Thus, isospin was introduced as a concept well before the development in the 1960s of the quark model which provides our modern understanding.
The nucleons, baryons of spin , 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 particle has two states, the two were said to be of isospin . The proton and neutron were then associated with different isospin projections Iz = + and − respectively. When constructing a physical theory of nuclear forces, 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 pions in 1947. The three pions (, ) could be assigned to an isospin triplet with I = 1 and Iz = +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 = of K mesons, a triplet I = 1 of baryons, a single I = 0 , four I = baryons, and so on. This multiplet structure was combined with strangeness in Murray Gell-Mann's Eightfold Way, ultimately leading to the quark model and quantum chromodynamics.
However, because the up and down quarks have different charges ( e and − e respectively), the four Deltas also have different charges ((uuu), (uud), (udd), (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 Iz or I3) 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 (giving the projections Iz = , , −, −. Thus, all the Deltas were said to have isospin I = and each individual charge had different Iz (e.g. the was associated with Iz = +).
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 Iz = (Nu − Nd) where Nu and Nd 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.
In quantum mechanics, when a Hamiltonian has a symmetry, that symmetry manifests itself through a set of states that have the same energy; that is, the states are 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 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.
Just as is the case for regular spin, isospin is described by two quantum numbers, I, the total isospin, and Iz, the component of the spin vector in some direction.
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 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.
|Baryon group||I||Iz = +||Iz = +1||Iz = +||Iz = 0||Iz = −||Iz = −1||Iz = −|
|Double charmed Xis||(ucc)||(dcc)|
|Charmed bottom Xis||(ucb)||(dcb)|
|Double bottom Xis||(ubb)||(dbb)|
|Double charmed Omegas||0||(scc)|
|Charmed bottom Omegas||0||(scb)|
|Double bottom Omegas||0||(sbb)|
|Triple Charmed Omegas||0||(''ccc}|
|Double charmed bottom Omegas||0||(ccb)|
|Charmed double bottom Omegas||0||(cbb)|
|Triple bottom Omegas||0||(bbb)|
and the (spin-up) neutron by
Here, is the up quark flavour eigenstate, and is the down quark flavour eigenstate, while and are the eigenstates of . 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:
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. 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 kaons. The coupling to the pions 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 mesons 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 quarks 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