Wilkins spectrum 6-j symbol 9-j symbol
Spouse(s)	Amelia Frank (1936–1937; her death) Mary Annette Wheeler (1941–1977; her death; 2 children) Eileen Clare-Patton Hamilton (1979–d. 2010; his death; 1 child)
Awards	Medal for Merit (1946) Franklin Medal (1950) Enrico Fermi Award (1958) Atoms for Peace Award (1959) Max Planck Medal (1961) Nobel Prize in Physics (1963) National Medal of Science (1969) Albert Einstein Award (1972) Wigner Medal (1978)
Scientific career
Fields	Theoretical physics Atomic physics Nuclear physics Solid-state physics
Institutions	University of Göttingen University of Wisconsin–Madison Princeton University Manhattan Project
Doctoral advisor	Michael Polanyi
Other academic advisors	László Rátz Richard Becker
Doctoral students	John Bardeen Victor Frederick Weisskopf Marcos Moshinsky Abner Shimony Edwin Thompson Jaynes Frederick Seitz Conyers Herring Frederick Tappert J O Hirschfelder
Signature

Eugene Paul "E. P." Wigner (Hungarian: Wigner Jenő Pál, pronounced [ˈviɡnɛr ˈjɛnøː ˈpaːl]; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist and mathematician. He received the Nobel Prize in Physics in 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".

A graduate of the Technical University of Berlin, Wigner worked as an assistant to Karl Weissenberg and Richard Becker at the Kaiser Wilhelm Institute in Berlin, and David Hilbert at the University of Göttingen. Wigner and Hermann Weyl were responsible for introducing group theory into physics, particularly the theory of symmetry in physics. Along the way he performed ground-breaking work in pure mathematics, in which he authored a number of mathematical theorems. In particular, Wigner's theorem is a cornerstone in the mathematical formulation of quantum mechanics. He is also known for his research into the structure of the atomic nucleus. In 1930, Princeton University recruited Wigner, along with John von Neumann, and he moved to the United States.

Wigner participated in a meeting with Leo Szilard and Albert Einstein that resulted in the Einstein-Szilard letter, which prompted President Franklin D. Roosevelt to initiate the Manhattan Project to develop atomic bombs. Wigner was afraid that the German nuclear weapon project would develop an atomic bomb first. During the Manhattan Project, he led a team whose task was to design nuclear reactors to convert uranium into weapons grade plutonium. At the time, reactors existed only on paper, and no reactor had yet gone critical. Wigner was disappointed that DuPont was given responsibility for the detailed design of the reactors, not just their construction. He became Director of Research and Development at the Clinton Laboratory (now the Oak Ridge National Laboratory) in early 1946, but became frustrated with bureaucratic interference by the Atomic Energy Commission, and returned to Princeton.

In the postwar period he served on a number of government bodies, including the National Bureau of Standards from 1947 to 1951, the mathematics panel of the National Research Council from 1951 to 1954, the physics panel of the National Science Foundation, and the influential General Advisory Committee of the Atomic Energy Commission from 1952 to 1957 and again from 1959 to 1964. In later life, he became more philosophical, and published The Unreasonable Effectiveness of Mathematics in the Natural Sciences, his best-known work outside technical mathematics and physics.

Early life

Werner Heisenberg and Eugene Wigner (1928)

Wigner Jenő Pál was born in Budapest, Austria-Hungary on November 17, 1902, to middle class Jewish parents, Elisabeth (Einhorn) and Anthony Wigner, a leather tanner. He had an older sister, Bertha, known as Biri, and a younger sister Margit, known as Manci, who later married British theoretical physicist Paul Dirac. He was home schooled by a professional teacher until the age of 9, when he started school at the third grade. During this period, Wigner developed an interest in mathematical problems. At the age of 11, Wigner contracted what his doctors believed to be tuberculosis. His parents sent him to live for six weeks in a sanatorium in the Austrian mountains, before the doctors concluded that the diagnosis was mistaken.

Wigner's family was Jewish, but not religiously observant, and his Bar Mitzvah was a secular one. From 1915 through 1919, he studied at the secondary grammar school called Fasori Evangélikus Gimnázium, the school his father had attended. Religious education was compulsory, and he attended classes in Judaism taught by a rabbi. A fellow student was János von Neumann, who was a year behind Wigner. They both benefited from the instruction of the noted mathematics teacher László Rátz. In 1919, to escape the Béla Kun communist regime, the Wigner family briefly fled to Austria, returning to Hungary after Kun's downfall. Partly as a reaction to the prominence of Jews in the Kun regime, the family converted to Lutheranism. Wigner explained later in his life that his family decision to convert to Lutheranism "was not at heart a religious decision but an anti-communist one". On religious views, Wigner was an atheist.

After graduating from the secondary school in 1920, Wigner enrolled at the Budapest University of Technical Sciences, known as the Műegyetem. He was not happy with the courses on offer, and in 1921 enrolled at the Technische Hochschule Berlin (now Technical University of Berlin), where he studied chemical engineering. He also attended the Wednesday afternoon colloquia of the German Physical Society. These colloquia featured such luminaries as Max Planck, Max von Laue, Rudolf Ladenburg, Werner Heisenberg, Walther Nernst, Wolfgang Pauli, and Albert Einstein. Wigner also met the physicist Leó Szilárd, who at once became Wigner's closest friend. A third experience in Berlin was formative. Wigner worked at the Kaiser Wilhelm Institute for Physical Chemistry and Electrochemistry (now the Fritz Haber Institute), and there he met Michael Polanyi, who became, after László Rátz, Wigner's greatest teacher. Polanyi supervised Wigner's DSc thesis, Bildung und Zerfall von Molekülen ("Formation and Decay of Molecules").

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 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 for two semesters. Wigner explored quantum mechanics, studying the work of Erwin Schrödinger. He also delved into the group theory of Ferdinand Frobenius and Eduard Ritter von Weber.

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. 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. 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.

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 symmetries in quantum mechanics. Wigner's theorem proved by Wigner 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 states. According to the theorem, any symmetry transformation is represented by a linear and unitary or antilinear and antiunitary transformation of Hilbert space. The representation of a symmetry group on a Hilbert space is either an ordinary representation or a projective representation.

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. 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 Nazis soon rose to power in Germany. At Princeton in 1934, Wigner introduced his sister Manci to the physicist Paul Dirac, whom she married.

Princeton did not rehire Wigner when his contract ran out in 1936. Through Gregory Breit, Wigner found new employment at the 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 a 1938 offer from Princeton to return there. Wigner became a naturalized citizen of the United States on January 8, 1937, and he brought his parents to the United States.

Manhattan Project

Wigner receiving the Medal for Merit for his work on the Manhattan Project from Robert P. Patterson (left), March 5, 1946

Although he was a professed political amateur, on August 2, 1939, he participated in a meeting with Leó Szilárd and Albert Einstein that resulted in the Einstein–Szilárd letter, which prompted President Franklin D. Roosevelt to initiate the Manhattan Project to develop atomic bombs. Wigner was afraid that the German nuclear weapon project would develop an atomic bomb first, and even refused to have his fingerprints taken because they could be used to track him down if Germany won. "Thoughts of being murdered," he later recalled, "focus your mind wonderfully."

On June 4, 1941, Wigner married his second wife, Mary Annette Wheeler, a professor of physics at Vassar College, who had completed her Ph.D. at Yale University in 1932. After the war she taught physics on the faculty of Rutgers University's Douglass College in New Jersey until her retirement in 1964. They remained married until her death in November 1977. They had two children, David Wigner and Martha Wigner Upton.

During the Manhattan Project, Wigner led a team that included Alvin M. Weinberg, Katharine Way, Gale Young and Edward Creutz. The group's task was to design the production nuclear reactors that would convert uranium into weapons grade plutonium. At the time, reactors existed only on paper, and no reactor had yet gone critical. In July 1942, Wigner chose a conservative 100 MW design, with a graphite neutron moderator and water cooling. Wigner was present at a converted rackets court under the stands at the University of Chicago's abandoned Stagg Field on December 2, 1942, when the world's first atomic reactor, Chicago Pile One (CP-1) achieved a controlled nuclear chain reaction.

The Chianti fiasco purchased by Wigner to help celebrate the first self-sustaining, controlled chain reaction. It was signed by the participants.

Wigner was disappointed that DuPont was given responsibility for the detailed design of the reactors, not just their construction. He threatened to resign in February 1943, but was talked out of it by the head of the Metallurgical Laboratory, Arthur Compton, who sent him on vacation instead. As it turned out, a design decision by DuPont to give the reactor additional load tubes for more uranium saved the project when neutron poisoning became a problem. Without the additional tubes, the reactor could have been run at 35% power until the boron impurities in the graphite were burned up and enough plutonium produced to run the reactor at full power; but this would have set the project back a year. During the 1950s, he would even work for DuPont on the Savannah River Site. Wigner did not regret working on the Manhattan Project, and sometimes wished the atomic bomb had been ready a year earlier.

An important discovery Wigner made during the project was the Wigner effect. This is a swelling of the graphite moderator caused by the displacement of atoms by neutron radiation. The Wigner effect was a serious problem for the reactors at the Hanford Site in the immediate post-war period, and resulted in production cutbacks and a reactor being shut down entirely. It was eventually discovered that it could be overcome by controlled heating and annealing.

Through Manhattan project funding, Wigner and Leonard Eisenbud also developed an important general approach to nuclear reactions, the Wigner–Eisenbud R-matrix theory, which was published in 1947.

Later years

Wigner accepted a position as the Director of Research and Development at the Clinton Laboratory (now the Oak Ridge National Laboratory) in Oak Ridge, Tennessee in early 1946. Because he did not want to be involved in administrative duties, he became co-director of the laboratory, with James Lum handling the administrative chores as executive director. When the newly created Atomic Energy Commission (AEC) took charge of the laboratory's operations at the start of 1947, Wigner feared that many of the technical decisions would be made in Washington. He also saw the Army's continuation of wartime security policies at the laboratory as a "meddlesome oversight", interfering with research. One such incident occurred in March 1947, when the AEC discovered that Wigner's scientists were conducting experiments with a critical mass of uranium-235 when the Director of the Manhattan Project, Major General Leslie R. Groves, Jr., had forbidden such experiments in August 1946 after the death of Louis Slotin at the Los Alamos Laboratory. Wigner argued that Groves's order had been superseded, but was forced to terminate the experiments, which were completely different from the one that killed Slotin.

Feeling unsuited to a managerial role in such an environment, he left Oak Ridge in 1947 and returned to Princeton University, although he maintained a consulting role with the facility for many years. In the postwar period he served on a number of government bodies, including the National Bureau of Standards from 1947 to 1951, the mathematics panel of the National Research Council from 1951 to 1954, the physics panel of the National Science Foundation, and the influential General Advisory Committee of the Atomic Energy Commission from 1952 to 1957 and again from 1959 to 1964. He also contributed to civil defense.

Near the end of his life, Wigner's thoughts turned more philosophical. In 1960, he published a now classic article on the philosophy of mathematics and of physics, which has become his best-known work outside technical mathematics and physics, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". He argued that biology and cognition could be the origin of physical concepts, as we humans perceive them, and that the happy coincidence that mathematics and physics were so well matched, seemed to be "unreasonable" and hard to explain. His original paper has provoked and inspired many responses across a wide range of disciplines. These included Richard Hamming in Computer Science, Arthur Lesk in Molecular Biology, Peter Norvig in data mining, Max Tegmark in Physics, Ivor Grattan-Guinness in Mathematics, and Vela Velupillai in Economics.

In November 1963, Wigner called for the allocation of 10% of the national defense budget to be spent on nuclear blast shelters and survival resources, arguing that such an expenditure would be less costly than disarmament. Wigner considered a recent Woods Hole study's conclusion that a nuclear strike would kill 20% of Americans to be a very modest projection and that the country could recover from such an attack more quickly than Germany had recovered from the devastation of World War II.

Wigner was awarded the Nobel Prize in Physics in 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". The prize was shared that year, with the other half of the award divided between Maria Goeppert-Mayer and J. Hans D. Jensen. Wigner professed that he had never considered the possibility that this might occur, and added: "I never expected to get my name in the newspapers without doing something wicked." He also won the Franklin Medal in 1950, the Enrico Fermi award in 1958, the Atoms for Peace Award in 1959, the Max Planck Medal in 1961, the National Medal of Science in 1969, the Albert Einstein Award in 1972, the Golden Plate Award of the American Academy of Achievement in 1974, and the eponymous Wigner Medal in 1978. In 1968 he gave the Josiah Willard Gibbs lecture.

Mary died in November 1977. In 1979, Wigner married his third wife, Eileen Clare-Patton (Pat) Hamilton, the widow of physicist Donald Ross Hamilton, the Dean of the Graduate School at Princeton University, who had died in 1972. In 1992, at the age of 90, he published his memoirs, The Recollections of Eugene P. Wigner with Andrew Szanton. In it, Wigner said: "The full meaning of life, the collective meaning of all human desires, is fundamentally a mystery beyond our grasp. As a young man, I chafed at this state of affairs. But by now I have made peace with it. I even feel a certain honor to be associated with such a mystery." In his collection of essays 'Philosophical Reflections and Syntheses' (1995), he commented: "It was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to consciousness."

Wigner died of pneumonia at the University Medical Center in Princeton, New Jersey on 1 January 1995. He was survived by his wife Eileen (died 2010) and children Erika, David and Martha, and his sisters Bertha and Margit.

Publications

• 1958 (with Alvin M. Weinberg). Physical Theory of Neutron Chain Reactors University of Chicago Press. ISBN 0-226-88517-8
• 1959. Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. New York: Academic Press. Translation by J. J. Griffin of 1931, Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren, Vieweg Verlag, Braunschweig.
• 1970 Symmetries and Reflections: Scientific Essays. Indiana University Press, Bloomington ISBN 0-262-73021-9
• 1992 (as told to Andrew Szanton). The Recollections of Eugene P. Wigner. Plenum. ISBN 0-306-44326-0
• 1995 (with Jagdish Mehra and Arthur S. Wightman, eds.). Philosophical Reflections and Syntheses. Springer, Berlin ISBN 3-540-63372-3