Continuum Subsystems and Local Algebras
A subsystem in continuum QFT is specified by the observables accessible in a spacetime region. Restricting a state to those observables is always meaningful; replacing that restriction by a reduced density matrix requires additional structure.
Enter this chapter
Section titled “Enter this chapter”Begin with Regions, Causal Complements, and Nets of Observables to construct a local subsystem from a region and its observable algebra.
Helpful background. Operator algebras and positive functionals supply the algebraic language; microcausality fixes the locality input.
Topics
Section titled “Topics”- Regions, Causal Complements, and Nets of Observables — Defines the locality and inclusion properties of a net of observables.
- Restricted States and Subregion Observables — Restricts a state to the observables accessible in a region.
- Why Continuum QFT Does Not Factorize Naively — Explains the obstruction to sharp spatial tensor factors.
- Von Neumann Factors and Type-III Local Algebras — Connects factor type to the absence of an intrinsic local trace.
- Standard Form, Cyclic and Separating Vectors — States the algebra–state hypotheses needed for modular theory.
- The Split Property and Approximate Tensor Products — Introduces type-I intermediate algebras for separated regions.
- Nuclearity, Phase-Space Bounds, and Split Distance — Relates phase-space control to the split construction.
- Additivity, Haag Duality, and Information Completeness — Distinguishes inclusion, additivity, and completeness of complement algebras.
- Local Preparation, State Dependence, and Operational Independence — Explains when independent local preparations define a joint state.
- Reeh–Schlieder Property and Limits of Localization — Separates dense local state preparation from finite-resource control.
- Choosing a Continuum Subsystem: Algebra, Split, or Regulator — Chooses a subsystem description suited to the physical task.
- Local Operations, Separability, and Distillability in QFT — Defines bipartite resources without assuming sharp tensor factors.
How the ideas fit together
Section titled “How the ideas fit together”Physically relevant continuum local algebras are generally type III, so ordinary trace-based density-matrix formulas cannot be assigned intrinsically to a sharp region Araki 1964, pp. 956–965. Algebraic state restriction nevertheless preserves the local predictions.
The split property supplies a type-I algebra between suitably separated region algebras, enabling tensor-product methods under additional assumptions Doplicher and Longo 1984, pp. 493–536. The separation and representation remain part of the construction: taking a collar to zero does not automatically preserve an ordinary reduced density matrix.
The comparison below separates the assumptions and tests needed for these different conclusions.
What this chapter establishes
Section titled “What this chapter establishes”| Information task | Declared subsystem | Licensed conclusion | Essential limitation |
|---|---|---|---|
| Compare continuum states locally | 𝔄(O) with normal functionals | Agreement, distinguishability, relative entropy, and modular data on the chosen algebra | No intrinsic trace or reduced density matrix is implied |
| Prepare separated regions independently | 𝔄(O₁) ⊂ 𝔑 ⊂ 𝔄(O₂) with a split collar | Normal product states and type-I tensor-product methods | The collar, reference representation, and phase-space assumptions remain part of the result |
| Compute a regulated entropy | Lattice, cutoff, or mode algebra | A density matrix and its Rényi or von Neumann entropy | Cutoff dependence and the continuum limiting prescription must be reported |
| Analyze a communication or detector protocol | Selected modes or a probe-generated operation algebra | Channel, noise, success probability, and resource claims for that protocol | The selected operations need not exhaust the sharp local algebra |
Where to go next
Section titled “Where to go next”Continue to Regulated Entropy and Replica Methods to define the regulated region and its reduced-state entropy.