Schr\"odinger-Feynman quantization and composition of observables in general boundary quantum field theory

TL;DR

This paper introduces a rigorous, functorial quantization scheme for linear and affine field theories within the general boundary formulation, incorporating observables and their composition via path integrals, applicable even without a metric spacetime background.

Contribution

It establishes a new quantization framework combining Schr"odinger and Feynman methods, including a novel way to compose observables through spacetime gluing, extending previous geometric quantization approaches.

Findings

Quantization scheme is equivalent to a holomorphic geometric quantization.

Includes a consistent method for quantizing and composing observables.

Reproduces the correct composition of observables in Minkowski space.

Abstract

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.