Introduction to coherent quantization

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Referee this paper: arXiv:1804.01400 by Arnold Neumaier, Arash Ghaani Farashahi

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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.