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There is a natural connection between
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
and
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, as first noted in the 1930s by
Eugene Wigner Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his con ...
. It links the properties of
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, an ...
s to the structure of
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s and
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s. According to this connection, the different
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
s of an elementary particle give rise to an
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
of the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
. 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 Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, any particular one-particle state is represented as a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
in a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
\mathcal H. To help understand what types of particles can exist, it is important to classify the possibilities for \mathcal H allowed by
symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, and their properties. Let \mathcal H be a Hilbert space describing a particular quantum system and let G be a group of symmetries of the quantum system. In a relativistic quantum system, for example, G might be the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
, while for the hydrogen atom, G might be the
rotation group SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a tr ...
. The particle state is more precisely characterized by the associated
projective Hilbert space In mathematics and the foundations of quantum mechanics, the projective Hilbert space P(H) of a complex Hilbert space H is the set of equivalence classes of non-zero vectors v in H, for the relation \sim on H given by :w \sim v if and only if v = \ ...
\mathrm P\mathcal H, also called
ray space Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (gra ...
, since two vectors that differ by a nonzero scalar factor correspond to the same physical
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
represented by a ''ray'' in Hilbert space, which is an
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
in \mathcal H and, under the natural projection map \mathcal H \rightarrow \mathrm P \mathcal H, an element of \mathrm P\mathcal H. By definition of a symmetry of a quantum system, there is a
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
on \mathrm P\mathcal H. For each g\in G, there is a corresponding transformation V(g) of \mathrm P\mathcal H. More specifically, if g is some symmetry of the system (say, rotation about the x-axis by 12°), then the corresponding transformation V(g) of \mathrm P\mathcal H is a map on ray space. For example, when rotating a ''stationary'' (zero momentum) spin-5 particle about its center, g is a rotation in 3D space (an element of \mathrm), while 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 \mathrm P\mathcal H associated with an 11-dimensional complex Hilbert space \mathcal H. Each map V(g) preserves, by definition of symmetry, the ray product on \mathrm P\mathcal H induced by the inner product on \mathcal H; according to
Wigner's theorem Wigner's theorem, proved by Eugene 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 are represented on the Hilbert spac ...
, this transformation of \mathrm P\mathcal H comes from a unitary or anti-unitary transformation U(g) of \mathcal H. Note, however, that the U(g) associated to a given V(g) is not unique, but only unique ''up to a phase factor''. The composition of the operators U(g) should, therefore, reflect the composition law in G, but only up to a phase factor: :U(gh)=e^U(g)U(h), where \theta will depend on g and h. Thus, the map sending g to U(g) is a ''projective unitary representation'' of G, or possibly a mixture of unitary and anti-unitary, if G is disconnected. In practice, anti-unitary operators are always associated with
time-reversal symmetry T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal, : T: t \mapsto -t. Since the second law of thermodynamics states that entropy increases as time flows toward the futur ...
.


Ordinary versus projective representations

It is important physically that in general U(\cdot) does not have to be an ordinary representation of G; it may not be possible to choose the phase factors in the definition of 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 \mathbb^3 with values in a two-dimensional spinor space. The action of \mathrm on the spinor space is only projective: It does not come from an ordinary representation of \mathrm. There is, however, an associated ordinary representation of the universal cover \mathrm of \mathrm on spinor space. For many interesting classes of groups G, Bargmann's theorem tells us that every projective unitary representation of G comes from an ordinary representation of the universal cover \tilde of G. Actually, if \mathcal H is finite dimensional, then regardless of the group G, every projective unitary representation of G comes from an ordinary unitary representation of \tilde. If \mathcal H is infinite dimensional, then to obtain the desired conclusion, some algebraic assumptions must be made on 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. (See
Wigner's classification In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ~ (~E \ge 0~)~ energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (Since this ...
of the representations of the universal cover of the Poincaré group.) The requirement referred to above is that the Lie algebra \mathfrak g does not admit a nontrivial one-dimensional central extension. This is the case if and only if the second cohomology group of \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 G by discrete groups are covers of G. For instance, the universal cover \tilde G is related to G through the quotient G \approx \tilde G/\Gamma with the central subgroup \Gamma being the center of \tilde G itself, isomorphic to the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of the covered group. Thus, in favorable cases, the quantum system will carry a unitary representation of the universal cover \tilde of the symmetry group G. This is desirable because \mathcal H is much easier to work with than the non-vector space \mathrm P\mathcal H. If the representations of \tilde can be classified, much more information about the possibilities and properties of \mathcal H are available.


The Heisenberg case

An example in which Bargmann's theorem does not apply comes from a quantum particle moving in \mathbb R^n. The group of translational symmetries of the associated phase space, \mathbb R^, is the commutative group \mathbb R^. In the usual quantum mechanical picture, the \mathbb R^ symmetry is not implemented by a unitary representation of \mathbb R^. 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 \mathbb R^, but it does not come from an ordinary representation of \mathbb R^, even though \mathbb R^ is simply connected. In this case, to obtain an ordinary representation, one has to pass to the
Heisenberg group In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ' ...
, which is a nontrivial one-dimensional central extension of \mathbb R^.


Poincaré group

The group of translations and
Lorentz transformations In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
form the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
, and this group should be a symmetry of a relativistic quantum system (neglecting
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
effects, or in other words, in
flat space In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...
).
Representations of the Poincaré group ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
are in many cases characterized by a nonnegative
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
and a half-integer
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
(see
Wigner's classification In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ~ (~E \ge 0~)~ energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (Since this ...
); this can be thought of as the reason that particles have quantized spin. (Note that there are in fact other possible representations, such as
tachyon A tachyon () or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such partic ...
s,
infraparticle An infraparticle is an electrically charged particle and its surrounding cloud of soft photons—of which there are infinite number, by virtue of the infrared divergence of quantum electrodynamics. That is, it is a dressed particle rather than a ...
s, etc., which in some cases do not have quantized spin or fixed mass.)


Other symmetries

While the
spacetime symmetries Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacetime symmetries are used in the study of exact ...
in the Poincaré group are particularly easy to visualize and believe, there are also other types of symmetries, called
internal symmetries In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be ''continuo ...
. One example is
color Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are associ ...
SU(3) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
, an exact symmetry corresponding to the continuous interchange of the three
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
colors.


Lie algebras versus Lie groups

Many (but not all) symmetries or approximate symmetries form
Lie groups In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
. Rather than study the
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of these Lie groups, it is often preferable to study the closely related
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of the corresponding Lie algebras, which are usually simpler to compute. Now, representations of the Lie algebra correspond to representations of the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
of the original group. Section 5.7 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 Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials a ...
, 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) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a tr ...
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 Strong may refer to: Education * The Strong, an educational institution in Rochester, New York, United States * Strong Hall (Lawrence, Kansas), an administrative hall of the University of Kansas * Strong School, New Haven, Connecticut, United Sta ...
and
weak force Weak may refer to: Songs * "Weak" (AJR song), 2016 * "Weak" (Melanie C song), 2011 * "Weak" (SWV song), 1993 * "Weak" (Skunk Anansie song), 1995 * "Weak", a song by Seether from '' Seether: 2002-2013'' Television episodes * "Weak" (''Fear t ...
s, but differently under the
electromagnetic force In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
.


Example: isospin symmetry

An example from the real world is isospin symmetry, an
SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
group corresponding to the similarity between
up quark The up quark or u quark (symbol: u) is the lightest of all quarks, a type of elementary particle, and a significant constituent of matter. It, along with the down quark, forms the neutrons (one up quark, two down quarks) and protons (two up quark ...
s and
down quark The down quark or d quark (symbol: d) is the second-lightest of all quarks, a type of elementary particle, and a major constituent of matter. Together with the up quark, it forms the neutrons (one up quark, two down quarks) and protons (two up q ...
s. This is an approximate symmetry: while up and down quarks are identical in how they interact under the
strong force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the n ...
, they have different masses and different electroweak interactions. Mathematically, there is an abstract two-dimensional vector space :\text \rightarrow \begin 1 \\ 0 \end, \qquad \text \rightarrow \begin 0 \\ 1 \end and the laws of physics are ''approximately'' invariant under applying a determinant-1
unitary transformation In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. Formal definition More precisely, ...
to this space:Lecture notes by Prof. Mark Thomson
/ref> :\begin x \\ y \end \mapsto A \begin x \\ y \end, \quad \text A \text SU(2) For example, A=\begin 0&1 \\ -1&0 \end 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 A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
, it can be transformed into a
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
, 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 In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of . The spin number describes how many symmetrical facets a particle has in one ful ...
particle behaves under ordinary rotation. *When these unitary transformations are applied to any of the three
pion In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gene ...
s (, , and ), 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 The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
, 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 In particle physics, flavour or flavor refers to the ''species'' of an elementary particle. The Standard Model counts six flavours of quarks and six flavours of leptons. They are conventionally parameterized with ''flavour quantum numbers'' th ...
, an
SU(3) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
group corresponding to the similarity between
up quark The up quark or u quark (symbol: u) is the lightest of all quarks, a type of elementary particle, and a significant constituent of matter. It, along with the down quark, forms the neutrons (one up quark, two down quarks) and protons (two up quark ...
s,
down quark The down quark or d quark (symbol: d) is the second-lightest of all quarks, a type of elementary particle, and a major constituent of matter. Together with the up quark, it forms the neutrons (one up quark, two down quarks) and protons (two up q ...
s, and
strange quark The strange quark or s quark (from its symbol, s) is the third lightest of all quarks, a type of elementary particle. Strange quarks are found in subatomic particles called hadrons. Examples of hadrons containing strange quarks include kaons ( ...
s. 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 Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American physicist who received the 1969 Nobel Prize in Physics for his work on the theory of elementary particles. He was the Robert Andrews Millikan Professor of Theoretical ...
and independently by
Yuval Ne'eman Yuval Ne'eman ( he, יובל נאמן, 14 May 1925 – 26 April 2006) was an Israeli theoretical physicist, military scientist, and politician. He was Minister of Science and Development in the 1980s and early 1990s. He was the President o ...
.


See also

*
Charge (physics) In physics, a charge is any of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges correspond to the time-invariant generators of a symmetry group, and specifically, ...
*
Representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
: ** Of Lie algebras ** Of Lie groups *
Projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where GL(' ...
*
Special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...


Notes


References

*Coleman, Sidney (1985) ''Aspects of Symmetry: Selected Erice Lectures of Sidney Coleman''. Cambridge Univ. Press. . *Georgi, Howard (1999) ''Lie Algebras in Particle Physics''. Reading, Massachusetts: Perseus Books. . *. * . *Sternberg, Shlomo (1994) ''Group Theory and Physics''. Cambridge Univ. Press. . Especially pp. 148–150. * Especially appendices A and B to Chapter 2.


External links

* {{Industrial and applied mathematics Lie algebras Representation theory of Lie groups Theoretical physics Conservation laws Quantum field theory