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The split property for quantum field theories in flat and curved spacetimes

Christopher J. Fewster

TL;DR

The paper addresses how spacelike-separated regions in quantum field theory can be treated as essentially independent by detailing the split property, which is stronger than Einstein causality and ties to the thermodynamic notion of locality via nuclearity. It surveys the original Minkowski-space results, elucidates the nuclearity criteria and their connection to quantum energy inequalities, and then extends the discussion to curved spacetimes using the locally covariant QFT framework, yielding a robust reduction to the ultrastatic case. A key contribution is a streamlined curved-spacetime strategy that preserves splitness along chains of Cauchy morphisms, enabling a general proof strategy for standard split inclusions and Reeh–Schlieder-type properties. The work highlights deep structural aspects of local algebras (e.g., type ${\rm III}_1$ factors), links between thermodynamic stability and QEIs, and a rigorous backbone for understanding locality in QFT across flat and curved spacetimes.

Abstract

The split property expresses a strong form of independence of spacelike separated regions in algebraic quantum field theory. In Minkowski spacetime, it can be proved under hypotheses of nuclearity. An expository account is given of nuclearity and the split property, and connections are drawn to the theory of quantum energy inequalities. In addition, a recent proof of the split property for quantum field theory in curved spacetimes is outlined, emphasising the essential ideas.

The split property for quantum field theories in flat and curved spacetimes

TL;DR

The paper addresses how spacelike-separated regions in quantum field theory can be treated as essentially independent by detailing the split property, which is stronger than Einstein causality and ties to the thermodynamic notion of locality via nuclearity. It surveys the original Minkowski-space results, elucidates the nuclearity criteria and their connection to quantum energy inequalities, and then extends the discussion to curved spacetimes using the locally covariant QFT framework, yielding a robust reduction to the ultrastatic case. A key contribution is a streamlined curved-spacetime strategy that preserves splitness along chains of Cauchy morphisms, enabling a general proof strategy for standard split inclusions and Reeh–Schlieder-type properties. The work highlights deep structural aspects of local algebras (e.g., type factors), links between thermodynamic stability and QEIs, and a rigorous backbone for understanding locality in QFT across flat and curved spacetimes.

Abstract

The split property expresses a strong form of independence of spacelike separated regions in algebraic quantum field theory. In Minkowski spacetime, it can be proved under hypotheses of nuclearity. An expository account is given of nuclearity and the split property, and connections are drawn to the theory of quantum energy inequalities. In addition, a recent proof of the split property for quantum field theory in curved spacetimes is outlined, emphasising the essential ideas.

Paper Structure

This paper contains 10 sections, 8 theorems, 76 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The split property in Minkowski space
  3. Nuclearity and Quantum Energy Inequalities
  4. The split property in curved spacetime
  5. The split property
  6. (Partial) Reeh--Schlieder property
  7. Distal split
  8. Concluding remarks
  9. Acknowledgement
  10. Proof of Theorem \ref{['thm:mbound']}

Key Result

Theorem 2.1

Suppose that the net of von Neumann algebras $O\mapsto {\mathfrak R}(O)$ obeys isotony, that the Hamiltonian is nonnegative, with zero eigenspace spanned by $\Omega$, and that the above nuclearity criterion holds. Then for any open bounded regions with $O_1\Subset O_2$, the inclusion ${\mathfrak R}(

Figures (5)

  • Figure 1: Three spacetime regions in Minkowski space. Time runs up the page, and rays travelling at the speed of light have unit gradient. Thus there is no possibility of communication between regions $O_2$ and $O_3$.
  • Figure 2: Regular Cauchy pairs with $(S_1,T_1)\prec_\Mb (S_2,T_2)$.
  • Figure 3: Illustration of the objects involved in defining the kinematic net
  • Figure 4: The chain of Cauchy morphisms in the proof of Theorem \ref{['thm:rigid_split']}.
  • Figure 5: Illustration of the 'inflationary' proof of the distal split bound.

Theorems & Definitions (17)

  • Theorem 2.1: BucDAnFre:1987
  • Proposition 3.1
  • proof : Outline of the proof
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Definition 4.1
  • Definition 4.2
  • ...and 7 more