Rotation generators and SU(2) representations

Choose active right-handed rotations with $U(\alpha,\mathbf n)=\exp(-i\alpha\mathbf n\cdot\mathbf J/\hbar)$. The components of angular momentum satisfy cyclic commutation relations. They cannot all have definite values simultaneously, but their squared magnitude commutes with each component. We choose joint eigenstates of squared magnitude and the z component. The labels $j$ and $m$ are dimensionless; the measured angular momenta contain the appropriate powers of Planck's reduced constant.

The raising and lowering combinations change the magnetic label by one. Their norms must be nonnegative, and a finite-dimensional unitary irreducible representation must terminate at both ends. These conditions give $m=-j,-j+1,\ldots,j$ and $2j+1$ states, with $j$ a nonnegative integer or half integer. This is a quantum-algebra result, not a picture of a rigid object secretly rotating about a selected axis. An orbital wavefunction on ordinary three-dimensional space has additional single-valuedness requirements giving integer orbital labels.

For spin one half, $\mathbf J=\hbar\boldsymbol\sigma/2$ in a two-component spin basis. Squaring the Pauli matrix along any unit axis gives the identity, which separates the exponential into cosine and sine terms. A rotation by a full turn gives minus the identity; two full turns give the identity. Thus spinors give an ordinary representation of SU(2), the double cover of SO(3), and a projective representation of the spatial rotation group.

The overall minus sign is unobservable for a spinor considered alone. In an interferometer where only one coherent branch undergoes the rotation, it becomes a relative phase between branches. The Bloch vector, which consists of the three Pauli expectations, returns after one full turn because it describes the ray. Consequently a moving arrow on a sphere cannot by itself display the spinor's full phase information. A ray diagram and a state-vector calculation answer related but different questions.

A rotationally invariant Hamiltonian commutes with all three generators. Within a single irreducible multiplet it acts identically on every magnetic component, so these components share an energy. This guarantees at least the multiplet degeneracy when that multiplet occurs as an energy subspace. It does not force different values of total angular momentum to have the same energy. Additional radial labels and multiple copies of an irreducible representation can carry different energies, with mixing possible between equivalent copies.

A Hamiltonian commuting only with the z generator preserves axial symmetry and the magnetic label but need not preserve the full multiplet energy. A Zeeman term proportional to the z generator is the simplest counterexample. Conserved angular momentum components therefore have to be inferred from the actual symmetry group of the apparatus. The ladder algebra remains available for representing states even when an external field breaks the symmetry that would otherwise relate their energies.