  We show that the Feynman path integral together with the Schr\"odinger
representation gives rise to a rigorous and functorial quantization scheme for
linear and affine field theories. Since our target framework is the general
boundary formulation, the class of field theories that can be quantized in this
way includes theories without a metric spacetime background. We also show that
this quantization scheme is equivalent to a holomorphic quantization scheme
proposed earlier and based on geometric quantization. We proceed to include
observables into the scheme, quantized also through the path integral. We show
that the quantized observables satisfy the canonical commutation relations, a
feature shared with other quantization schemes also discussed. However, in
contrast to other schemes the presented quantization also satisfies a
correspondence between the composition of classical observables through their
product and the composition of their quantized counterparts through spacetime
gluing. In the special case of quantum field theory in Minkowski space this
reproduces the operationally correct composition of observables encoded in the
time-ordered product. We show that the quantization scheme also generalizes
other features of quantum field theory such as the generating function of the
S-matrix.
