Local Nets, States, and Representations

Algebraic QFT begins with regions, bounded observable algebras, and inclusion maps; a state then selects a Hilbert-space representation and its local von Neumann closures. This order prevents three common conflations: an abstract C*-algebra is not one of its representations, a normal state is not an intrinsic sharp-region density matrix, and Einstein causality is not Haag duality. The chapter develops those distinctions through free, thermal, Maxwell, and factorizing models.

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.

Enter this chapter

Unless a page says otherwise, the basic region class is the set of bounded double cones in Minkowski spacetime. The symbol $O'$ denotes the interior of the causal complement, $\mathcal A(O)$ an abstract local C*-algebra, and

$$\mathfrak A_\pi(O)=\pi(\mathcal A(O))''$$

its von Neumann closure in a named representation. Wedges, ring-like regions, and Cauchy neighborhoods are introduced only when their different geometry matters. Lorentzian formulas use the site-wide $(+---)$ metric convention.

The main construction has a fixed direction. Coherent local embeddings give a quasilocal C*-algebra. A positive normalized functional on that algebra gives a GNS representation. Only then do normality, commutants, folia, factor type, and affiliation become available. Locality and covariance constrain the net; positive energy and vacuum cyclicity constrain a selected state and representation.

The dependency map shows that no later conclusion repairs a missing earlier map. Inspect the two branches after GNS: local restriction compares folia, while spectrum and weak additivity support Reeh–Schlieder. Neither branch turns locality into duality or supplies a type-I tensor factor.

Injective coherent inclusions assemble the abstract quasilocal algebra, and a positive normalized state selects its cyclic representation. Local restrictions then support folium and quasiequivalence comparisons; positive energy with weak additivity supports Reeh–Schlieder; strong field commutativity can generate a net; and scaling plus phase-space hypotheses sharpen local factor type. Every arrow is one-way unless an additional theorem is named. The diagram is schematic and not to scale. Structured description and source data (JSON)

The foundational net axioms were separated from state and representation assumptions by Haag and Kastler 1964, §§ 2–3, pp. 850–856. A modern account with the free scalar, GNS theorem, Reeh–Schlieder proof, and local-type qualifications is Fewster and Rejzner 2020, §§ 2.3 and 4–6, pp. 6–32.

The chapter sequence

Read the pages in this order.

1. Haag–Kastler nets and locality defines the region assignment and separates abstract C*-nets from represented von Neumann nets.
2. Quasilocal C*-algebras and inductive limits completes coherent bounded-region observables without choosing a Hilbert space.
3. States, GNS representations, and folia constructs the cyclic representation and distinguishes purity, normality, folia, and disjointness.
4. Local normality, quasiequivalence, and folia explains how vacuum and thermal representations can agree locally while remaining globally disjoint.
5. The Reeh–Schlieder theorem derives local cyclicity from positive-energy analyticity and states its resource boundary.
6. Isotony, additivity, duality, and primitive causality records the valid implications and exhibits the Maxwell flux obstruction to naive duality.
7. From pointlike fields to nets and affiliated operators identifies the domain and strong-commutativity hypotheses needed to generate bounded local algebras.
8. Factorizing S-matrices, wedge-local fields, and Borchers constructions follows an analytic two-particle amplitude to wedge algebras and conditionally nontrivial double-cone intersections.
9. Representation types, factors, and local-algebra structure separates type III, type III₁, factoriality, and hyperfiniteness and names the extra hypotheses for each.

The order is not circular. A state may be studied on the abstract quasilocal algebra before its local von Neumann algebras are classified. A wedge-local construction may supply a model before compact localization is known to be nontrivial. A field family may generate a net without being a unique coordinatization of it.

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) preserves the caption, scoped headers, rows, and reading order.

The table is deliberately directional. For example, a Hadamard theorem can give local quasiequivalence without global equivalence, and a type-III proof need not identify type III$_1$. Similarly, wedge locality precedes compact locality. Lechner’s construction makes the last distinction exact: strip analyticity proves the reflected-wedge commutator, while modular nuclearity proves nontrivial double-cone algebras only for the stated regular class Lechner 2008, § 3 and Theorems 5.6, 5.8, pp. 832–851.

The failure map groups the most consequential illegal upgrades. Each lower branch preserves a weaker object, but blocks the stronger conclusion named above it.

Incompatible embeddings destroy the faithful quasilocal limit; a divergent Gibbs trace does not invalidate the KMS state but places it outside the proposed vacuum folium; local quasiequivalence stops short of a global unitary; locality stops short of Haag duality in the Maxwell ring; and real-axis scattering or bare type-III information stops short of compact locality or hyperfinite type III₁. Dashed branches mark the first unavailable inference. The diagram is schematic and not to scale. Structured description and source data (JSON)

Scope boundaries

This chapter states the nuclearity input needed by the wedge-local and hyperfiniteness examples, but develops modular and nuclearity theorems separately. It also does not reconstruct superselection sectors or global gauge groups. Physical protocols for subregions, thermal selection, and curved-spacetime deployment remain distinct from the theorem-level algebra and representation comparisons developed here.

The classification claims are correspondingly qualified. Araki’s free-field result establishes type III; the universal hyperfinite type-III$_1$ conclusion uses additional scaling, nuclearity, and factoriality hypotheses Buchholz, D’Antoni, and Fredenhagen 1987, pp. 126–134. A cutoff calculation is useful evidence for local observables but necessarily has type-I regional algebras.

Review the chapter

For any proposed algebraic QFT, answer these questions in order.

1. What is the region class, and are causal complements and causal completions still in it?
2. Are all bonding maps injective, isometric, and coherent around every inclusion diagram?
3. Is the stated object an abstract C*-algebra, a represented von Neumann algebra, or an unbounded affiliated field?
4. Which state selects the representation, and which other states are normal in its folium?
5. Is a comparison local or global, and what topology or Hilbert–Schmidt test supports it?
6. Which exact hypotheses yield cyclicity, separation, additivity, duality, or the time-slice property?
7. If fields generate the net, where are closure, core, and strong-commutativity proofs?
8. If a factor type or local model is claimed, where do scaling, phase-space, and compact-localization inputs enter?

Synthesis exercise

A translation-invariant state on an isotonic local C*-net is locally normal to the vacuum on every double cone. The net obeys Einstein causality. May one conclude that the state is a vacuum density matrix, that the two GNS representations are globally unitarily equivalent, or that Haag duality holds?

Solution

No. Local normality places each restricted state in the corresponding vacuum folium; it neither supplies a trace-class operator representing the global state on vacuum Hilbert space nor makes the local intertwiners compatible at infinite volume. Einstein causality gives only $\mathfrak A(O')\subseteq\mathfrak A(O)'$. Haag duality is equality and requires a separate maximality theorem sensitive to topology and the observable net. The licensed conclusions are the stated local normality and causal commutation only.

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.