Isospin, SU(2), and Flavor SU(3)

The proton and neutron differ in mass by little more than a tenth of a percent, and the three pions by a few percent. To the strong force, which supplies almost all of a hadron's mass, these differences are negligible: it treats proton and neutron as two states of one object, and the three pions as three states of another. That statement is a continuous internal symmetry — isospin, an $SU(2)$ acting not on space but on an abstract charge space — and it comes with the full apparatus of a non-Abelian symmetry: multiplets, ladder operators, and conserved quantum numbers. Enlarging it to include strangeness gives the approximate flavor $SU(3)$ whose weight diagrams are the stage on which the quark model is built. This lesson develops that machinery operationally, from the nucleon doublet to the isospin–hypercharge plane.

Isospin as an SU(2) symmetry

Heisenberg observed in 1932 that the strong interaction is charge-independent: the nuclear force between two protons, two neutrons, or a proton and neutron is the same once the electromagnetic contribution is removed. The proton and neutron are then two states of a single nucleon, distinguished by a label that behaves exactly like the projection of a spin-$\frac{1}{2}$ angular momentum.¹ This label is isospin $I$ — the name borrows the formalism of spin while having nothing to do with rotations in space.

The isospin operators satisfy the same algebra as angular momentum, the Lie algebra of $SU(2)$,

so a multiplet of total isospin $I$ contains $2I+1$ states labeled by the third component $I_3=-I,\dots ,+I$. The nucleon is an isospin doublet, $I=\frac{1}{2}$:

The pions form an isospin triplet, $I=1$, with ${\pi }^{+}$, ${\pi }^0$, ${\pi }^{-}$ carrying $I_3=+1,0,-1$. Because isospin is an exact symmetry of the strong force alone, it is conserved in strong reactions and broken — by the electromagnetic force, which does see charge, and by the up–down quark mass difference — only at the percent level.

Ladder operators and multiplet structure

The non-Abelian structure gives more than a counting rule: it connects the members of a multiplet. Define the raising and lowering operators

which shift $I_3$ by one unit,

For the nucleon, $I_{+}$ turns a neutron into a proton and $I_{-}$ does the reverse; for the pion triplet the ladder walks ${\pi }^{-}\to {\pi }^0\to {\pi }^{+}$. Because these operators commute with the strong Hamiltonian, states connected by the ladder are degenerate up to isospin-breaking corrections — this is precisely why the multiplet members share a mass.

Isospin conservation makes quantitative predictions about strong-interaction rates. Because the initial and final states must be decomposed into total-isospin eigenstates using Clebsch–Gordan coefficients, the cross sections for related reactions stand in fixed ratios. The classic case is pion–nucleon scattering through the $\Delta$ resonance ($I=\frac{3}{2}$): at the resonance the ratios

follow purely from isospin, and are confirmed by experiment — direct evidence that the symmetry is real and not merely bookkeeping.²

From SU(2) to flavor SU(3)

The discovery of strange particles added a new quantum number, strangeness $S$, conserved by the strong force. The natural move is to enlarge the symmetry: if the strong interaction is nearly blind to the difference between up and down quarks (isospin $SU(2)$), it is also — though less exactly, because the strange quark is heavier — blind to the difference among up, down, and strange. That approximate symmetry is flavor $SU(3)$, observed independently by Gell-Mann and Ne'eman in 1961 and named the eightfold way.³

$SU(3)$ has rank two: two of its generators commute simultaneously and can be diagonalized together. One is the isospin third component $I_3$; the other is the hypercharge $Y=B+S$ (for the light hadrons). A hadron is therefore labeled by the pair $(I_3,Y)$, and a multiplet becomes a two-dimensional pattern — a weight diagram — rather than a one-dimensional ladder. The single ladder of $SU(2)$ is replaced by three sets of raising and lowering operators, one for each $SU(2)$ subgroup embedded in $SU(3)$, and these move a state along three directions in the $(I_3,Y)$ plane.

The Gell-Mann–Nishijima relation

The link between these internal labels and the observable electric charge is the Gell-Mann–Nishijima relation,

which holds for every hadron. It was first found empirically as a way to organize the strange particles, and it fixes the electric charge of each node on a weight diagram: the isospin third component sets the horizontal position, the hypercharge sets the vertical, and their combination gives the charge. Sloping lines of constant $Q$ cut across the diagram at a fixed angle.

Read the other way, the relation is a constraint: given the electric charge and the isospin assignment of a candidate state, its hypercharge — and so its strangeness — is fixed. This same relation determines the quark charges themselves. The up quark sits at $(I_3,Y)=(+\frac{1}{2},+\frac{1}{3})$ giving $Q=+\frac{2}{3}$, the down at $(-\frac{1}{2},+\frac{1}{3})$ giving $Q=-\frac{1}{3}$, and the strange at $(0,-\frac{2}{3})$ giving $Q=-\frac{1}{3}$ — the fundamental triplet of flavor $SU(3)$.

Weight diagrams: the language of the quark model

The payoff is that the entire hadron spectrum is organized by these plots. The three light quarks form the fundamental $SU(3)$ triplet at the corners of a small triangle in the $(I_3,Y)$ plane; the antiquarks form the inverted antitriplet. Combining a quark with an antiquark, or three quarks together, builds the meson and baryon multiplets as larger patterns in the same plane — the hexagons, triangles, and central multiplicities that the next module derives in full.

The logic runs in one direction throughout. A symmetry of the strong force — exact for isospin $SU(2)$, approximate for flavor $SU(3)$ — organizes the hadrons into multiplets; the multiplets are patterns on a weight diagram; and the regularity of those patterns, together with a gap where the ${\Omega }^{-}$ was later found, is what demanded that hadrons be built from a few fundamental constituents. Isospin and its $SU(3)$ enlargement are thus not just conservation laws but the direct predecessors of the quark model, whose construction is the subject of the next module.