Abstract: In this talk, we report a mathematical formalism of quantum mechanics based on the notion of a prototype. The main novelty of this formalism is that the theory includes both vector and singular states as quantum states for a quantum system and thus extends the conventional formulation of quantum mechanics merely involving vector states. That the prototype is introduced as a prime concept in quantum mechanics reflects the postulation of Bohr's complementary principle that one cannot make a measurement on two incompatible observables at a particular moment. Mathematically, for a quantum system Q with an associated Hilbert space H, a prototype of Q is defined by an orthonormal basis of H. The evolution of the system Q as described by prototypes is governed by Schrödinger's equation for the associated bases of H. Given a prototype with its associated basis (en), a quantum state is defined as a valuation of all self-adjoint operators on H diagonal under the basis (en). Although a quantum state is defined in a certain prototype, it can be uniquely extended to the whole system as proved respectively as being a vector state by Kadison and Singer in 1959, and as a singular state by Marcus, Spielman, and Srivastava recently. Consequently, we can apply the Copenhagen interpretation to a prototype for regarding a quantum state as an external observation, and thus obtain the Born rule of random outcomes. Moreover, a mathematical method for constructing some singular states is given by the so-called Banach limit. We expect that the singular states have helpful implications in the application of quantum mechanics to understanding the nature.
