General boundary quantum field theory: Timelike hypersurfaces in Klein-Gordon theory

TL;DR

This paper extends the general boundary framework of quantum field theory by demonstrating how the Klein-Gordon theory can be described using states on timelike hypersurfaces, including novel finite-region amplitudes.

Contribution

It provides the first explicit examples of amplitudes for finite regions bounded by timelike hypersurfaces in Klein-Gordon theory, expanding the boundary quantum field theory framework.

Findings

Explicit construction of states on timelike hypersurfaces

First examples of finite-region amplitudes in this context

Generalized probability interpretation beyond standard quantum mechanics

Abstract

We show that the real massive Klein-Gordon theory admits a description in terms of states on various timelike hypersurfaces and amplitudes associated to regions bounded by them. This realizes crucial elements of the general boundary framework for quantum field theory. The hypersurfaces considered are hyperplanes on the one hand and timelike hypercylinders on the other hand. The latter lead to the first explicit examples of amplitudes associated with finite regions of space, and admit no standard description in terms of ``initial'' and ``final'' states. We demonstrate a generalized probability interpretation in this example, going beyond the applicability of standard quantum mechanics.