09-20-2021-1102 - gauge group

A gauge group is a group of gauge symmetries of the Yang – Mills gauge theory of principal connections on a principal bundle. Given a principal bundle  with a structure Lie group , a gauge group is defined to be a group of its vertical automorphisms. This group is isomorphic to the group  of global sections of the associated group bundle  whose typical fiber is a group  which acts on itself by the adjoint representation. The unit element of  is a constant unit-valued section  of .

At the same time, gauge gravitation theory exemplifies field theory on a principal frame bundle whose gauge symmetries are general covariant transformations which are not elements of a gauge group.

In the physical literature on gauge theory, a structure group of a principal bundle often is called the gauge group.

In quantum gauge theory, one considers a normal subgroup  of a gauge group  which is the stabilizer

of some point  of a group bundle . It is called the pointed gauge group. This group acts freely on a space of principal connections. Obviously, . One also introduces the effective gauge group  where  is the center of a gauge group . This group  acts freely on a space of irreducible principal connections.

If a structure group  is a complex semisimple matrix group, the Sobolev completion  of a gauge group  can be introduced. It is a Lie group. A key point is that the action of  on a Sobolev completion  of a space of principal connections is smooth, and that an orbit space  is a Hilbert space. It is a configuration space of quantum gauge theory.

https://en.wikipedia.org/wiki/Gauge_group_(mathematics)

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09-20-2021-1101 - graviton xi baryon lambda baryon delta baryon delta resonances magnon roton electron hole J/ψ (J/psi) meson psion gravitino

In theoretical physics, the dual graviton is a hypothetical elementary particle that is a dual of the graviton under electric-magnetic duality, as an S-duality, predicted by some formulations of supergravity in eleven dimensions.^[3]

The dual graviton was first hypothesized in 1980.^[4] It was theoretically modeled in 2000s,^[1][2] which was then predicted in eleven-dimensional mathematics of SO(8) supergravity in the framework of electric-magnetic duality.^[3] It again emerged in the E₁₁ generalized geometry in eleven dimensions,^[5] and the E₇ generalized vielbein-geometry in eleven dimensions.^[6] While there is no local coupling between graviton and dual graviton, the field introduced by dual graviton may be coupled to a BF model as non-local gravitational fields in extra dimensions.^[7]

A massive dual gravity of Ogievetsky-Polubarinov model^[8] can be obtained by coupling the dual graviton field to the curl of its own energy-momentum tensor.^[9][10]

The previously mentioned theories of dual graviton are in flat space. In de Sitter and anti-de Sitter spaces (A)dS, the massless dual graviton exhibits less gauge symmetries dynamics compared with those of Curtright field in flat space, hence the mixed-symmetry field propagates in more degrees of freedom.^[11] However, the dual graviton in (A)dS transforms under GL(D) representation, which is identical to that of massive dual graviton in flat space.^[12] This apparent paradox can be resolved using the unfolding technique in Brink, Metsaev, and Vasiliev conjecture.^[13][14] For the massive dual graviton in (A)dS, the flat limit is clarified after expressing dual field in terms of the Stueckelberg coupling of a massless spin-2 field with a Proca field.^[11]

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

The Xi baryons /ksaɪ bæriənz/ or cascade particles are a family of subatomic hadron particles which have the symbol Ξ and may have an electric charge (Q) of +2 e, +1 e, 0, or −1 e, where e is the elementary charge.

Like all conventional baryons, Ξ particles contain three quarks. Ξ baryons, in particular, contain either one up or one down quark and two other, more massive quarks. The two more massive quarks are any two of strange, charm, or bottom (doubles allowed). For notation, the assumption is that the two heavy quarks in the Ξ are both strange; subscripts "c" and "b" are added for each even heavier charm or bottom quark that replaces one of the two presumed strange quarks.

They are historically called the cascade particles because of their unstable state; they are typically observed to decay rapidly into lighter particles, through a chain of decays (cascading decays).^[1] The first discovery of a charged Xi baryon was in cosmic ray experiments by the Manchester group in 1952.^[2] The first discovery of the neutral Xi particle was at Lawrence Berkeley Laboratory in 1959.^[3] It was also observed as a daughter product from the decay of the omega baryon (
Ω−
) observed at Brookhaven National Laboratory in 1964.^[1] The Xi spectrum is important to nonperturbative quantum chromodynamics (QCD), such as lattice QCD.^[why?]

^{https://en.wikipedia.org/wiki/Xi_baryon}

^{The lambda baryons (Λ) are a family of subatomic hadron particles containing one up quark, one down quark, and a third quark from a higher flavour generation, in a combination where the quantum wave function changes sign upon the flavour of any two quarks being swapped (thus differing from a sigma baryon). They are thus baryons, with total isospin of 0, and have either neutral electric charge or the elementary charge +1.}

^{https://en.wikipedia.org/wiki/Lambda_baryon}

The Delta baryons (or Δ baryons, also called Delta resonances) are a family of subatomic particle made of three up or down quarks (u or d quarks).

Four closely related Δ baryons exist: 
Δ++
 (constituent quarks: uuu), 
Δ+
 (uud), 
Δ0
 (udd), and 
Δ−
 (ddd), which respectively carry an electric charge of +2 e, +1 e, 0 e, and −1 e. The Δ baryons have a mass of about 1232 MeV/c², a spin of 3⁄2, and an isospin of 3⁄2. Ordinary protons and neutrons (nucleons (symbol N)), by contrast, have a mass of about 939 MeV/c², a spin of 1⁄2, and an isospin of 1⁄2. The 
Δ+
 (uud) and 
Δ0
 (udd) particles are higher-mass excitations of the proton (
N+
, uud) and neutron (
N0
, udd), respectively. However, the 
Δ++
 and 
Δ−
 have no direct nucleon analogues.

The states were established experimentally at the University of Chicago cyclotron^[1][2] and the Carnegie Institute of Technology synchro-cyclotron^[3] in the mid-1950s using accelerated positive pions on hydrogen targets. The existence of the 
Δ++
, with its unusual +2 charge, was a crucial clue in the development of the quark model.

The Delta states discussed here are only the lowest-mass quantum excitations of the proton and neutron. At higher masses, additional Delta states appear, all defined by having 3⁄2 units of isospin, but with a spin quantum numbers including 1⁄2, 3⁄2, 5⁄2, ... 11⁄2. A complete listing of all properties of all these states can be found in Beringer et al. (2013).^[4]

There also exist antiparticle Delta states with opposite charges, made up of the corresponding antiquarks.

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

A magnon is a quasiparticle, a collective excitation of the electrons' spin structure in a crystal lattice. In the equivalent wave picture of quantum mechanics, a magnon can be viewed as a quantized spin wave. Magnons carry a fixed amount of energy and lattice momentum, and are spin-1, indicating they obey boson behavior.

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

In theoretical physics, a roton is an elementary excitation, or quasiparticle, seen in superfluid helium-4 and Bose–Einstein condensates with long-range dipolar interactions or spin-orbit coupling. The dispersion relation of elementary excitations in this superfluid shows a linear increase from the origin, but exhibits first a maximum and then a minimum in energy as the momentum increases. Excitations with momenta in the linear region are called phonons; those with momenta close to the minimum are called rotons. Excitations with momenta near the maximum are called maxons.

The term "roton" is also used for the quantized eigenmode of a freely rotating molecule.^{[citation needed]}

^{https://en.wikipedia.org/wiki/Roton}

In physics, chemistry, and electronic engineering, an electron hole (often simply called a hole) is the lack of an electron at a position where one could exist in an atom or atomic lattice. Since in a normal atom or crystal lattice the negative charge of the electrons is balanced by the positive charge of the atomic nuclei, the absence of an electron leaves a net positive charge at the hole's location. 

Holes in a metal^[1] or semiconductor crystal lattice can move through the lattice as electrons can, and act similarly to positively-charged particles. They play an important role in the operation of semiconductor devices such as transistors, diodesand integrated circuits. If an electron is excited into a higher state it leaves a hole in its old state. This meaning is used in Auger electron spectroscopy (and other x-ray techniques), in computational chemistry, and to explain the low electron-electron scattering-rate in crystals (metals, semiconductors). Although they act like elementary particles, holes are not actually particles, but rather quasiparticles; they are different from the positron, which is the antiparticle of the electron. (See also Dirac sea.)

In crystals, electronic band structure calculations lead to an effective mass for the electrons, which is typically negative at the top of a band. The negative mass is an unintuitive concept,^[2] and in these situations, a more familiar picture is found by considering a positive charge with a positive mass.

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

The 
J/ψ
 (J/psi) meson /ˈdʒeɪ ˈsaɪ ˈmiːzɒn/ or psion^[1] is a subatomic particle, a flavor-neutral meson consisting of a charm quark and a charm antiquark. Mesons formed by a bound state of a charm quark and a charm anti-quark are generally known as "charmonium". The 
J/ψ
 is the most common form of charmonium, due to its spin of 1 and its low rest mass. The 
J/ψ
 has a rest mass of 3.0969 GeV/c², just above that of the 
η
c (2.9836 GeV/c²), and a mean lifetime of 7.2×10⁻²¹ s. This lifetime was about a thousand times longer than expected.^[2]

Its discovery was made independently by two research groups, one at the Stanford Linear Accelerator Center, headed by Burton Richter, and one at the Brookhaven National Laboratory, headed by Samuel Ting of MIT. They discovered they had actually found the same particle, and both announced their discoveries on 11 November 1974. The importance of this discovery is highlighted by the fact that the subsequent, rapid changes in high-energy physics at the time have become collectively known as the "November Revolution". Richter and Ting were awarded the 1976 Nobel Prize in Physics.

https://en.wikipedia.org/wiki/J/psi_meson

In supergravity theories combining general relativity and supersymmetry, the gravitino (
G͂
) is the gauge fermion supersymmetric partner of the hypothesized graviton. It has been suggested as a candidate for dark matter.

If it exists, it is a fermion of spin 3/2 and therefore obeys the Rarita-Schwinger equation. The gravitino field is conventionally written as ψ_μα with μ = 0, 1, 2, 3 a four-vector index and α = 1, 2 a spinor index. For μ = 0 one would get negative norm modes, as with every massless particle of spin 1 or higher. These modes are unphysical, and for consistency there must be a gauge symmetry which cancels these modes: δψ_μα = ∂_με_α, where ε_α(x) is a spinor function of spacetime. This gauge symmetry is a local supersymmetry transformation, and the resulting theory is supergravity.

Thus the gravitino is the fermion mediating supergravity interactions, just as the photon is mediating electromagnetism, and the graviton is presumably mediating gravitation. Whenever supersymmetry is broken in supergravity theories, it acquires a mass which is determined by the scale at which supersymmetry is broken. This varies greatly between different models of supersymmetry breaking, but if supersymmetry is to solve the hierarchy problem of the Standard Model, the gravitino cannot be more massive than about 1 TeV/c².

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

N = 4 supersymmetric Yang–Mills (SYM) theory is a mathematical and physical model created to study particles through a simple system, similar to  string theory, with conformal symmetry. It is a simplified toy theory based on Yang–Mills theory that does not describe the real world, but is useful because it can act as a proving ground for approaches for attacking problems in more complex theories.^[1] It describes a universe containing boson fields and fermion fields which are related by 4 supersymmetries (this means that swapping boson, fermion and scalar fields in a certain way leaves the predictions of the theory invariant). It is one of the simplest (because it has no free parameters except for the gauge group) and one of the few finite quantum field theories in 4 dimensions. It can be thought of as the most symmetric field theory that does not involve gravity.

https://en.wikipedia.org/wiki/N_%3D_4_supersymmetric_Yang–Mills_theory

09-20-2021-1058 - Clebsch–Gordan coefficients expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis.

In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. Mathematically, they specify the decomposition of the tensor product of two irreducible representations into a direct sum of irreducible representations, where the type and the multiplicities of these irreducible representations are known abstractly. The name derives from the German mathematicians Alfred Clebsch (1833–1872) and Paul Gordan (1837–1912), who encountered an equivalent problem in invariant theory.

Generalization to SU(3) of Clebsch–Gordan coefficients is useful because of their utility in characterizing hadronic decays, where a flavor-SU(3) symmetry exists (the eightfold way) that connects the three light  quarks: up, down, and strange.

https://en.wikipedia.org/wiki/Clebsch–Gordan_coefficients_for_SU(3)

In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n 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 case.

The group operation is matrix multiplication. The special unitary group is a normal subgroup of the unitary group U(n), consisting of all n×n unitary matrices. As a compact classical group, U(n) is the group that preserves the standard inner product on .^[a] It is itself a subgroup of the general linear group, .

The SU(n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.^[1]

The simplest case, SU(1), is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}.^[b] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations.

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

In particle physics, the Georgi–Glashow model^[1] is a particular grand unified theory (GUT) proposed by Howard Georgi and Sheldon Glashow in 1974. In this model the standard model gauge groups SU(3) × SU(2) × U(1) are combined into a single simple gauge group SU(5). The unified group SU(5) is then thought to be spontaneously broken into the standard model subgroup below a very high energy scale called the grand unification scale.

Since the Georgi–Glashow model combines leptons and quarks into single irreducible representations, there exist interactions which do not conserve baryon number, although they still conserve the quantum number B – Lassociated with the symmetry of the common representation. This yields a mechanism for proton decay, and the rate of proton decay can be predicted from the dynamics of the model. However, proton decay has not yet been observed experimentally, and the resulting lower limit on the lifetime of the proton contradicts the predictions of this model. However, the elegance of the model has led particle physicists to use it as the foundation for more complex models which yield longer proton lifetimes, particularly SO(10) in basic and SUSY variants.

(For a more elementary introduction to how the representation theory of Lie algebras are related to particle physics, see the article Particle physics and representation theory.)

This model suffers from the doublet–triplet splitting problem.

https://en.wikipedia.org/wiki/Georgi–Glashow_model

A Grand Unified Theory (GUT) is a model in particle physics in which, at high energies, the three gauge interactions of the Standard Model comprising the electromagnetic, weak, and strong forces are merged into a single force. Although this unified force has not been directly observed, the many GUT models theorize its existence. If unification of these three interactions is possible, it raises the possibility that there was a grand unification epoch in the very early universein which these three fundamental interactions were not yet distinct.

Experiments have confirmed that at high energy the electromagnetic interaction and weak interaction unify into a single electroweak interaction. GUT models predict that at even higher energy, the strong interaction and the electroweak interaction will unify into a single electronuclear interaction. This interaction is characterized by one larger gauge symmetry and thus several force carriers, but one unified coupling constant. Unifying gravity with the electronuclear interaction would provide a more comprehensive theory of everything (TOE) rather than a Grand Unified Theory. Thus, GUTs are often seen as an intermediate step towards a TOE.

The novel particles predicted by GUT models are expected to have extremely high masses—around the GUT scale of  GeV (just a few orders of magnitude below the Planck scale of  GeV)—and so are well beyond the reach of any foreseen particle collider experiments. Therefore, the particles predicted by GUT models will be unable to be observed directly, and instead the effects of grand unification might be detected through indirect observations such as proton decay, electric dipole moments of elementary particles, or the properties of neutrinos.^[1] Some GUTs, such as the Pati-Salam model, predict the existence of magnetic monopoles.

While GUTs might be expected to offer simplicity over the complications present in the Standard Model, realistic models remain complicated because they need to introduce additional fields and interactions, or even additional dimensions of space, in order to reproduce observed fermion masses and mixing angles. This difficulty, in turn, may be related to an existence^{[clarification needed]} of family symmetries beyond the conventional GUT models. Due to this, and the lack of any observed effect of grand unification so far, there is no generally accepted GUT model.

Models that do not unify the three interactions using one simple group as the gauge symmetry, but do so using semisimple groups, can exhibit similar properties and are sometimes referred to as Grand Unified Theories as well.

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