Isospin

In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions of baryons and mesons.

The name of the concept contains the term ''spin'' because its quantum mechanical description is mathematically similar to that of angular momentum (in particular, in the way it couples; for example, a proton–neutron pair can be coupled either in a state of total isospin 1 or in one of 0). But unlike angular momentum, it is a dimensionless quantity and is not actually any type of spin.

Etymologically, the term was derived from isotopic spin, a confusing term to which nuclear physicists prefer isobaric spin, which is more precise in meaning. Before the concept of quarks was introduced, particles that are affected equally by the strong force but had different charges (e.g. protons and neutrons) were considered different states of the same particle, but having isospin values related to the number of charge states.. A close examination of isospin symmetry ultimately led directly to the discovery and understanding of quarks and to the development of Yang–Mills theory. Isospin symmetry remains an important concept in particle physics.

Quark content and isospin

In the modern formulation, isospin () is defined as a vector quantity in which up and down quarks have a value of  = 1/2, with the 3rd-component (₃) being +1/2 for up quarks, and −1/2 for down quarks, while all other quarks have  = 0. Therefore, for hadrons in general, where _u and _d are the numbers of up and down quarks respectively,
: $I_3 = \frac(n_u - n_d).$

In any combination of quarks, the 3rd component of the isospin vector (₃) could either be aligned between a pair of quarks, or face the opposite direction, giving different possible values for total isospin for any combination of quark flavours. Hadrons with the same quark content but different total isospin can be distinguished experimentally, verifying that flavour is actually a vector quantity, not a scalar (up vs down simply being a projection in the quantum mechanical  axis of flavour space).

For example, a strange quark can be combined with an up and a down quark to form a baryon, but there are two different ways the isospin values can combine either adding (due to being flavour-aligned) or cancelling out (due to being in opposite flavour directions). The isospin-1 state (the ) and the isospin-0 state (the ) have different experimentally detected masses and half-lives.

Isospin and symmetry

Isospin is regarded as a symmetry of the strong interaction under the action of the Lie group SU(2), the two states being the up flavour and down flavour. In quantum mechanics, when a Hamiltonian has a symmetry, that symmetry manifests itself through a set of states that have the same energy (the states are described as being ''degenerate''). In simple terms, the energy operator for the strong interaction gives the same result when an up quark and an otherwise identical down quark are swapped around.

Like the case for regular spin, the isospin operator I is vector-valued: it has three components I_''x'', I_''y'', I_''z'', which are coordinates in the same 3-dimensional vector space where the 3 representation acts. Note that this vector space has nothing to do with the physical space, except similar mathematical formalism. Isospin is described by two quantum numbers: the total isospin, and ₃ an eigenvalue of the I_''z'' projection for which flavor states are eigenstates, . In other words, each ₃ state specifies certain flavor state of a multiplet. The third coordinate (), to which the "3" subscript refers, is chosen due to notational conventions that relate bases in 2 and 3 representation spaces. Namely, for the spin-1/2 case, components of I are equal to Pauli matrices divided by 2, and so I_''z'' =  ₃, where

: $\tau_3 = \begin 1 & 0 \\ 0 & -1 \end.$

While the forms of these matrices are isomorphic to those of spin, ''these'' Pauli matrices only act within the Hilbert space of isospin, not that of spin, and therefore is common to denote them with τ rather than σ to avoid confusion.

Although isospin symmetry is actually 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, which would hold if all six quarks were identical. However, the very much larger masses of the charm, bottom, and top quarks means that SU(6) flavour symmetry is very badly broken in nature (at least at low energies), and assuming this symmetry leads to qualitatively and quantitatively incorrect predictions. In modern applications, such as lattice QCD, isospin symmetry is often treated as exact for the three light quarks (uds), while the three heavy quarks (cbt) must be treated separately.

Hadron nomenclature

Hadron nomenclature is based on isospin.

* Particles of total isospin 3/2 are named Delta baryons and can be made by a combination of any three up or down quarks (but only up or down quarks).
* Particles of total isospin 1 can be made from two up quarks, two down quarks, or one of each:
** certain mesons further differentiated by total spin into pions (total spin 0) and rho mesons (total spin 1)
** with an additional quark of higher flavour Sigma baryons
* Particles of total isospin 1/2 can be made from:
** a single up or down quark together with an additional quark of higher flavour strange (kaons), charm ( D meson), or bottom ( B meson)
** a single up or down quark together with two additional quarks of higher flavour Xi baryon
** an up quark, a down quark, and either an up or a down quark nucleons. Note that three identical quarks would be forbidden by the Pauli exclusion principle due to requirement of anti-symmetric wave function
* Particles of total isospin 0 can be made from
** a neutral quark-antiquark pair: $u\bar$ or $d\bar$ eta mesons
** one up quark and one down quark, with an additional quark of higher flavour Lambda baryons
** anything not involving any up or down quarks

History

Original motivation for isospin

Isospin was introduced as a concept in 1932, well before the 1960s development of the quark model. The man who introduced it, Werner Heisenberg, did so to explain symmetries of the then newly discovered neutron (symbol n):
* The mass of the neutron and the proton (symbol p) are almost identical: they are nearly degenerate, and both are thus often called nucleons. Although the proton has a positive electric charge, and the neutron is neutral, they are almost identical in all other aspects.
* 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.

This behavior is not unlike the electron, where there are two possible states based on their spin. Other properties of the particle are conserved in this case. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, Eugene Wigner introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.

Protons and neutrons were then grouped together as nucleons because they both have nearly the same mass and interact in nearly the same way, if the (much weaker) electromagnetic interaction is neglected. In particle physics, the near mass-degeneracy of the neutron and proton points to an approximate symmetry of the Hamiltonian describing the strong interactions. It was thus convenient to treat them as being different states of the same particle.

Heisenberg's particular 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. The neutron and the proton are assigned to the doublet