A measure-free approach to coherent states

Abstract

In the process of generalizing coherent states, the situation when the measure—which is customarily incorporated in their definition—is indeterminate becomes unavoidable. A more dramatic situation may occur if there is no measure which makes the reproducing kernel Hilbert space, involved in the construction of coherent states, isometrically included in an space. Therefore, it appears that there is a need to redefine coherent states making the definition measure-free. Starting out with the reproducing kernel property, we ensure the basic feature of coherent states—resolution of the identity—to be maintained, cf (15) and remark 2. The only investment in the whole undertaking is a sequence (Φn)dn = 0 satisfying (4); the rest, including the aforesaid resolution of the identity, is a consequence of our choice. The approach is supported by examples which make the circumstances clear under which the sequence (Φn)dn = 0 appears; moment problems or rather orthogonal polynomials are one of them.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.