The quantum counterpart of a phase space is a Hilbert space \(\HC\): a complex vector space with an inner product. If infinite dimensional it must be complete—Cauchy sequences have limits. But for our purposes finite-dimensional spaces will suffice for discussing the major conceptual difficulties of quantum theory and how the histories approach resolves them. (Some examples use a harmonic oscillator with an infinite dimensional Hilbert space simply because it is relatively simple and familiar.) We use Dirac notation in which an element of \(\HC\), a “ket”, is denoted by \(\ket{\psi}\), where \(\psi\) is a label, and \(\inpd{\phi}{\psi}\) denotes the inner product of \(\ket{\phi}\) with \(\ket{\psi}\). The simplest quantum physical property, the counterpart of a point \(\gamma\) in the classical phase space, is a one-dimensional subspace of the Hilbert space, or ray, consisting of all multiples \(c\ket{\psi}\) of some nonzero \(\ket{\psi}\), with \(c\) an arbitrary complex number. The ray uniquely determines and is uniquely determined by the corresponding projector