Observables in the General Boundary Formulation

TL;DR

This paper introduces a new notion of quantum observable tailored for the general boundary formulation, extending the concept to spacetime regions and integrating various quantization schemes.

Contribution

It develops a generalized quantum observable concept compatible with spacetime regions and provides multiple quantization methods within the general boundary framework.

Findings

Standard quantum observables are recovered as a special case.

Various quantization schemes yield different types of observables.

A generalized expectation value concept is proposed.

Abstract

We develop a notion of quantum observable for the general boundary formulation of quantum theory. This notion is adapted to spacetime regions rather than to hypersurfaces and naturally fits into the topological quantum field theory like axiomatic structure of the general boundary formulation. We also provide a proposal for a generalized concept of expectation value adapted to this type of observable. We show how the standard notion of quantum observable arises as a special case together with the usual expectation values. We proceed to introduce various quantization schemes to obtain such quantum observables including path integral quantization (yielding the time-ordered product), Berezin-Toeplitz (antinormal ordered) quantization and normal ordered quantization, and discuss some of their properties.