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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
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.