The metric known to be relevant for standard quantization procedures receives a natural interpretation and its explicit use simultaneously gives both physical and mathematical meaning to a (coherent-state) phase-space path integral, and at the same time establishes a fully satisfactory, geometric procedure of quantization.

In a recent paper, a “distance” function, $\cal D$ , was defined which measures the distance between pure classical and quantum systems. In this work, we present a new definition of a “distance”, D, which measures the distance between either pure or impure classical and quantum states. We also compare the new distance formula with the previous formula, when the latter is applicable. To illustrate these distances, we have used 2 × 2 matrix examples and two-dimensional vectors for simplicity and (...) clarity. Several specific examples are calculated. (shrink)

From the very beginning, coherent state path integrals have always relied on a coherent state resolution of unity for their construction. By choosing an inadmissible fiducial vector, a set of “coherent states” spans the same space but loses its resolution of unity, and for that reason has been called a set of weak coherent states. Despite having no resolution of unity, it is nevertheless shown how the propagator in such a basis may admit a phase-space path integral representation in essentially (...) the same form as if it had a resolution of unity. Our examples are toy models of similar situations that arise in current studies of quantum gravity. (shrink)

Perturbations of quantum systems ranging from oscillators to fields can be either continuous or discontinuous functions of the coupling. The system under consideration is the familiar harmonic oscillator in one degree of freedom. Previous studies have shown that when the harmonic oscillator is subjected to a perturbation with a power law singularity, a permanent change in the system characteristics is observed for a specific range of power law values. The introduction of a logarithmic singularity into the power law potential fine (...) tunes the singularity power. (shrink)

For several examples of Hermitian operators, the issues involved in their possible self-adjoint extension are shown to conform with recognizable properties in the solutions to the associated classical equations of motion. This result confirms the assertion made in an earlier paper (Ref. 1) that there are sufficient classical “symptoms” to diagnose any quantum “illness.”.