In quantum mechanics, symmetry groups can be realized by projective, as well as by ordinary unitary, representations. For the permutation symmetry relevant to quantum statistics of N indistinguishable particles, the simplest properly projective representation is highly non-trivial, of dimension
2(N−1)/2, and is most easily realized starting with spinor geometry. Quasiparticles in the Pfaffian quantum Hall state realize this representation. Projective statistics is a consistent theoretical possibility in any dimension.
We first give a quick survey of the realization of symmetries of quantum systems in the various formalisms of quantum mechanics: traditional (Hilbert space), algebraic (C*-algebras), rigged Hilbert spaces, *-algebras of unbounded operators, partial *-algebras of closable operators. Then we describe in some detail the concept of partial inner product spaces (PIP-spaces) and operators on them. Finally, we examine various classes of operators on PIP-spaces that allow a correct realization of symmetries.