- Isospin
In

physics , and specifically,particle physics ,**isospin**(**"isotopic spin", "isobaric spin**") is aquantum number related to thestrong interaction . This term was derived from**"isotopic spin**", but the term**"isotopic spin**" is confusing as two isotopes of a nucleus have different numbers of nucleons; in contrast, rotations of isospin maintain the number of nucleons. Nuclear physicists prefer**"isobaric spin**", which is more precise in meaning. Isospin symmetry is a subset of theflavour symmetry seen more broadly in the interactions ofbaryon s andmeson s. Isospin symmetry remains an important concept in particle physics, and a close examination of this symmetry historically led directly to the discovery and understanding ofquark s and of the development ofYang-Mills theory .**Motivation for isospin**[

Combinations of three u, d or s-quarks forming baryons withspin -frac|3|2 form the "baryon decuplet".]Isospin was introduced by

Werner Heisenberg in 1932 [*"Über den Bau der Atomkerne" (Zeitschrift für Physik 77: 1-11)*] (although it was named byEugene Wigner in 1937 [*Physical Review 51: 106-119*] ) to explain symmetries of the then newly discoveredneutron :

* Themass of the neutron and the proton are almost identical: they are nearly degenerate, and both are thus often callednucleon s. Although the proton has a positive charge, and the neutron is neutral, they are almost identical in all other respects.

* The strength of the strong interaction between any pair of nucleons is the same, independent of whether they are interacting as protons or as neutrons.Thus, isospin was introduced as a concept well before the development in the 1960s of thequark model which provides our modern understanding.The

nucleon s, baryons of spin frac|1|2, were grouped together because they both have nearly the same mass and interact in nearly the same way. Thus, it was convenient to treat them as being different states of the same particle. Since a spin frac|1|2 particle has two states, the two were said to be of isospin frac|1|2. The proton and neutron were then associated with different isospin projections "I_{z}" = +frac|1|2 and −frac|1|2 respectively. When constructing a physical theory ofnuclear force s, one could then simply assume that it does not depend on isospin.These considerations would also prove useful in the analysis of

meson -nucleon interactions after the discovery of thepion s in 1947. The three pions ("SubatomicParticle|pion+", "SubatomicParticle|pion0", "SubatomicParticle|pion-") could be assigned to an isospin triplet with "I" = 1 and "I_{z}" = +1, 0 or −1. By assuming that isospin was conserved by nuclear interactions, the new mesons were more easily accommodated by nuclear theory.As further particles were discovered, they were assigned into isospin multiplets according to the number of different charge states seen: a doublet "I" = frac|1|2 of "K" mesons, a triplet "I" = 1 of SubatomicParticle|sigma baryons, a single "I" = 0 SubatomicParticle|lambda, four "I" = frac|3|2 SubatomicParticle|delta baryons, and so on. This multiplet structure was combined with

strangeness inMurray Gell-Mann 'sEightfold Way , ultimately leading to thequark model andquantum chromodynamics .**Modern understanding of isospin**Observation of the light

baryon s (those made of up, down andstrange quark s) lead us to believe that some of these particles are so similar in terms of theirstrong interaction s that they can be treated as different states of the same particle. In the modern understanding ofquantum chromodynamics , this is because up and down quarks are very similar in mass, and have the same strong interactions. Particles made of the same numbers of up and down quarks have similar masses and are grouped together. For examples, the particles known as theDelta baryon s — baryons of spin frac|3|2 made of a mix of three up and down quarks — are grouped together because they all have nearly the same mass (approximately val|1232|ul=MeV/c2), and interact in nearly the same way.However, because the up and down quarks have different charges (frac|2|3 "e" and −frac|1|3 "e" respectively), the four Deltas also have different charges (SubatomicParticle|Delta++ ("uuu"), SubatomicParticle|Delta+ ("uud"), SubatomicParticle|Delta0 ("udd"), SubatomicParticle|Delta- ("ddd")). These Deltas could be treated as the same particle and the difference in charge being due to the particle being in different states. Isospin was devised as a parallel to spin to associate an isospin projection (denoted "I

_{z}" or "I_{3}") to each charged state. Since there were four Deltas, four projections were needed. Because isospin was modeled on spin, the isospin projections were made to vary in increments of 1 and to have four increments of 1, you needed an isospin value of frac|3|2 (giving the projections "I_{z}" = frac|3|2, frac|1|2, −frac|1|2, −frac|3|2. Thus, all the Deltas were said to have isospin "I" = frac|3|2 and each individual charge had different "I_{z}" (e.g. the SubatomicParticle|Delta++ was associated with "I_{z}" = +frac|3|2).After the quark model was elaborated, it was noted that the isospin projection was related to the up and down quark content of particles. The relation is "I

_{z}" = frac|1|2("N_{u}" − "N_{d}") where "N_{u}" and "N_{d}" are the number of up and down quarks respectively.In the isospin picture, the four Deltas and the two nucleons were thought to be the different states of two particles. In the quark model, the Deltas can be thought of as the excited states of the nucleons.

**Isospin symmetry**In

quantum mechanics , when aHamiltonian has a symmetry, that symmetry manifests itself through a set of states that have the same energy; that is, the states are degenerate. Inparticle physics , the near mass-degeneracy of the neutron and proton points to an approximate symmetry of the Hamiltonian describing the strong interactions. The neutron does have a slightly higher mass due to isospin breaking; this is due to the difference in the masses of the up and down quarks and the effects of the electromagnetic interaction. However, the appearance of an approximate symmetry is still useful, since the small breakings can be described by aperturbation theory , which gives rise to slight differences between the near-degenerate states.**U(2)**Heisenberg's contribution was to note that the mathematical formulation of this symmetry was in certain respects similar to the mathematical formulation of spin, whence the name "isospin" derives. To be precise, the isospin symmetry is given by the invariance of the Hamiltonian of the strong interactions under the action of the

Lie group SU(2) . The neutron and the proton are assigned to the doublet (the spin-frac|1|2,**2**, orfundamental representation ) of SU(2). The pions are assigned to the triplet (the spin-1,**3**, oradjoint representation ) of SU(2).Just as is the case for regular spin, isospin is described by two

quantum number s, "I", the total isospin, and "I"_{z}, the component of the spin vector in some direction.**Relationship to flavor**The discovery and subsequent analysis of additional particles, both

meson s andbaryon s, made it clear that the concept of isospin symmetry could be broadened to an even larger symmetry group, now calledflavor symmetry . Once thekaon s and their property of strangeness became better understood, it started to become clear that these, too, seemed to be a part of an enlarged symmetry that contained isospin as a subgroup. The larger symmetry was named the Eight-fold Way byMurray Gell-Mann , and was promptly recognized to correspond to the adjoint representation ofSU(3) . To better understand the origin of this symmetry, Gell-Mann proposed the existence of up, down and strangequark s which would belong to the fundamental representation of the SU(3) flavor symmetry.Although isospin symmetry is very slightly broken, SU(3) symmetry is more badly broken, due to the much higher mass of the strange quark compared to the up and down. The discovery of

charm ,bottomness andtopness could lead to further expansions up toSU(6) flavour symmetry, but the very large masses of these quarks makes such symmetries almost useless. In modern applications, such aslattice QCD , isospin symmetry is often treated as exact while the heavier quarks must be treated separately.**Quark content and isospin**Up and down quarks each have isospin "I" = frac|1|2, and isospin z-components ("I

_{z}") of frac|1|2 and −frac|1|2 respectively. All other quarks have "I" = 0. A composite particle made of quarks must have "I_{z}" equal to the sum of the "I_{z}" of its quarks and "I" less than or equal*Particles of isospin frac|3|2 can only be made by a mix of three "u" and "d" quarks (Δ's).

*Particles of isospin 1 are made of a mix of two "u" quarks and "d" quarks (Σ's, "π**s, "ρ**s, etc.).

*Particles of isospin frac|1|2 can be made of a mix of three "u" and "d" quarks ("N**s) or from one "u" or "d" quark (Ξ's, "K**s, "D"'s, etc.)

*Particles of isospin 0 can be made of one "u" and one "d" quark (Λ's, "η**s, "ω**s, etc.), or from no "u" or "d" quarks at all (Ω's, "φ"'s, etc.).**Isospin symmetry of quarks**In the framework of the

Standard Model , the isospin symmetry of the proton and neutron are reinterpreted as the isospin symmetry of the up anddown quark s. Technically, the nucleon doublet states are seen to be linear combinations of products of 3-particle isospin doublet states and spin doublet states. That is, the (spin-up) protonwave function , in terms of quark-flavour eigenstates, is described by:$vert\; puparrow\; angle\; =\; frac\; 1\{3sqrt\; 2\}left(egin\{array\}\{ccc\}\; vert\; duu\; angle\; vert\; udu\; angle\; vert\; uud\; angle\; end\{array\}\; ight)\; left(egin\{array\}\{ccc\}\; 2\; -1\; -1\backslash \backslash \; -1\; 2\; -1\backslash \backslash \; -1\; -1\; 2\; end\{array\}\; ight)\; left(egin\{array\}\{c\}\; vertdownarrowuparrowuparrow\; angle\backslash \backslash \; vertuparrowdownarrowuparrow\; angle\backslash \backslash \; vertuparrowuparrowdownarrow\; angle\; end\{array\}\; ight)$Cite book|last=Greiner|first=Walter|coauthors=Müller, Berndt|title=Quantum mechanics: symmetries|year=1989|publisher=Springer Verlag|isbn=3540580808|pages=p. 279|url=http://books.google.com/books?id=gCfvWx6vuzUC&pg=RA4-PA279&sig=TDMI0NRc3MaCxXunKmRpdmlnQg0#PRA4-PA279,M1]

and the (spin-up) neutron by

:$vert\; nuparrow\; angle\; =\; frac\; 1\{3sqrt\; 2\}left(egin\{array\}\{ccc\}\; vert\; udd\; angle\; vert\; dud\; angle\; vert\; ddu\; angle\; end\{array\}\; ight)\; left(egin\{array\}\{ccc\}\; 2\; -1\; -1\backslash \backslash \; -1\; 2\; -1\backslash \backslash \; -1\; -1\; 2\; end\{array\}\; ight)\; left(egin\{array\}\{c\}\; vertdownarrowuparrowuparrow\; angle\backslash \backslash \; vertuparrowdownarrowuparrow\; angle\backslash \backslash \; vertuparrowuparrowdownarrow\; angle\; end\{array\}\; ight)$

Here, $vert\; u\; angle$ is the

up quark flavour eigenstate, and $vert\; d\; angle$ is thedown quark flavour eigenstate, while $vertuparrow\; angle$ and $vertdownarrow\; angle$ are the eigenstates of $S\_z$. Although these superpositions are the technically correct way of denoting a proton and neutron in terms of quark flavour and spin eigenstates, for brevity, they are often simply referred to as**uud**and**udd**. Note also that the derivation above assumes exact isospin symmetry and is modified by SU(2)-breaking terms.Similarly, the isopsin symmetry of the pions are given by:

:$vert\; pi^+\; angle\; =\; vert\; uoverline\; \{d\}\; angle$:$vert\; pi^0\; angle\; =\; frac\{1\}\{sqrt\{2left(vert\; uoverline\; \{u\}\; angle\; -\; vert\; d\; overline\{d\}\; angle\; ight)$:$vert\; pi^-\; angle\; =\; -vert\; doverline\; \{u\}\; angle$

**Weak isospin**Isospin is similar to, but should not be confused with

weak isospin . Briefly, weak isospin is the gauge symmetry of theweak interaction which connects quark and lepton doublets of left-handed particles in all generations; for example, up and down quarks, top and bottom quarks, electrons and electron neutrinos. Isospin connects only up and down quarks, acts on both chiralities (left and right) and is a global (not a gauge) symmetry.**Gauged isospin symmetry**Attempts have been made to promote isospin from a global to a local symmetry. In 1954,

Chen Ning Yang and Robert Mills suggested that the notion of protons and neutrons, which are continuously rotated into each other by isospin, should be allowed to vary from point to point. To describe this, the proton and neutron direction in isospin space must be defined at every point, giving local basis for isospin. Agauge connection would then describe how to transform isospin along a path between two points.This

Yang-Mills theory describes interacting vector bosons, like thephoton of electromagnetism. Unlike the photon, the SU(2) gauge theory would contain self-interacting gauge bosons. The condition ofgauge invariance suggests that they have zero mass, just as in electromagnetism.Ignoring the massless problem, as Yang and Mills did, the theory makes a firm prediction: the vector particle should couple to all particles of a given isospin "universally". The coupling to the nucleon would be the same as the coupling to the

kaon s. The coupling to thepion s would be the same as the self-coupling of the vector bosons to themselves.When Yang and Mills proposed the theory, there was no candidate vector boson.

J. J. Sakurai in 1960 predicted that there should be a massive vector boson which is coupled to isospin, and predicted that it would show universal couplings. Therho meson s were discovered a short time later, and were quickly identified as Sakurai's vector bosons. The couplings of the rho to the nucleons and to each other were verified to be universal, as best as experiment could measure. The fact that the diagonal isospin current contains part of the electromagnetic current led to the prediction of rho-photon mixing and the concept ofvector meson dominance , ideas which led to successful theoretical pictures of GeV-scale photon-nucleus scattering.Although the discovery of the

quark s led to reinterpretation of the rho meson as a vector bound state of a quark and an antiquark, it is sometimes still useful to think of it as the gauge boson of a hidden local symmetry [*Bando et al., "Is the ρ Meson a Dynamical Gauge Boson of Hidden Local Symmetry?" (1985) PRL54 1215.*]**References*** Claude Itzykson and Jean-Bernard Zuber, "Quantum Field Theory" (1980) McGraw-Hill Inc. New York. ISBN 0-07-032071-3

* David Griffiths, "Introduction to Elementary Particles" (1987) John Wiley & Sons Inc. New York. ISBN 0-471-60386-4

* Particle Physics Booklet - Particle Data Group CERN ( Version July 2000 )

*Wikimedia Foundation.
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### Look at other dictionaries:

**Isospin**— ist eine mit einer inneren Symmetrie verbundene Quantenzahl in der Theorie der Elementarteilchen. Allgemeiner wird das Konzept (so auch in der Festkörperphysik) verwendet, um Zweizustandssysteme zu beschreiben. Hierbei werden die beiden… … Deutsch Wikipedia**isospin**— [ izospin ] n. m. • XXe; en angl. 1963; contract. de spin isotopique ♦ Phys. Grandeur quantique associée au fait que l interaction entre deux nucléons est indépendante de la charge (SYN. spin isotopique). ● isospin nom masculin Grandeur physique… … Encyclopédie Universelle**isospin**— [ī′sō spin΄, ī′səspin΄] n. Particle Physics a quantum number used to distinguish the various states of electrical charge of a subatomic particle … English World dictionary**Isospín**— En física, y específicamente, en la física de partículas, el isospín (espín isotópico o espín isobárico) es un número cuántico relacionado a la interacción fuerte y aplicado a las interacciones del neutrón y el protón. Este término se deriva de… … Wikipedia Español**isospin**— /uy seuh spin /, n. Physics. See isotopic spin. [by shortening] * * * or isobaric spin or isotopic spin Property characteristic of families of related subatomic particles differing mainly in the values of their electric charge. The families are… … Universalium**Isospin**— En physique des particules, l isospin (contraction de spin isotopique) est une symétrie de l interaction forte qui est mathématiquement analogue au spin. On dit aussi que c est un nombre quantique. Sommaire 1 Historique 2 Symétrie 3 SU(2) … Wikipédia en Français**Isospin**— Iso|spin 〈m. 6; Phys.〉 Quantenzahl der Elementarteilchen zur Charakterisierung von Teilchen verschiedener Ladung bei annähernd gleicher Masse (z. B. Protonen u. Neutronen) [<grch. isos „gleich + engl. spin „Drehung“] * * * I|so|spin ↑ Spin. *… … Universal-Lexikon**isospín**— ► sustantivo masculino FÍSICA NUCLEAR Número cuántico indicador del estado de carga de una partícula elemental. * * * o espín isobárico o espín isotópico Propiedad característica de familias de partículas subatómicas relacionadas que difieren… … Enciclopedia Universal**isospin**— izotopinis sukinys statusas T sritis fizika atitikmenys: angl. isobaric spin; isospin; isotopic spin vok. Isobarenspin, m; Isotopenspin, m; Ladungsspin, m rus. изобарический спин, m; изоспин, m; изотопический спин, m pranc. isospin, m; spin… … Fizikos terminų žodynas**Isospin**— Iso|spin* der; s, s <aus gleichbed. engl. amerik. isospin zu ↑iso... u. engl. spin »schnelle Drehung«> Quantenzahl zur Klassifizierung von Elementarteilchen (Phys.) … Das große Fremdwörterbuch