Gauss-Law Infrasectors and Asymptotic Charge Classes

In a theory with massless gauge quanta, a charged state generally differs from the vacuum at arbitrarily large distances. Gauss flux, velocity-dependent fields, and soft photon clouds defeat double-cone and often spacelike-cone localization. The resulting inequivalent representations are called infrasectors. Useful coarser classifications restrict which measurements count—for example to the algebra inside a future light cone—and group many globally disjoint infrared sectors into one charge class. Total electric charge alone is too coarse, while a global superselection sector is often too fine.

Required background. Infraparticles and Velocity Superselection supplies the spectral obstruction; Gauss-Law Charges and Infrared Sectors supplies asymptotic flux; BF Sectors, Spacelike Cones, and Massive Charges supplies the closest massive comparison.

Helpful background. Dressed States and Infrared-Finite Scattering explains coherent dressings; Dressed Charges, Soft Sectors, and Infrared Information explains what asymptotic data can encode.

Infrared representations and asymptotic data

Let $\mathcal A$ be the electromagnetic observable algebra. In a coherent-state model for the asymptotic photon field, a dressing is specified by a classical displacement $f$ on one-photon phase space. Two coherent representations are unitarily equivalent exactly when the displacement difference belongs to the one-photon Hilbert space. Coulombic profiles behave singularly as $|\mathbf k|\to0$; different angular flux or velocity profiles can have a difference outside that Hilbert space and therefore define disjoint representations even when their total flux agrees.

Schematically, an asymptotic electric field determines a function $q(\hat{\mathbf n})$ on the sphere with total charge

$$Q=\int_{S^2}q(\hat{\mathbf n})\,d\Omega.$$

Equality of $Q$ does not imply equality of $q$, and neither equality proves unitary equivalence of full photon-cloud representations. The norm of the soft displacement, its angular profile, and the observable algebra retained in the comparison all matter. This explains velocity superselection: distinct asymptotic velocities support different long-range Liénard–Wiechert fields.

Light-cone charge classes

Fix a future light cone $V$ and the algebra $\mathcal A(V)$ of observables localized within it. Radiation emitted before the apex and moving outward never enters $V$; restricting to $\mathcal A(V)$ therefore forgets part of the uncontrollable infrared past. Buchholz and Roberts define charge classes of elemental states by mutual norm approximation under inner automorphisms of the type-III factor generated by $\mathcal A(V)$. A simple charge class admits morphisms localized in arbitrary hypercones $C\subset V$, acts trivially on the causal complement within $V$, and has an appropriate fullness property. These morphisms can be composed after controlled extensions, possess conjugate classes, and, under the paper’s assumptions, yield Bose/Fermi statistics and a compact symmetry group Buchholz and Roberts 2014, §4, preprint pp. 10–21.

The mechanism is operational. Restriction to $V$ coalesces globally distinct soft sectors that cannot be distinguished by future-directed measurements. Hypercone localization then recovers enough transportability to define morphisms. This is a classification relative to $\mathcal A(V)$, not a proof that the global representations became equivalent.

Two electron dressings with equal charge

Compare electron states with the same total charge $e$ but angular flux profiles $q_1(\hat{\mathbf n})$ and $q_2(\hat{\mathbf n})$, each integrating to $e$. If the associated coherent displacement difference is not square integrable in the infrared, the global GNS representations are disjoint. The two states can nevertheless belong to the same light-cone charge class if the radiation distinguishing them escapes the chosen $V$ and their restrictions satisfy the norm-closure criterion. They need not do so: equal charge is necessary for this comparison but is not sufficient. The physical long-range-force setting is developed in Charges, Screening, and Long-Range Forces.

Domain, status, and excluded converse

The coherent-state equivalence test is exact for the specified asymptotic Weyl algebra; it is a model of infrared representation theory, not a full nonperturbative classification of four-dimensional QED. The light-cone construction is a theorem for nets satisfying its type, covariance, hypercone, and simplicity assumptions. It classifies simple charge classes within $V$ and does not establish that all QED charges are simple or that all relevant classes occur.

No converse from total charge exists: $Q_1=Q_2$ neither forces equal angular flux nor unitary equivalence nor membership in the same charge class. Conversely, membership in one light-cone charge class does not make the corresponding global sectors identical.

Adversarial failure: classification by total charge

Take $q_1-q_2$ with zero sphere integral but a nonzero angular harmonic and an infrared displacement whose norm diverges logarithmically. Both states have charge $e$, yet their coherent representations are disjoint. A table indexed only by $Q$ assigns them one sector and loses measurable asymptotic field data. The counterexample also shows why “same global charge” cannot replace a localization or norm-equivalence test.

Independent checks

Compute both the sphere integral and the infrared one-particle norm; they test different information. State the observable algebra before declaring equivalence. Verify hypercone transporters within $V$ rather than globally. Check conjugation by composing a class with its opposite and testing that the vacuum class occurs.

Exercises

1. Same charge, different profile. Let $q_2=q_1+aY_{20}$ on $S^2$. Show the total charges agree.

Solution

Every nonconstant spherical harmonic integrates to zero, so $\int q_2d\Omega=\int q_1d\Omega+a\int Y_{20}d\Omega=\int q_1d\Omega$.

2. Coherent equivalence. If $f_1-f_2$ has infinite one-photon norm, can a Fock-space Weyl unitary implement the displacement?

Solution

No. Weyl implementers exist for vectors in the one-photon Hilbert space. An infinite norm places the two coherent states in disjoint representations even if their total charge agrees.

3. Restriction. Explain why equivalence on $\mathcal A(V)$ need not extend to equivalence on $\mathcal A$.

Solution

The global algebra contains observables outside $V$ that can detect radiation omitted by the restriction. An intertwiner for the smaller algebra has no obligation to intertwine those additional observables.

References

• Buchholz, Detlev. “Gauss’ Law and the Infraparticle Problem.” Physics Letters B 174 (1986): 331–334. DOI.
• Buchholz, Detlev, and John E. Roberts. “New Light on Infrared Problems: Sectors, Statistics, Symmetries and Spectrum.” Communications in Mathematical Physics 330 (2014): 935–972. DOI. Open PDF.