# AG Lie-Gruppen: A. Neumaier, Universität Wien: Coherent Quantization I: Coherent Spaces for Linear Fields

Coherent Quantization I: Coherent Spaces for Linear Fields – Vortragender:

Arnold Neumaier, Universität Wien – Einladender: Karl-Hermann Neeb

This is the first of three lectures on Coherent Quantization and Field Theory

to be given 11.-18.12.2023 in Erlangen (Germany).

(Lecture one and three = AG Lie-Gruppen; lecture two = AG Mathematische

Physik)

Abstract: The notion of a coherent space is a nonlinear version of the notion

of a complex Hilbert space: The vector space axioms are dropped while the

notion of inner product, now called a coherent product, is kept.

Every coherent space can be uniquely embedded into a Hilbert space, its

completed quantum space, by suitably extending the coherent product to an

inner product. In the interesting examples, the coherent space is an extended

classical phase space, and there is a quantization functor that turns the

symmetries of the coherent space into unitary operators in the corresponding

quantum space. Thus the quantum space is a representation space for quantum

dynamics.

This provides a universal framework for quantization, extending the

traditional geometric quantization of finite-dimensional symplectic manifolds

to more general situations, and in particular to the quantization of certain

classical field theories.

Fields defined by linear field equations can be quantized by means of Klauder

spaces, a class of coherent spaces discussed by Neumaier and Ghani Farashahi

in Anal. Math. Phys. 12 (2022), 1-47.

We describe another quantization of linear field equations in terms of

symplectic and orthogonal Hua spaces (for bosons and fermions), a new class of

coherent spaces based on the geometric analysis by Loo-Keng Hua in Trans.

Amer. Math. Soc. 57 (1945), 441-481. Their symmetry groups are infinite-

dimensional metaplectic or metagonal (spin) groups. They allow one to describe

the full quantum scattering behavior of linear field equations in terms of

classical scattering and Maslov corrections for the phase of the S-matrix.

The slides of the lecture will probably become available \- at least three

hours before the lecture – at <https://arnold-

neumaier.at/cohErlangen2023.html> where one can also find background material

(abstracts and some references) for all lectures.