8.1: The First Symmetry - Isospin

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

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

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,

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

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

$$Q= e(I_{3}+B), \nonumber$$

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

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

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