accommodated by a new mathematical theory which is similar to the standard (von Neumann) quantum mechanics (nonrelativistic or relativistic) but uses a different mathematical idealization [@R11a].

The interpretation of this new quantum theory is, like the Hilbert space idealization, based on the Copenhagen interpretation of quantum mechanics, but it makes a much more distinct separation between the [*state*]{} and the [*observable*]{}. The state is defined by a preparation apparatus that prepares the state and is described mathematically by a statistical operator (density matrix) or a state vector. The observable is defined by a registration apparatus that measures its values in the state and is mathematically described by self-adjoint operators and their projectors (in place of the projection operator $\mid\psi\rangle\langle\psi\mid$ one can also take the vector $\psi$ up to a phase factor to describe this observable).

The mathematical formulation of this new quantum theory uses also a linear topological space, but instead of von Neumann’s Hilbert space it uses the Gelfand triplet (also called rigged Hilbert space (RHS)).

Rigging the Hilbert space may turn many people away from this subject because it may appear to some as an unnecessary mathematical complication (or even a disreputable practice).

This is really not the 