  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.
