Sector Completeness, Field Algebras, and Classification Limits

A sector analysis is complete only relative to a declared representation class and selection criterion. Reconstructing a field algebra from a full finite-statistics DHR category can be rigorous without proving that every physical representation is DHR, that every extension is captured, or that the observable net is classified by its fusion data. Completeness, reconstruction, and classification are three different claims. Scaling limits add another separation: charges can disappear, split, merge, or emerge as the observation scale tends to zero.

Required background. Counterexamples, Nonconverses, and Hypothesis Stress Tests supplies the claim discipline; Superselection Sectors and DHR Reconstruction supplies the selected category; Doplicher–Roberts Compact-Gauge Reconstruction supplies field reconstruction; Solitonic, Topological, and Boundary Sectors supplies a known class outside compact localization.

Helpful background. Actions, Generalized Charges, and Selection Rules explains categorical actions; Topological Order, Invertible Phases, and Matter Diagnostics explains why excitation data do not exhaust a phase.

Three claims that require different evidence

Let $\mathcal C_{\mathrm{DHR}}^f(\mathfrak A)$ denote the finite-statistics DHR category of a vacuum net.

Sector completeness says that every representation in a specified admissible class is equivalent to a direct sum or integral of listed sectors. Its quantifier must name the class: locally normal positive-energy DHR representations, BF representations, boundary sectors, or something else. No theorem about one class exhausts the others.

Field reconstruction says that specified categorical data produce an extension $\mathfrak F$ with an action of $G$ and $\mathfrak F^G=\mathfrak A$. Doplicher–Roberts reconstruction covers the full symmetric rigid finite-statistics DHR category under its closure hypotheses. It is an existence-and-uniqueness result for a normal complete field system in that domain Doplicher and Roberts 1990, §§3–5, pp. 65–94.

Net classification says that some invariant determines the local net up to isomorphism. An abstract fusion ring is not enough; even a unitary modular tensor category need not be a complete invariant of a conformal net. The concrete braided action on local algebras contains strictly more information. Giorgetti and Rehren state this gap and formulate a stronger action-based program Giorgetti and Rehren 2017, §§1 and 3–4, pp. 1–4 and 13–23.

Finite-group fixed points as a controlled test

Let a field net $\mathfrak F$ have a faithful action of a finite group $G$ and set $\mathfrak A=\mathfrak F^G$. Suppose the field net has no additional relevant sectors and the observable net satisfies the DHR and duality hypotheses. Then the reconstructed finite-statistics DHR simples correspond to irreducible representations of $G$, with fusion given by representation tensor product. A disciplined completeness test has four parts:

1. construct every predicted sector from charged fields;
2. prove any irreducible representation satisfying the DHR criterion is one of them;
3. verify that the reconstructed field extension is equivalent to $\mathfrak F$;
4. state explicitly that BF, infrared, twisted, solitonic, and boundary representations were not quantified over.

The physical fixed-point operation is related to Gauging Continuous and Finite Symmetries, but the AQFT conclusion concerns representations of the observable net and should not be replaced by a path-integral slogan.

Scaling limits can change the sector theory

The scaling-algebra construction associates to each bounded region $O$ functions $\underline A(\lambda)\in\mathcal A(\lambda O)$ that vary uniformly under scaled Poincaré transformations. Weak-* limit points of the lifted vacuum states as $\lambda\to0$ produce scaling-limit states and nets $\mathcal A_{0,\iota}$. Different limit points need not yield isomorphic nets. Buchholz and Verch distinguish classical, unique quantum, and degenerate scaling limits Buchholz and Verch 1995, §4, pp. 20–27.

Sector comparison across scales therefore needs explicit families of localized charge creators with uniform phase-space control. A charge present at finite scale may be confined or disappear in the limit; one sector may split into several limit sectors; distinct finite-scale sectors may coalesce; a scaling-limit theory may contain DHR charges absent as finite-energy sectors of the underlying theory. The original paper gives an instructive two-dimensional current example in which charged sectors occur in the scaling limit although no corresponding finite-energy charged sectors exist before taking the limit Buchholz and Verch 1995, §7, pp. 39–41.

Thus a finite-cutoff sector list cannot establish ultraviolet completeness, and a unique scaling limit of nets does not automatically identify sector categories. Charge preservation is an additional theorem with additional uniform hypotheses.

Exact conclusion and nonconverses

A complete finite-statistics DHR analysis licenses Doplicher–Roberts reconstruction within that category. A proof that its field algebra has no further DHR sectors licenses DHR completeness. Neither conclusion classifies all representations or the net. Classification requires an invariant shown to be injective on a stated class of nets.

The nonconverses are concrete. Equal fusion rings do not imply equivalent tensor categories; equivalent tensor categories need not have equivalent braidings; equivalent braided categories need not give isomorphic concrete actions; isomorphic sector actions need extra hypotheses before they classify local nets. Likewise, agreement of sectors at one scale does not imply agreement in every scaling-limit net.

Adversarial failure: fusion rules only

Suppose a proposed classification records simple labels and integers $N_{ab}^{c}$ but omits associators, braiding, conjugate solutions, and the functor into localized endomorphisms. Distinct categories can share those integers, and one abstract category can act inequivalently on local factors. The data also say nothing about an infinite-statistics sector or a kink excluded by DHR localization. Therefore “the fusion table matches” proves none of sector completeness, field uniqueness, or net isomorphism.

Independent checks

Write the quantified representation class beside every completeness claim. Compare not only simple labels but arrow spaces, tensor products, associators, braiding, and the concrete action on each local algebra. Search explicitly for representations failing the chosen localization condition. For scaling, compare all limit points or prove uniqueness, then construct charge families with bounds uniform in $\lambda$.

Exercises

1. Logical separation. A net has DHR category $\operatorname{Rep}(G)$. Which extra statement is needed before concluding that a proposed $G$-field algebra is complete?

Solution

One must show that the category used is the full admissible finite-statistics DHR category and that the field system realizes every object with the required normal commutation relations. The abstract equivalence alone neither proves realization nor excludes sectors outside the selected class.

2. Scaling ambiguity. Why does one scaling-limit subsequence not determine a degenerate scaling limit?

Solution

Degeneracy means different weak-* limit points yield nonisomorphic nets. One subsequence samples only one limit point; all limit points must be compared or uniqueness proved.

3. Missing action. Give one datum present in a concrete DHR action but absent from fusion coefficients.

Solution

For example, it specifies the actual endomorphism $\rho:\mathfrak A(O)\to\mathfrak A(\widetilde O)$ and the local algebra containing each intertwiner. Fusion coefficients record only multiplicities in object products.

References

• Buchholz, Detlev, and Rainer Verch. “Scaling Algebras and Renormalization Group in Algebraic Quantum Field Theory.” Reviews in Mathematical Physics 7 (1995): 1195–1239. DOI. Open PDF.
• Doplicher, Sergio, and John E. Roberts. “Why There Is a Field Algebra with a Compact Gauge Group Describing the Superselection Structure in Particle Physics.” Communications in Mathematical Physics 131 (1990): 51–107. DOI.
• Giorgetti, Luca, and Karl-Henning Rehren. “Braided Categories of Endomorphisms as Invariants for Local Quantum Field Theories.” Communications in Mathematical Physics 357 (2018): 3–41. DOI. Open PDF.