Affine holomorphic quantization

TL;DR

This paper develops a rigorous, functorial quantization scheme for affine field theories using a combination of holomorphic geometric quantization and path integrals within the general boundary framework, enabling local and background-independent quantum field theory constructions.

Contribution

It introduces a novel quantization approach for affine field theories that integrates holomorphic geometric quantization with path integrals, tailored for the general boundary formulation.

Findings

Established a functorial quantization scheme for affine theories.

Developed an adapted coherent state and vacuum state framework.

Derived a factorization identity for the amplitude in linear theories with sources.

Abstract

We present a rigorous and functorial quantization scheme for affine field theories, i.e., field theories where local spaces of solutions are affine spaces. The target framework for the quantization is the general boundary formulation, allowing to implement manifest locality without the necessity for metric or causal background structures. The quantization combines the holomorphic version of geometric quantization for state spaces with the Feynman path integral quantization for amplitudes. We also develop an adapted notion of coherent states, discuss vacuum states, and consider observables and their Berezin-Toeplitz quantization. Moreover, we derive a factorization identity for the amplitude in the special case of a linear field theory modified by a source-like term and comment on its use as a generating functional for a generalized S-matrix.