 The quantum-probabilistic formalism, as developed by von Neumann [1932], assumes that each physical system is associated with a (separable) Hilbert space \(\mathbf{H}\), the unit vectors of which correspond to possible physical states of the system. Each “observable” real-valued random quantity is represented by a self-adjoint operator \(A\) on \(\mathbf{H}\), the spectrum of which is the set of possible values of \(A\). If \(u\) is a unit vector in the domain of \(A\), representing a state, then the expected value of the observable represented by \(A\) in this state is given by the inner product \(\langle Au,u\rangle\). The observables represented by two operators \(A\) and \(B\) are commensurable iff \(A\) and \(B\) commute, i.e., AB = BA. (For further discussion, see the  entry on quantum mechanics.)