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.