Isospin

In physics, and specifically, particle physics, isospin (isotopic spin, isobaric spin) is a symmetry of the strong interaction as it applies to the interactions of the neutron and proton. Isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions of baryons and mesons. Isospin symmetry remains an important concept in particle physics, and a close examination of this symmetry historically lead directly to the discovery and understanding of quarks and of the development of Yang-Mills theory.

Flavour in particle physics

Flavour quantum numbers

Lepton number: L
Baryon number: B
Electric charge: Q
Weak hypercharge: Y_W
Weak isospin: T_z
Isospin: I, I_z
Hypercharge: Y
Strangeness: S
Charm: C
Bottomness: B'
Topness: T

Y=B+S+C+B'+T
Q=I_z+Y/2
Q=T_z+Y_W/2
B−L

Symmetry

Isospin was introduced by Werner Heisenberg to explain several related symmetries:

The mass of the neutron and the proton are almost identical: they are nearly degenerate, and are thus often called nucleons. 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.
The mass of the pions which mediate the strong interaction between the nucleons are the same. In particular, the mass of the postively-charged pion is identical to that of the negatively-charged pion, and both have nearly the same mass as the neutral pion.

In quantum mechanics, when a Hamiltonian has a symmetry, that symmetry manifests itself through a set of states that have (almost) the same energy; that is, the states are degenerate. In particle physics, mass is the same thing as energy (since E=mc²), and so the near mass-degeneracy of the neutron and proton points at a symmetry of the Hamiltonian describing the strong interactions. The neutron does have a slightly higher mass: the mass degeneracy is not exact. The proton is charged, the neutron is not. However, here, as the case would be in general for quantum mechanics, the appearance of a symmetry can be imperfect, as it is perturbed by other forces, which give rise to slight differences between states.

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, from 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-1/2 or fundamental representation) of SU(2). The pions are assigned to the triplet (the spin-1 or adjoint representation) of SU(2).

Just as is the case for regular spin, isospin is described by two numbers, I, the total isospin, and I₃, the component of the spin vector in a given direction. The proton and neutron both have I=1/2, as they belong to the doublet. The proton has I₃=+1/2 or 'isospin-up' and the neutron has I₃=−1/2 or 'isospin-down'. The pions, belonging to the triplet, have I=1, and π⁺, π⁰ and π⁻ have, respectively, I₃=+1, 0, −1.

Yang-Mills

Isospin symmetry was central to the original formulation of Yang-Mills theory. The pions were proposed to be the SU(2) gauge bosons of this theory. While it is now understood that isospin symmetry is not a true gauge symmetry, this initial confusion was historically important for the development of the overall ideas of gauge invariance.

Relationship to flavour

The discovery and subsequent close analysis of additional particles, both mesons and baryons, made it clear that the concept of isospin symmetry could be broadened to an even larger symmetry group, now called flavour symmetry. Once the kaons and their property of strangeness became better understood, it started to become clear that these, too, seemed to be a part of an enlarged, more general symmetry that contained isospin as a subset. The larger symmetry was named the Eight-fold Way by Murray Gell-Mann, and was promptly recognized to correspond to the adjoint representation of SU(3). This immediately lead to Gell-Mann's proposal of the existence of quarks. The quarks would belong to the fundamental representation of the flavour SU(3) symmetry, and it is from the fundamental rep, and its conjugate (the quarks and the anti-quarks) that the higher representation (the mesons and baryons) could be assembled. In short, the theory of Lie groups and Lie algebras modelled the physical reality of particles in the most exceptional and unexpected way.

The discovery of the J/ψ meson and charm lead to the expansion of flavour symmetry to SU(4), and the discovery of the upsilon meson (and the corresponding top and bottom quarks) lead to the current SU(6) flavour symmetry. Isospin symmetry is just one little corner of this broader symmetry. There are strong theoretical reasons, confirmed by experiment, that lead one to believe that things stop there, and that there are no further quarks to be found.

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 and down quarks. Technically, the nucleon doublet is seen to be the product of a single quark (thus, a doublet) and a pair of quarks in a singlet state. That is, the proton wave function, in terms of quark flavor eigenstates, is described by

$\vert p\rangle = \vert u\rangle \frac{1}{\sqrt{2}} \left( \vert ud\rangle + \vert du\rangle \right) + \mbox {perms.}$

and the neutron by

$\vert n\rangle = \vert d\rangle \frac{1}{\sqrt{2}} \left( \vert ud\rangle + \vert du\rangle \right) + \mbox {perms.}$

where perms stands for permutations. Here, $\vert u \rangle$ is the up quark flavour eigenstate, and $\vert d \rangle$ is the down quark flavour eigenstate. Although the above is the technically correct way of denoting a proton and neutron in terms of quark flavour eigenstates, this is almost always glossed over, and these are more simply referred to as uud and udd.

Similarly, the isopsin symmetry of the pions are given by:

$\vert \pi^+\rangle = \vert u\overline {d}\rangle$

$\vert \pi^0\rangle = \frac{1}{\sqrt{2}} \left(\vert u\overline {u}\rangle - \vert d \overline{d} \rangle \right)$

$\vert \pi^-\rangle = \vert d\overline {u}\rangle$

The overline denotes, as usual, the complex conjugate representation of SU(2), or, equivalently, the antiquark.

Weak isospin

The quarks also feel the weak interaction, however, the mass eigenstates of the strong interaction are not exactly the same as the eigenstates of the weak interaction. Thus, while there are still a pair of quarks u and d that take part in the weak interaction, they are not quite the same as the strong u and d quarks. The difference is given by a rotation, whose magnitude is called the Cabibbo angle or more generally, the CKM matrix.