Mathematical formulation of quantum mechanics
The mathematical formulation of quantum mechanics in general use is based on an identification of the bra-ket notation of Dirac, with the abstract notion of Hilbert space used in functional analysis. This formulation is often attributed to von Neumann.
There is given a separable Hilbert space H. The states are the projective rays of H. An operator is a linear map from a dense subspace of H to H (we cannot assume that the operator is defined on the whole of H because of the Hellinger-Toeplitz theorem). If this operator is continuous, then this map can be uniquely extended to a bounded linear map from H to H. By tradition, observables are identified with operators, although this is rather questionable, especially in the presence of symmetries leading to superselection sectors. This is why some people prefer the density state formulation.
See also Schrödinger picture, Heisenberg picture, Born principle, relative state interpretation.
In this framework, Heisenberg's uncertainty principle becomes a theorem about noncommuting operators. Furthermore, both continuous and discrete observables may be accommodated; in the former case, the Hilbert space is a space of square-integrable wavefunctions.
In the Everett many-worlds interpretation of quantum mechanics, postulate (3) is demoted to a phenomenological principle; see [[quantum
In this formulation, there is given a C* algebra, the associative algebra of operators. Positive elements of its dual vector space is are called states and they describe the quantum states. This is related to the density matrix.
Given a state, we can construct a unitary representation of it using the Gelfand-Naimark-Segal construction. Two unitarily inequivalent representations are said to belong to different superselection sectors. Relative phases between superselection sectors are not observable.C* formulation