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{{Refimprove|date=February 2008}}
{{Flavour quantum numbers}}
In [[particle physics]], '''isospin''' ('''''isotopic spin''''', '''''isobaric spin''''') is a [[quantum number]] related to the [[strong interaction]]. Particles that are affected equally by the strong force but have different charges (e.g. protons and neutrons) can be treated as being different states of the same particle with isospin values related to the number of charge states.<ref>http://www.thefreedictionary.com/isospin</ref>

Isospin does not have the units of angular momentum and is not a type of spin. It is a dimensionless quantity and the name derives from the fact that the mathematical structures used to describe it are very similar to those used to describe spin.

This term was derived from '''''isotopic spin''''' which is confusing and nuclear physicists prefer '''''isobaric spin''''', which is more precise in meaning. Isospin symmetry is a subset of the [[flavour symmetry]] seen more broadly in the interactions of [[baryon]]s and [[meson]]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 of [[quark]]s and of the development of [[Yang-Mills theory]].

==Motivation for isospin==
[[Image:Baryon-decuplet-small.svg|thumb|200px|Combinations of three u, d or s-quarks forming baryons with [[spin (physics)|spin]]-{{frac|3|2}} form the ''[[Eightfold way (physics)|baryon decuplet]]''.]]
[[Image:Baryon-octet-small.svg|thumb|200px|Combinations of three u, d or s-quarks forming baryons with spin-{{frac|1|2}} form the ''[[Eightfold way (physics)|baryon octet]]'']]

Isospin was introduced by [[Werner Heisenberg]] in 1932<ref>
{{cite journal
 |last=Heisenberg |first=W. |authorlink=Werner Heisenberg
 |year=1932
 |title=Über den Bau der Atomkerne
 |journal=[[Zeitschrift für Physik]]
 |volume=77 |pages=1–11
 |doi=10.1007/BF01342433
|bibcode = 1932ZPhy...77....1H }} {{De icon}}</ref> to explain symmetries of the then newly discovered [[neutron]]:
* The [[mass]] of the neutron and the proton are almost identical: they are nearly degenerate, and both are thus often called [[nucleon]]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 the [[quark model]] which provides our modern understanding.
The name ''isospin'' however, was introduced by [[Eugene Wigner]] in 1937.<ref>
{{cite journal
 |last=Wigner |first=E. |authorlink=Eugene Wigner
 |year=1937
 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei
 |journal=[[Physical Review]]
 |volume=51
 |pages=106–119
 |doi=10.1103/PhysRev.51.106
|bibcode = 1937PhRv...51..106W
 |issue=2 }}</ref>

Protons and neutrons, baryons of spin {{frac|1|2}}, were grouped together as [[nucleon]]s 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''<sub>3</sub>&nbsp;=&nbsp;+{{frac|1|2}} and −{{frac|1|2}} respectively. When constructing a physical theory of [[nuclear 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 the [[pion]]s in 1947. The three pions ({{SubatomicParticle|pion+}}, {{SubatomicParticle|pion0}}, {{SubatomicParticle|pion-}}) could be assigned to an isospin triplet with ''I''&nbsp;=&nbsp;1 and ''I''<sub>3</sub>&nbsp;=&nbsp;+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 multiplet]]s according to the number of different charge states seen: 2 doublets, ''I''&nbsp;=&nbsp;{{frac|-1|2}} and ''I''&nbsp;=&nbsp;{{frac|-1|2}} of [[kaon|K mesons]] ({{SubatomicParticle|Kaon-}}, {{SubatomicParticle|Antikaon0}}),({{SubatomicParticle|Kaon+}}, {{SubatomicParticle|Kaon0}}), a triplet ''I''&nbsp;=&nbsp;1 of Sigma baryons ({{SubatomicParticle|Sigma+}}, {{SubatomicParticle|Sigma0}}, {{SubatomicParticle|Sigma-}}) a singlet ''I''&nbsp;=&nbsp;0 Lambda baryon ({{SubatomicParticle|Lambda0}}), a quartet ''I''&nbsp;=&nbsp;{{frac|3|2}} Delta baryons ({{SubatomicParticle|Delta++}}, {{SubatomicParticle|Delta+}}, {{SubatomicParticle|Delta0}}, {{SubatomicParticle|Delta-}}), and so on. This multiplet structure was combined with [[Strangeness (particle physics)|strangeness]] in [[Murray Gell-Mann]]'s [[Eightfold way (physics)|eightfold way]], ultimately leading to the [[quark model]] and [[quantum chromodynamics]].

==Modern understanding of isospin==

Observation of the light [[baryon]]s (those made of [[up quark|up]], [[down quark|down]] and [[strange quark]]s) lead us to believe that some of these particles are so similar in terms of their [[strong interaction]]s that they can be treated as different states of the same particle. In the modern understanding of [[quantum 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 the [[Delta baryon]]s—baryons of [[spin (physics)|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}}&thinsp;''e'' and −{{frac|1|3}}&thinsp;''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<sub>3</sub>'') 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''<sub>3</sub>&nbsp;=&nbsp;{{frac|3|2}}, {{frac|1|2}}, −{{frac|1|2}}, −{{frac|3|2}}). Thus, all the Deltas were said to have isospin ''I''&nbsp;=&nbsp;{{frac|3|2}} and each individual charge had different ''I''<sub>3</sub> (e.g. the {{SubatomicParticle|Delta++}} was associated with ''I''<sub>3</sub>&nbsp;=&nbsp;+{{frac|3|2}}). 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.

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

:<math>I_\mathrm{3} = \frac{1}{2}\left[\left(n_\mathrm{u} - n_\mathrm{\bar u}) - (n_\mathrm{d} - n_\mathrm{\bar d}\right)\right]</math>

where ''n''<sub>u</sub> and ''n''<sub>d</sub> are the numbers of up and down quarks respectively, and ''n''<sub>{{SubatomicParticle|Up antiquark}}</sub> and ''n''<sub>{{SubatomicParticle|down antiquark}}</sub> are the numbers of up and down antiquarks respectively.

By this, the value of ''I''<sub>3</sub> of the nucleons [[proton]] (symbol p) and [[neutron]] (symbol n) is determined by their quark composition, ''uud'' for the proton and ''udd'' for the neutron.

==Isospin symmetry==

In [[quantum mechanics]], when a [[Hamiltonian (quantum mechanics)|Hamiltonian]] has a symmetry, that symmetry manifests itself through a set of [[eigenstate|states]] that have the same energy; that is, the states are [[degenerate energy level|degenerate]]. In [[particle 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 [[Symmetry breaking|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 a [[perturbation theory]], which gives rise to slight differences between the near-degenerate states.

===SU(2)===
{{see also|Representation theory of SU(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 (physics)|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 [[group action|action]] of the [[Lie group]] [[SU(2)]]. The neutron and the proton are assigned to the [[doublet (physics)|doublet]] (the spin-{{frac|1|2}}, '''2''', or [[fundamental representation]]) of SU(2). The pions are assigned to the [[spin triplet|triplet]] (the spin-1, '''3''', or [[Adjoint representation of a Lie group|adjoint representation]]) of SU(2). Though, there is a difference from the theory of spin: the group action does not preserve [[flavor (particle physics)|flavor]].

Like the case for regular spin, the isospin [[operator (physics)|operator]] {{math|'''I'''}} is [[vector space|vector]]-valued: it has three components {{math|'''I'''<sub>''x''</sub>, '''I'''<sub>''y''</sub>, '''I'''<sub>''z''</sub>}} which are coordinates in the same 3-dimensional vector space where the '''3''' representation acts. Note that it has nothing to do with the physical space, except similar mathematical formalism. Isospin is described by two [[quantum number]]s: ''I'', the total isospin, and ''I''<sub>3</sub>, an eigenvalue of the {{math|'''I'''<sub>''z''</sub>}} [[vector projection|projection]] for which flavor states are [[eigenstate]]s, not an ''arbitrary projection'' as in the case of spin. In other words, each ''I''<sub>3</sub> state specifies certain flavor state of a [[multiplet]]. The third coordinate (''z''), to which the "3" subscript refers, is chosen due to notational conventions which relate [[basis (linear algebra)|bases]] in '''2''' and '''3''' representation spaces. Namely, for the spin-{{frac|1|2}} case components of {{math|'''I'''}} are equal to [[Pauli matrices]] divided by 2 and {{math|1='''I'''<sub>''z''</sub> = {{sfrac|1|2}}''&sigma;''<sub>3</sub>}}, where the matrix <math>\sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}</math>.

==Relationship to flavor==

The discovery and subsequent analysis of additional particles, both [[meson]]s and [[baryon]]s, made it clear that the concept of isospin symmetry could be broadened to an even larger symmetry group, now called [[flavor symmetry]]. Once the [[kaon]]s and their property of [[Strangeness (particle physics)|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 [[Eightfold way (physics)|Eightfold Way]] by [[Murray Gell-Mann]], and was promptly recognized to correspond to the adjoint representation of [[SU(3)]]. To better understand the origin of this symmetry, Gell-Mann proposed the existence of up, down and strange [[quark]]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 (quantum number)|charm]], [[bottomness]] and [[topness]] could lead to further expansions up to [[SU(6)]] flavour symmetry, but the very large masses of these quarks makes such symmetries almost useless. In modern applications, such as [[lattice 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''&nbsp;=&nbsp;{{frac|1|2}}, and isospin 3-components (''I''<sub>3</sub>) of {{frac|1|2}} and &minus;{{frac|1|2}} respectively. All other quarks have ''I''&nbsp;=&nbsp;0. In general
:<math>I_3=\frac{1}{2}(n_u-n_d).\ </math>

===Hadron nomenclature===
{{Main|Baryon|Mesons}}
Hadron nomenclature is based on isospin.<ref name=PDGBaryonsymbols>
{{cite journal
 |author=C. Amsler ''et al.''
 |coauthors=([[Particle Data Group]])
 |year=2008
 |title=Review of Particle Physics: Naming scheme for hadrons
 |url=http://pdg.lbl.gov/2008/reviews/namingrpp.pdf
 |journal=[[Physics Letters B]]
 |volume=667 |pages=1
 |doi=10.1016/j.physletb.2008.07.018
|bibcode = 2008PhLB..667....1P }}</ref>{{Failed verification|date=September 2013|reason=I did not find the statement actually said in the source.}}
*Particles of isospin {{frac|3|2}} can only be made by a mix of three u and d quarks (Delta baryons).
*Particles of isospin 1 are made of a mix of two u and d quarks (Pi mesons, Rho mesons, Sigma baryons with one heavier quark, etc.).
*Particles of isospin {{frac|1|2}} can be made of a mix of three u and d quarks (nucleons) or from one u or d quark with heavier quarks (K mesons, D mesons, Xi baryons, etc.)
*Particles of isospin 0 can be made of one u and one d quark (Eta mesons, Omega mesons, Lambda baryons, etc.), or from no u or d quarks at all (Omega baryons, Phi mesons, etc.), with heavier quarks in all cases.

===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 quark|up]] and [[down 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) proton [[wave function]], in terms of quark-flavour eigenstates, is described by

:<math>\vert p\uparrow \rangle = \frac 1{3\sqrt 2}\left(\begin{array}{ccc} \vert duu\rangle & \vert udu\rangle & \vert uud\rangle \end{array}\right) \left(\begin{array}{ccc} 2 & -1 & -1\\ -1 & 2 & -1\\ -1 & -1 & 2 \end{array}\right) \left(\begin{array}{c} \vert\downarrow\uparrow\uparrow\rangle\\ \vert\uparrow\downarrow\uparrow\rangle\\ \vert\uparrow\uparrow\downarrow\rangle \end{array}\right)</math><ref name="greiner">
{{cite book
 |last1=Greiner |first1=W. |authorlink1=
 |last2=Müller |first2=B. |authorlink2=
 |year=1989
 |title=Quantum Mechanics: Symmetries
 |url=http://books.google.com/?id=gCfvWx6vuzUC&pg=RA4-PA279
 |page=279
 |publisher=[[Springer-Verlag]]
 |isbn=3-540-58080-8
 }}</ref>

and the (spin-up) neutron by

:<math>\vert n\uparrow \rangle = \frac 1{3\sqrt 2}\left(\begin{array}{ccc} \vert udd\rangle & \vert dud\rangle & \vert ddu\rangle \end{array}\right) \left(\begin{array}{ccc} 2 & -1 & -1\\ -1 & 2 & -1\\ -1 & -1 & 2 \end{array}\right) \left(\begin{array}{c} \vert\downarrow\uparrow\uparrow\rangle\\ \vert\uparrow\downarrow\uparrow\rangle\\ \vert\uparrow\uparrow\downarrow\rangle \end{array}\right)</math><ref name="greiner"/>

Here, <math>\vert u \rangle</math> is the [[up quark]] flavour eigenstate, and <math>\vert d \rangle</math> is the [[down quark]] flavour eigenstate, while <math>\vert\uparrow\rangle</math> and <math>\vert\downarrow\rangle</math> are the eigenstates of <math>S_z</math>. 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 [[pion]]s are given by:

:<math>\vert \pi^+\rangle = \vert u\overline {d}\rangle</math>
:<math>\vert \pi^0\rangle = \frac{1}{\sqrt{2}}\left(\vert u\overline {u}\rangle - \vert d \overline{d} \rangle \right)</math>
:<math>\vert \pi^-\rangle = -\vert d\overline {u}\rangle</math>

===Weak isospin===

{{main|weak isospin}}
Isospin is similar to, but should not be confused with [[weak isospin]]. Briefly, weak isospin is the gauge symmetry of the [[weak 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. By contrast (strong) isospin connects only up and down quarks, acts on both [[chirality (physics)|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 (physicist)|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. A [[gauge connection]] would then describe how to transform isospin along a path between two points.

This [[Yang-Mills]] theory describes interacting vector bosons, like the [[photon]] of electromagnetism. Unlike the photon, the SU(2) gauge theory would contain self-interacting gauge bosons. The condition of [[gauge 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 the [[pion]]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. The [[rho 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 of [[vector 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<ref>
{{cite journal
 |last1=Bando
 |first1=M.
 |last2=Kugo
 |first2=T.
 |last3=Uehara
 |first3=S.
 |last4=Yamawaki
 |first4=K.
 |last5=Yanagida
 |first5=T.
 |year=1985
 |title=Is the &rho; Meson a Dynamical Gauge Boson of Hidden Local Symmetry?
 |journal=[[Physical Review Letters]]
 |volume=54
 |issue=12
 |pages=1215–1218
 |doi=10.1103/PhysRevLett.54.1215
 |bibcode=1985PhRvL..54.1215B
 |pmid=10030967
}}</ref>

==References==
<references/>

==Further reading==
*{{cite book
 |last1=Itzykson |first1=C.
 |last2=Zuber |first2=J.-B.
 |year=1980
 |title=Quantum Field Theory
 |publisher=[[McGraw-Hill]]
 |isbn=0-07-032071-3
}}
*{{cite book
 |last=Griffiths |first=D. |authorlink=David Griffiths (physicist)
 |year=1987
 |title=Introduction to Elementary Particles
 |publisher=[[John Wiley & Sons]]
 |isbn=0-471-60386-4
}}

== External links ==
* [[Image:Queryensdf.jpg]] '''[http://www-nds.iaea.org/queryensdf Nuclear Structure and Decay Data - IAEA ]''' Nuclides' Isospin

[[Category:Baryons]]
[[Category:Hadrons]]
[[Category:Nuclear physics]]
[[Category:Particle physics]]
[[Category:Quarks]]
[[Category:Particle physics flavour quantum number]]