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."