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Abstract:

This paper is the second in a series of papers on coherent spaces and

their applications. It begins the study of coherent quantization - the

way operators in a quantum 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 to 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.