Superselection makes sense not abstractly in an arbitrary Hilbert space but only in Hilbert spaces structured by the introduction of an algebra of distinguished observables of interest, with prescribed commutation rules. These define the physics that is possible in the class of models considered, and superselection is a concept defined relative to these.
Typical examples are the universal enveloping algebras of Lie algebras, or the $C^*$ algebras generated by the corresponding Lie groups. For example Heisenberg algebras and Heisenberg (or Weyl) groups correspond to canonical commutation rules, which are the basis of much of quantum physics.
The Hilbert spaces of interest are the (continuous unitary) irreducible representation spaces of these algebras, Lie algebras, or groups. These (or more precisely the classes of equivalent such representation spaces) are called the (superselection) sectors of the theory. Since they constitute different Hilbert spaces it makes no sense to superimpose vectors of the different sectors. One can define an inner product on the direct sum of these Hilbert spaces, but the algebra of operators still map each sector into itself, hence there is no way to create (in a physically relevant way) a superposition from pure states within the sectors.
For finite-dimensional Heisenberg algebras/groups, all continuous unitary irreducible representations are equivalent (Stone-von Neumann theorem); hence for nonrelativistic N-particle theories, there are no superselection rules (that would specify superselection sectors).
Once one also accounts for spin, the situation becomes more complicated: a mixture of a fermionic and a bosonic state makes no longer physical sense since the two state vectors behave differently under a rotation by 360 degree - though formally it is still defined. No amount of new physics will change that.
For infinite-dimensional Heisenberg algebras/groups as they occur in (relativistic or nonrelativistic) quantum field theory, the Stone-von Neumann theorem is no longer valid, and there are uncountably many inequivalent continuous unitary irreducible representations, hence there are uncountably many superselection sectors, distinguished by their essentially different behavior at spacelike infinity.
In more technical terms: The most interesting superselection rules, accounting for superconductivity, charge, baryon number, etc., arise due to unimplementable Bogoliubov transformations, involving limits that are so singular that they lead out of the Hilbert space representing the vacuum sector. In particular, charged states have a sufficiently different asymptotic structure from uncharged ones, since the Coulomb field is long range, and they belong to different superselection sectors. No amount of new physics will change that.
Under certain conditions, superselection sectors can be classified; see, e.g., the article DHR superselection theory from nLab. DHR stands for Doplicher, Haag, and Roberts; see, e.g.,
- Doplicher, S., & Roberts, J. E. (1990). Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Communications in Mathematical Physics, 131(1), 51-107.