# 8.1: The First Symmetry - Isospin

- Page ID
- 15045

The first particles that show an interesting symmetry are actually the nucleon and the proton. Their masses are remarkably close,

\[M_{p}=939.566\;{\rm MeV/c^{2}}\quad M_{n}= 938.272\;{\rm MeV/c^{2}}.\]

If we assume that these masses are generated by the strong interaction there is more than a hint of symmetry here. Further indications come from the pions: they come in three charge states, and once again their masses are remarkably similar,

\[M_{\pi^{+}}=M_{\pi^{-}}=139.567\;{\rm MeV/c^{2}},\quad M_{\pi^{0}}=134.974\;{\rm MeV/c^{2}}.\]

This symmetry is reinforced by the discovery that the interactions between nucleon (\(p\) and \(n\)) is independent of charge, they only depend on the nucleon character of these particles – the strong interactions see only one nucleon and one pion. Clearly a continuous transformation between the nucleons and between the pions is a symmetry. The symmetry that was proposed (by Wigner) is an internal symmetry like spin symmetry called **isotopic spin **or **isospin**. It is an abstract rotation in isotopic space, and leads to similar type of states with isotopic spin \(I=1/2,1,3/2,\ldots\). One can define the third component of isospin as

\[Q= e(I_{3}+B),\]

where \(B\) is the baryon number (\(B=1\) for \(n,p\), \(0\) for \(\pi\)). We thus find

\[\begin{array}{lllll} &B&Q/e&I&I_{3}\\ \hline n & 1 & 0& 1/2 & -1/2\\ p & 1 & 1& 1/2 & 1/2 \\ \pi^{-} & 0 & -1 &1 & -1\\ \pi^{0} & 0 & 0 & 1 & 0\\ \pi^{+} & 0 & 1 & 1 & 1 \end{array}\]

Notice that the energy levels of these particles are split by a magnetic force, as ordinary spins split under a magnetic force.