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The helium-four atomic nucleus, containing two protons and two neutrons. Its ground-state spin and isospin are zero. A quantized cubic four-Skyrmion supplies a corresponding nuclear state in the soliton description.

An antibaryon is the antiparticle of a baryon. In the valence quark model it consists of three antiquarks in a colour singlet. Charge conjugation reverses its additive charges and third component of isospin, preserving total isospin and spin.

The spin-one-half octet of flavor symmetry contains the nucleons, Sigma baryons, Lambda baryon, and Xi baryons. Its weight diagram has six peripheral states and two states at $(I_3,Y)=(0,0)$: ${\Sigma }^0$ and ${\Lambda }^0$, with different total isospin.

For an isospin singlet quark $h$ of charge $q_h$, the antisymmetric light-flavour pair gives an $I=0,J=1/2$ state with charge $q_h+1/3$. The symmetric light-flavour pairs $uu,\{ud\},dd$ give $I=1$ triplets of spins $1/2$ and $3/2$, with charges $q_h+4/3,q_h+1/3,q_h-2/3$. These follow from light-diquark spin-isospin correlation and addition of constituent electric charges, with zero orbital angular momentum.

Chiral symmetry allows independent flavor rotations of left-handed and right-handed fields. For two massless quark flavors, the non-Abelian chiral transformations form $SU(2{)}_L\times SU(2{)}_R$. The additional vector $U(1)$ symmetry of baryon number acts trivially on the pion field $U$. The low-energy pion field $U\in SU(2)$ transforms as $U\mapsto LU{R}^{†}$. Its derivative Skyrme model terms are invariant under constant $L,R$. A chosen vacuum $U=1$ preserves the diagonal subgroup $L=R$, which acts by conjugation and is the isospin symmetry. An ordinary common nonzero pion mass explicitly breaks the independent left/right symmetry to that vector subgroup; vacuum selection in the massless theory is instead spontaneous symmetry breaking. The singlet axial anomaly is a separate issue and is not part of this two-flavor non-anomalous group.

A rigid Skyrmion orientation is quantized on a cover of its Skyrmion collective-coordinate orbit. A lifted combined rotation in the static field's stabilizer subgroup identifies the same classical field and imposes ${\hat{D}}^J(R){\hat{D}}^I(A)\Psi ={\chi }_{\mathrm{FR}}\Psi$, where the sign is the chosen Finkelstein-Rubinstein constraints character of that configuration-space loop. Choosing the nontrivial character models unit baryons as fermions. A $2\pi$ spatial rotation or isorotation of charge $B$ then has sign $(-1{)}^B$: deforming the field to separated unit lumps adds the unit-lump loop classes in the fundamental group ${\mathbb{Z}}_2$, while the labelled orbital paths can be contracted in three dimensions. The resulting Finkelstein-Rubinstein constraints sign is the product of the unit-lump signs. Hence spin and isospin are half-integer for odd $B$ and integer for even $B$. Further stabilizer constraints restrict which pairs $(J,I)$ and which body-fixed states occur. The energy operator comes from the inertia tensors on the orbit; symmetry selects allowed states but does not by itself determine their energies. A bosonic choice of the trivial topological character is mathematically possible but does not model a fermionic nucleon.

Couple the nucleon isospin $1/2$ to the pion isospin $1$. The highest state $|3/2,3/2⟩=|p{\pi }^{+}⟩$ is normalized. One application of the total isospin lowering operator gives $|3/2,1/2⟩=\sqrt{2/3}|p{\pi }^0⟩+\sqrt{1/3}|n{\pi }^{+}⟩$. For a common isospin-invariant two-body decay amplitude, the partial decay rates relative to the ${\Delta }^{++}$ channel are $2/3$ and $1/3$. The lowest state gives ${\Delta }^{-}\to n{\pi }^{-}$. Exact relations assume isospin-degenerate masses and common kinematic factors; electromagnetic and quark-mass differences cause corrections.

A correlated pair of quarks. Its color, flavor, spin and orbital states are separate degrees of freedom and must collectively satisfy Fermi statistics. In flavor SU(3), the antisymmetric pair space is ${\Lambda }^{2}3\cong \overline{3}$; contraction with the invariant Levi-Civita symbol realizes the isomorphism. The antisymmetric $ud$ flavor pair is an isospin singlet with flavor hypercharge $2/3$. Combining two such flavor-antitriplet pairs with a strange antiquark permits the highest weight of the pentaquark antidecuplet; this flavor construction does not itself establish a bound state.

The zero-isospin, zero-flavor hypercharge state ${\eta }_8=(u\bar{u}+d\bar{d}-2s\bar{s})/\sqrt{6}$ in the pseudoscalar meson octet. It is a pure octet basis state; the physical eta may mix it with the flavor-singlet state.

The spin-independent vector contribution to an allowed beta decay. The isospin operator ${\tau }_k^{\pm }$ changes a neutron into a proton or the reverse; the sign convention depends on the assigned nucleon isospin projection. Its rotational rank is zero, so $J_f=J_i$ and the parity is unchanged. The phrase Fermi-type forbidden contribution extends the $S=0$ classification to higher orbital angular momentum.

For flavor symmetry acting on $(u,d,s)$, trace-orthonormal Hermitian diagonal generators are $h_1=\mathrm{diag}(1,-1,0)/\sqrt{2}$ and $h_2=\mathrm{diag}(1,1,-2)/\sqrt{6}$. The isospin and flavor hypercharge operators are $I_3=h_1/\sqrt{2}$ and $Y=\sqrt{2/3}{h}_2$. Their multiples by $i$ lie in the compact SU(3) Lie algebra.

The spin-dependent axial-vector contribution to an allowed beta decay. Its rotational rank is one, permitting $\Delta J=0,1$ with unchanged parity, but excluding $0\to 0$. Here ${\mathbf{\sigma }}_k$ acts on nucleon spin and ${\tau }_k^{\pm }$ on isospin. Higher-orbital spin-dependent contributions are conventionally described as Gamow-Teller-type forbidden transitions.

The product of charge conjugation and a half-turn in isospin is G parity. Both conjugations act on the isospin generators by $(J_1,J_2,J_3)\mapsto (-J_1,J_2,-J_3)$, so their product commutes with every generator. The Schur lemma therefore gives a common G-parity value throughout each irreducible isospin multiplet on which the operator acts internally. It can classify a charged multiplet even though individual charged states are not eigenstates of charge conjugation.

For integer isospin $I$ and neutral-state charge conjugation value $\eta$, the entire isospin multiplet has $G=\eta (-1{)}^I$. Both charge conjugation and the second-axis half-turn reverse the first and third generators, so their product commutes with the full isospin algebra. Evaluate it on the $M=0$ state and use ladder connectivity to obtain the value for every $M$.

Nuclear quantum states in different nuclear isobars belonging to the same isospin multiplet, with equal total isospin and approximately the same spatial and spin structure. An isospin raising or lowering operator connects the states in the limit of exact isospin symmetry. Nuclear Coulomb energy and other isospin-breaking effects shift their energies and modify this ideal relationship.