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Quantum Mechanics

Symmetry, Representations, and Superselection

A pure-state symmetry first acts on rays, preserving transition probabilities without choosing vector phases. Wigner’s theorem supplies its unitary or antiunitary implementation. Turning these individual implementations into a group action introduces multipliers, continuity, and global topology. Restricting the observable algebra introduces another issue: which relative phases remain physically distinguishable. This chapter keeps those steps separate.

Enter through transition probabilities

Section titled “Enter through transition probabilities”

The shared on-ramp is State Vectors → Projectors and Probability Amplitudes → Born Rule → Transition Probabilities. Then follow the theorem sequence:

  1. Physical States as Rays defines the projective state and its intrinsic overlap probability.
  2. Transition-Probability Preserving Maps separates bijective full preservers from embeddings and orthogonality-only maps.
  3. Wigner’s Theorem gives the unitary-or-antiunitary lift and its uniqueness up to one phase.

These hypotheses are not interchangeable. Wigner’s full-preserver theorem and Uhlhorn’s orthogonality theorem have different dimension boundaries. A kinematic ray symmetry also need not preserve a chosen Hamiltonian.

Turn individual lifts into a symmetry group

Section titled “Turn individual lifts into a symmetry group”

Projective Representations and Multipliers derives the associativity and rephasing conditions. For antiunitary group elements, scalar phases are conjugated and the cocycle identity is twisted.

Central Extensions retains those phases as group coordinates, making a unitary projective implementation ordinary on an enlarged group. The phase subgroup is central for unitary lifts; antiunitary elements generally act on it by conjugation.

Combining Ray Representations asks when two actions can coexist on a direct sum or tensor product. Direct sums require compatible multipliers on the full ray space, while tensor products multiply multiplier classes. This distinction is the mathematical input to the spin and mass examples.

Spin Representations and Descent to Rotations tests whether the central element of SU(2)SU(2) acts trivially, scalarly, or differently on different summands. Its role is to give the exact descent criterion; the physical rotation and spinor examples remain in Symmetry, Angular Momentum, and Spin.

Transport observables and dynamics with their domains

Section titled “Transport observables and dynamics with their domains”

Unitary and Antiunitary Operator Transport follows self-adjoint domains, spectral projections, and complex spectral functions through a symmetry. It gives a spectral justification for transforming an unbounded Hamiltonian’s exponential.

Time Reversal with Operator Domains uses quasi-periodic boundary conditions to distinguish invariance of one Hamiltonian from covariance between different models. A real differential expression alone does not settle the question.

Continuous Symmetries and Their Generators applies the existing Stone theorem. It keeps the exact one-parameter group law, strong continuity, additive generator ambiguity, and periodicity conditions explicit.

Distinguish symmetry type, multiplicity, and superselection

Section titled “Distinguish symmetry type, multiplicity, and superselection”

Symmetry Decompositions and Multiplicity Spaces develops the compact-group decomposition and its commutant. Equivalent irreducible copies may mix under symmetry-invariant operators, so a decomposition into copies does not itself establish superselection.

Superselection Rules and Charge Invariance instead starts from an allowed observable algebra. It distinguishes central charge projections from merely invariant subspaces, conservation from observable restriction, and local inaccessible phases from relational coherence in a larger system.

Galilei Symmetry and the Mass Superselection Argument combines the representation and observable questions. Its boost-translation phase gives a precise obstruction for the unextended group. Its physical interpretation remains conditional on the group, model, and observable algebra.

The resulting route connects rigorous symmetry statements to physics while retaining the assumptions that make each conclusion true. Neither a formal commutator nor a projective phase alone determines the complete physical symmetry structure.

References

Section titled “References”