Sector Selection, Localization, and Transportability

A superselection sector is not merely an inequivalent representation. It is an equivalence class of representations selected relative to a reference vacuum by a specified localization criterion. For a Haag–Kastler net in Minkowski space, the Doplicher–Haag–Roberts (DHR) criterion says that a charge can be hidden inside any double cone: outside that compact spacetime region its representation is indistinguishable from the vacuum. Transportability adds that the same charge type can be moved to every other double cone. Together these requirements isolate movable, compactly localizable charges and exclude thermal phases, infrared photon clouds, and topological excitations whose distinguishing data extend to infinity.

Required background. Haag–Kastler Nets and Locality supplies the net and quasilocal algebra; States, GNS Representations, and Folia supplies representations and local normality; Isotony, Additivity, Duality, and Primitive Causality supplies causal complements and the duality assumptions used below.

Helpful background. Continuous Symmetries, Generators, and Charges distinguishes charge generators from sector labels; Superselection Rules and Accessible Entanglement, Symmetry-Constrained Operations in QFT, and Reference Frames, Asymmetry, and Charged Resources explain operational consequences; What Is a Symmetry of a QFT? fixes the relation between observable symmetries and gauge redundancy; Charge-Resolved Entanglement and Charged Moments and Entanglement Asymmetry and Symmetry Restoration give information-theoretic applications.

Selection relative to the vacuum

Let $O\mapsto\mathcal A(O)$ be a local net on $(1+s)$-dimensional Minkowski space, let $\mathcal A$ be its quasilocal C*-algebra, and let $(\pi_0,\mathcal H_0,\Omega_0)$ be the vacuum representation. A representation $\pi$ obeys the DHR selection criterion in a double cone $O$ when

$$\pi\big\vert_{\mathcal A(O')}\simeq \pi_0\big\vert_{\mathcal A(O')} .$$

Here $O'$ is the spacelike complement, and $\simeq$ means unitary equivalence of representations of the exterior algebra—not equality of expectation values for a convenient subset of observables. The criterion is normally required for every double cone $O$, with a possibly different implementing unitary for each choice. One also assumes local normality and positive-energy covariance when interpreting the representation as a physical charged sector. The original analysis formulates precisely the idea that observations sufficiently far from a bounded charge preparation reproduce the vacuum and derives charge composition and permutation statistics from that premise Doplicher, Haag, and Roberts 1971, §2, pp. 204–209.

This criterion has three logically separate parts:

• Reference: the comparison is with the chosen vacuum representation $\pi_0$. A different vacuum phase can produce a different sector theory.
• Localization: equivalence is demanded on all observables in a causal exterior.
• Mobility: transportability requires representatives localized in arbitrary double cones.

A representation may satisfy the first two for one region yet fail mobility. Conversely, a covariant family of charges may be movable only in spacelike cones, not in double cones. Such cases motivate other selection criteria; they are not defective DHR sectors.

From representations to localized morphisms

Assume Haag duality in the vacuum representation, $\mathfrak A(O)=\mathfrak A(O')'$, with $\mathfrak A(O)=\pi_0(\mathcal A(O))''$, together with the usual property that isometries needed for subobjects can be found in slightly larger local algebras. Choose a unitary $V_O:\mathcal H_\pi\to\mathcal H_0$ implementing the exterior equivalence and put

$$\rho_O(A)=V_O\pi(A)V_O^*,\qquad A\in\mathcal A.$$

Then $\rho_O(A)=\pi_0(A)$ for $A\in\mathcal A(O')$. Identifying the faithful vacuum image with $\mathcal A$, duality and locality imply that $\rho_O$ maps each sufficiently large local algebra into itself. Thus a selected representation can be encoded by a transportable endomorphism localized in $O$. Changing $V_O$ changes $\rho_O$ only by a unitary intertwiner. The detailed passage, including the precise role of duality, is developed in Halvorson and Müger 2006, §§7.2–8.2, pp. 57–77.

The mechanism is worth keeping visible. Exterior equivalence first puts the charged and vacuum representations on one Hilbert space. Locality says that a localized morphism cannot alter spacelike observables. Duality then turns commutation with the exterior into membership in the interior algebra. Transporters compare different placements of the charge. None of these steps follows from inequivalence of representations alone.

A charged free-field example

Take the net $\mathcal F$ of a massive free complex scalar field and its global $U(1)$ action $\alpha_z(\phi)=z\phi$. The observable net is the fixed-point subnet $\mathcal A(O)=\mathcal F(O)^{U(1)}$. A field operator smeared in a double cone creates charge $+1$ from the vacuum; restricting the corresponding charged representation of $\mathcal F$ to $\mathcal A$ yields a representation $\pi_1$. Charge-neutral observables in $O'$ commute with the localized charged field, so $\pi_1|_{\mathcal A(O')}$ is vacuum-equivalent. Moving the smearing region with local charged fields gives transportable representatives. Iteration yields integer-labeled sectors $\pi_n$, and charge addition later becomes composition of the corresponding endomorphisms.

This is the sector-theoretic form of the multiplet and invariant-observable construction treated physically in Multiplets, Invariants, and Selection Rules. The mass assumption matters: it removes the long-range electromagnetic field that prevents a charged QED state from being vacuum-like outside a double cone.

Status, conclusion, and excluded converse

Under the stated net, vacuum, duality, local-normality, and transportability assumptions, the DHR criterion licenses a category of compactly localized charge types relative to $\pi_0$. With additional finite-statistics and covariance hypotheses, that category supports conjugates, statistics, and field reconstruction. It does not say that every physically meaningful representation is DHR, that every inequivalent representation is a charge, or that vacuum-like behavior on one bounded collection of tests proves exterior unitary equivalence.

The converse also fails: knowing that a state has the same total global charge as another does not imply that their exterior representations agree. Long-range flux profiles, soft radiation, boundary conditions, or a different thermodynamic phase can remain detectable arbitrarily far away.

Adversarial failure: a thermal representation

Let $\pi_\beta$ be an infinite-volume KMS representation at temperature $T=\beta^{-1}>0$. Local normality may hold, so every bounded laboratory can compare $\pi_\beta$ with the vacuum folium. But energy density and thermal correlations differ from their vacuum values in arbitrarily remote double cones. Hence no bounded $O$ can make $\pi_\beta|_{\mathcal A(O')}$ unitarily equivalent to $\pi_0|_{\mathcal A(O')}$. Calling $\pi_\beta$ a localized charge sector confuses a thermodynamic phase with a transportable excitation. This test is decisive because it probes the full exterior, not just a finite list of correlators.

Independent checks

Three checks catch common misclassifications. First, enlarge the exterior test: if a distinguishing observable can always be translated beyond every proposed localization region, compact localization fails. Second, move the charge: representatives in spacelike-separated double cones must be related by unitary intertwiners obeying coherent composition. Third, inspect the neutral limit: a charge–anticharge pair localized together should admit the vacuum among its fusion products when conjugates exist. These checks test localization, transportability, and charge interpretation independently.

Exercises

1. Exterior commutation. Let $F\in\mathcal F(O)$ have $U(1)$ charge $1$, and define $\omega_F(A)=\langle F\Omega_0,AF\Omega_0\rangle/\|F\Omega_0\|^2$ for $A\in\mathcal A$. Show that for $A\in\mathcal A(O')$, locality lets $A$ pass through $F$ and $F^*$, but does not by itself prove $\omega_F(A)=\omega_0(A)$.

Solution

Locality gives $[A,F]=[A,F^*]=0$, hence $\omega_F(A)=\langle\Omega_0,A F^*F\Omega_0\rangle/\|F\Omega_0\|^2$. This need not equal $\omega_0(A)$ because $F^*F$ can be correlated with $A$. The DHR statement is unitary equivalence of exterior representations; it is stronger and differently formulated than equality of the vector states produced by one bounded operator.

2. Localization is monotone. If $\rho$ is localized in $O_1$ and $O_1\subset O_2$, prove that it is localized in $O_2$.

Solution

Causal complementation reverses inclusion: $O_2'\subset O_1'$. Since $\rho(A)=A$ for every $A\in\mathcal A(O_1')$, the same equality holds on the smaller algebra $\mathcal A(O_2')$.

3. Why temperature fails. Suppose a translation-covariant state $\omega$ has a local observable $B$ with $\omega(\alpha_x(B))=c\ne\omega_0(B)$ for every spacelike translation $x$. Show that its representation cannot satisfy the DHR criterion.

Solution

For every bounded $O$, choose $x$ so that the localization region of $\alpha_x(B)$ lies in $O'$. Exterior vacuum equivalence would preserve all representation-theoretic expectation data in the selected folium, whereas the persistent translated distinction supplies an exterior discriminator for every $O$. Thus no compact $O$ localizes the difference.

References

• Doplicher, Sergio, Rudolf Haag, and John E. Roberts. “Local Observables and Particle Statistics I.” Communications in Mathematical Physics 23 (1971): 199–230. DOI.
• Haag, Rudolf. Local Quantum Physics: Fields, Particles, Algebras. 2nd ed. Berlin: Springer, 1996. DOI.
• Halvorson, Hans, and Michael Müger. “Algebraic Quantum Field Theory.” In Handbook of the Philosophy of Science, Vol. 2: Philosophy of Physics, edited by Jeremy Butterfield and John Earman, 731–922. Amsterdam: Elsevier, 2007. Open PDF, 2006 preprint.