Introduction to coherent quantization

TL;DR

This paper introduces a generalized framework for coherent quantization that extends geometric quantization to non-unitary cases and applies to quantum field theory, emphasizing minimal assumptions and broad group applicability.

Contribution

It develops a broad, topology-minimal approach to coherent quantization, generalizing geometric quantization and connecting to quantum field theory and Fock space operators.

Findings

Characterizes linear operators via coherent matrix elements.

Introduces coherent maps and symmetry groups for quantization.

Derives properties of creation and annihilation operators in Fock spaces.

Abstract

This paper is one of a series of papers on coherent spaces and their applications, defined in the recent book 'Coherent Quantum Mechanics' by the first author. The paper studies coherent quantization -- the way operators in the quantum space of a coherent space can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties. The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization of coherent maps. The importance of coherent maps for quantum mechanics is due to the fact that there is a quantization operator that associates homomorphically with every coherent map a linear operator from the quantum space into itself. This operator generalizes to general symmetry groups of coherent spaces the second quantization procedure for free classical fields. The latter is obtained by specialization to Klauder spaces, whose quantum spaces are the bosonic Fock spaces. A coordinate-free derivation is given of the basic properties of creation and annihilation operators in Fock spaces.