Coherent manifolds

TL;DR

Coherent manifolds provide a geometric generalization of quantum spaces by equipping coherent spaces with a smooth manifold structure and a coherent product $K$. The paper develops a full quantization framework: a quantization map $\Gamma$ from coherent maps to operators, coherent differential operators, and shadow operators, enabling Schrödinger dynamics to be computed via the coherent product. It also identifies concrete realizations, notably Klauder spaces, which recover Bosonic Fock spaces with Weyl relations and CCR in a geometric setting. The approach yields exact solvable cases when Hamiltonians generate coherent motions and offers a coherent-geometry-based path for time-dependent dynamics via the Dirac–Frenkel variational principle, with broad applicability to quantum mechanics and field theory.

Abstract

This paper defines coherent manifolds and discusses their properties and their application in quantum mechanics. Every coherent manifold with a large group of symmetries gives rise to a Hilbert space, the completed quantum space of $Z$, which contains a distinguished family of coherent states labeled by the points of the manifold. The second quantization map in quantum field theory is generalized to quantization operators on arbitrary coherent manifolds. It is shown how the Schrödinger equation on any such completed quantum space can be solved in terms of computations only involving the coherent product. In particular, this applies to a description of Bosonic Fock spaces as completed quantum spaces of a class of coherent manifolds called Klauder spaces.