Extension of the Sheaf-theoretic Structure to Algebraic Quantum Field Theory

TL;DR

This paper extends sheaf-theoretic structures from quantum mechanics to algebraic quantum field theory, analyzing locality, no-signalling, and Bell inequality violations in the context of infinite degrees of freedom and spacetime structure.

Contribution

It introduces a sheaf-theoretic framework for AQFT, connecting it with conventional quantum mechanics and proving no-signalling and Bell inequality results in this extended setting.

Findings

Sheaf-theoretic structures can be extended to AQFT.

Strict spacelike separation corresponds to locality in AQFT.

No-signalling can be proved via Split Property in AQFT.

Abstract

The sheaf-theoretic structure is useful in classifying no-go theorems related to non-locality and contextuality. It provides a new point of view different from conventional formularization of quantum mechanics. First, we examine a relationship between the conventional formularization and the innovative formularization. There exists an equivalence of their categories, and from the equivalence, one locality can be transformed to another as a concrete example. Next, we extend the quantum mechanics which has a finite-degree of freedom to the quantum filed theory with an infinite-degree of freedom, especially to the algebraic quantum field theory (AQFT for short). We consider about a violation of the Bell inequality in AQFT, and we show that the condition of strict spacelike separation has the same Cartesian product structure as locality of quantum mechanics. Also, we show that no-signalling property can be proved by Split Property. A local state is a sheaf which is defined by Split Property in AQFT. It induces the sheaf-theoretic structure. Finally, we show an extension of the No-Signalling theorem which depends on spacetime in AQFT.