Symmetry, Representations, and Superselection

A physical pure-state symmetry acts on rays, where global phase has already been removed. The invariant it preserves is the transition probability, not a phase-chosen inner product. Wigner’s theorem then asks when that projective map has a Hilbert-space implementation.

Reading sequence

Complete the shared on-ramp first: State Vectors → Projectors and Probability Amplitudes → the Born Rule → Transition Probabilities. Then read:

1. Physical States as Rays, which supplies projective pure states, rank-one projectors, and their intrinsic transition probability;
2. Transition-Probability Preserving Maps, which distinguishes a bijective full preserver from a nonsurjective isometric embedding and from an orthogonality-only map;
3. Wigner’s Theorem, which proves that a bijective full preserver is implemented by a unitary or antiunitary operator, unique up to one phase.

Distinctions the chapter enforces

• Wigner’s full-preserver theorem and Uhlhorn’s orthogonality theorem have different hypotheses and dimension boundaries.
• Antiunitary means conjugate-linear isometry, not a special unitary matrix.
• A lift of each group element can form only a projective representation until its phase cocycle is analyzed.
• A kinematic Wigner symmetry need not commute with a Hamiltonian.
• Strong continuity of a unitary lift is an extra requirement before Stone’s generator theorem applies.
• Superselection can restrict or partition the ray domain on which a symmetry acts.

Physical examples and first-encounter intuition belong to Symmetry, Spin, and Geometry. This chapter is self-contained for the exact projective hypotheses and theorem boundary.