Local Nets, States, and Representations

Algebraic QFT assigns observables to spacetime regions before choosing a Hilbert-space representation. This chapter follows that distinction from local nets to states, local factors, and constructive examples.

Enter this chapter

Read nets, quasilocal algebras, and GNS representations consecutively; the later topics depend on keeping the abstract algebra and its represented closure distinct. Helpful background. Operator Algebras and Positive Functionals supplies the C*-algebra, von Neumann algebra, and state vocabulary. Von Neumann Factors and Type-III Local Algebras previews the sharp-region consequence, and Local Field Algebras, Causality, and the Time-Slice Property shows how the same structure is deployed on globally hyperbolic spacetimes.

Topics

1. Haag–Kastler Nets and Locality — Defines a net and its causal compatibility conditions.
2. Quasilocal C*-Algebras and Inductive Limits — Assembles local observables through compatible C*-algebra inclusions.
3. States, GNS Representations, and Folia — Constructs representations from states and identifies their normal-state families.
4. Local Normality, Quasiequivalence, and Folia — Compares representations locally without assuming global unitary equivalence.
5. The Reeh–Schlieder Theorem — Explains the dense vacuum orbit of a local algebra.
6. Isotony, Additivity, Duality, and Primitive Causality — Separates inclusion, completeness of local generation, duality, and dynamics.
7. From Pointlike Fields to Nets and Affiliated Operators — Relates unbounded fields to bounded local observable algebras.
8. Factorizing S-Matrices, Wedge-Local Fields, and Borchers Constructions — Builds wedge-local models and states the extra conditions for compact localization.
9. Representation Types, Factors, and Local-Algebra Structure — Identifies what factor type says about a represented local algebra.

How the ideas fit together

A net specifies localization; a state supplies a representation through GNS. The norm-closed quasilocal algebra and the weak closures of represented local algebras answer different questions Haag and Kastler 1964, §§ 2–3. Local quasiequivalence does not imply a global unitary identification Fewster and Rejzner 2020, §§ 2.3 and 4–6 (PDF). Likewise, wedge locality needs further estimates before it yields nontrivial double-cone observables Lechner 2008, § 3 and Theorems 5.6, 5.8; identifying the precise local factor requires further structural hypotheses Buchholz, D’Antoni, and Fredenhagen 1987, pp. 126–134.

Hypotheses and licensed conclusions

Objects, hypotheses, licensed conclusions, and failure boundaries for local nets and representations

Object and domain	Required hypotheses	Licensed conclusion	Excluded converse or upgrade	Adversarial check
Regional C*-algebras on a directed family of double cones	Injective coherent inclusions; spacelike commutation and covariant region maps when those properties are claimed	An isotonic local net and its representation-independent quasilocal C*-completion	The construction is not a tensor product of sharp regions, and locality does not imply duality	Assign independent factors to nested or overlapping regions and find the missing inclusion or inconsistent overlap
Positive normalized functional on the quasilocal algebra	C*-positivity and normalization; invariance or KMS analyticity only when separately imposed	A cyclic GNS representation, with purity equivalent to irreducibility and a representation-dependent von Neumann closure	GNS does not supply locality, positive energy, faithfulness, or a density matrix in another representation	Write an infinite-volume thermal state as a vacuum-Fock Gibbs trace and expose the non-trace-class partition function
Two quasifree representations restricted to bounded local algebras	Equivalent induced topologies and regional Hilbert–Schmidt covariance difference; Hadamard hypotheses in the free curved-spacetime theorem	Equality of local folia and mutual normality on every stated bounded region	Local quasiequivalence does not produce a compatible global unitary or erase infrared phase differences	Let the region exhaust infinite volume and watch the thermal covariance difference fail the global Hilbert–Schmidt test
Vacuum representation and a nonempty open region	Positive translation spectrum, global cyclicity, weak additivity, covariance, and locality for the separating conclusion	The vacuum is cyclic locally and separating when the causal complement has nonempty interior	Orbit density gives no uniform operator norm, energy, success probability, or signaling protocol	Drive the approximation error to zero while recording norm and energy instead of assuming they remain bounded
Additive free-Maxwell net on a causally completed ring-like region	Specified contractible generators, region topology, causal complement, and the vacuum representation	Einstein causality survives while a linked flux gives a strict inclusion into the dual algebra	Commutation with the complement does not prove that an operator is generated in the original region	Compute the nonzero linking pairing of electric and magnetic fluxes on disjoint thickened tori
Smeared unbounded fields on a common invariant core	Closability, essential self-adjointness or controlled resolvents, support covariance, and strong spacelike commutativity	Spectral unitaries generate an isotonic local von Neumann net to which the closed fields are affiliated	A vanishing form commutator or an unclosable distribution does not license local exponentials	Change the self-adjoint extension or test spectral commutation beyond the proposed core
Scalar factorizing amplitude and its wedge algebra in two dimensions	Unitarity, hermitian analyticity, crossing, bounded physical-strip continuation, and modular nuclearity for compact localization	ZF operators give relatively wedge-local fields and, in the proved regular class, nontrivial local intersections; further scaling and phase-space input can fix factor type	A real-axis unitary phase is not a local QFT, and type III alone is not hyperfinite type III₁	Remove strip analyticity so the wedge contour shift fails, or remove scaling control so the Connes type is undetermined

Structured table data (JSON).

Review the chapter

Does Reeh–Schlieder allow a local unitary to prepare any distant state?

Solution

No. Cyclicity concerns vectors obtained using arbitrary local operators and their limits, not only unitaries. A unitary in one region commutes with observables in a spacelike-separated region and leaves their expectation values unchanged.

Where to go next

Continue to modular theory, nuclearity, and split inclusions to study finer structure of these represented local algebras.

References

• Buchholz, Detlev, Claudio D’Antoni, and Klaus Fredenhagen. “The Universal Structure of Local Algebras.” Communications in Mathematical Physics 111 (1987): 123–135. DOI.
• Fewster, Christopher J., and Kasia Rejzner. “Algebraic Quantum Field Theory—an Introduction.” In Progress and Visions in Quantum Theory in View of Gravity, 1–61. Cham: Birkhäuser, 2020. DOI; Open PDF.
• Haag, Rudolf, and Daniel Kastler. “An Algebraic Approach to Quantum Field Theory.” Journal of Mathematical Physics 5 (1964): 848–861. DOI.
• Lechner, Gandalf. “Construction of Quantum Field Theories with Factorizing S-Matrices.” Communications in Mathematical Physics 277 (2008): 821–860. DOI; Open PDF.