Gauss-Law Charges and Infrared Sectors

Gauss’ law makes electric charge visible at arbitrarily large distance: the charge inside a region equals electric flux through its boundary. Consequently no compactly localized observable or unitary can create a nonzero charged state from the neutral vacuum. Charged sectors are globally distinguished by asymptotic flux and are generally inequivalent representations of the observable algebra, even when they are locally indistinguishable in bounded regions.

Required background. Positivity, spectrum, covariance, and locality hypotheses supplies the observable net; massless scattering and radiation fields supplies the electromagnetic asymptote; and infraparticles and velocity superselection gives the charged spectral consequence.

Helpful background. Gauge orbits, Gauss constraints, and stabilizers derives the constraint, while continuous symmetries, generators, and charges gives the charge language.

Charge as flux at infinity

Formally at one time,

$$\nabla\cdot\mathbf E(\mathbf x)=j^0(\mathbf x).$$

After smearing so that all quantities are well-defined distributions, integration over a ball gives

$$Q_R =\int_{|\mathbf x|<R}\mathrm d^3\mathbf x\,j^0(\mathbf x) =\int_{|\mathbf x|=R}\mathbf E\cdot\mathrm d\mathbf S.$$

The limit $R\to\infty$ is not automatically a strong operator limit; it may be defined through commutators, weak limits, or sector-dependent asymptotic fields. What is invariant is that a nonzero total charge changes the boundary flux.

The order of operations is part of the assertion. One first fixes the time and radial smearing needed to define $Q_R$ on a common domain, evaluates its commutator with an observable localized in a fixed bounded region, and only then lets $R$ grow. Interchanging the large-radius limit with an ill-defined unsmeared charge integral can create spurious surface terms. The robust algebraic datum is the limiting action on local observables together with the sector’s asymptotic flux, not an assumed globally defined charge density operator.

If $A$ is localized in a fixed bounded region, the surface of a sufficiently large ball is spacelike separated from $A$ after suitable time smearing. Locality gives

$$[A,Q_R]=0$$

for sufficiently distant flux approximants, up to the controlled smearing tails. Passing to the charge action yields $[A,Q]=0$. Hence an observable localized in a bounded region cannot change total electric charge.

The compact-localization obstruction

Assume a compactly supported unitary $V$ creates a charged state $V\Omega$ with charge $q\neq0$. For large $R$, locality gives

$$\langle V\Omega,Q_RV\Omega\rangle =\langle\Omega,V^*Q_RV\Omega\rangle =\langle\Omega,Q_R\Omega\rangle=0,$$

contradicting the nonzero limiting flux. Therefore a charge-creating field must be nonlocal relative to the observable algebra—string-, cone-, or asymptotically localized—or must act between inequivalent representations rather than as a local observable on the vacuum Hilbert space.

This is the representation-theoretic content behind gauge-invariant and dressed observables. The dressing’s long-range field is required by Gauss’ law, not a removable gauge artifact.

The infrared and mass consequences are established under explicit algebraic assumptions in Buchholz 1986, pp. 331–334. Charged coherent representations and their asymptotic electromagnetic fields are analyzed in Fröhlich, Morchio, and Strocchi 1979, pp. 241–284.

Superselection and localization classes

Let $\pi_0$ be the vacuum representation of the quasilocal observable algebra and $\pi_q$ a charged representation. In a bounded laboratory region the representations may be locally normal and operationally comparable, but their asymptotic flux differs. No global unitary intertwining $\pi_q$ with $\pi_0$ can preserve every observable, so charge labels superselection sectors.

Doplicher–Haag–Roberts localization asks that a charged representation agree with the vacuum outside a bounded region. Electric charge fails this criterion because its flux is measurable arbitrarily far away. Spacelike-cone localization is appropriate for some massive charges with weaker long-range structure, while QED may require still more diffuse asymptotic localization. These alternatives have distinct hypotheses; “not DHR-localized” does not mean “unphysical.”

An independent dimensional check confirms the long-range tail. In four dimensions Coulomb $|\mathbf E|\sim q/(4\pi R^2)$ times sphere area $4\pi R^2$ gives an $R$-independent flux $q$. A compactly supported electric field would instead give zero flux for every sufficiently large sphere and hence zero charge.

Radiation versus Coulomb data

Radiation fields scale as $1/R$ and carry finite energy flux through large spheres. Coulomb fields scale as $1/R^2$ and carry finite charge flux. A neutral radiation theorem can discard the latter while a charged-sector classification cannot. Mixing these two limits is a common source of incorrect claims that soft radiation alone supplies a compact charged state.

The flux label can also depend on angular asymptotic data, leading to finer infrared sectors than total charge alone. Which distinctions survive depends on the observable algebra and spacetime region available to measurement.

Exercises

Why does local commutativity with $Q_R$ not imply that the electric field vanishes in a charged state?

Solution

It says a bounded-region observable cannot change the asymptotic flux. A charged representation can already have a nonzero expectation of the distant flux operator. The conclusion is superselection of that flux, not vanishing of the field.

References

• Buchholz, Detlev. 1986. “Gauss’ Law and the Infraparticle Problem.” Physics Letters B 174: 331–334. DOI.
• Fröhlich, Jürg, Giovanni Morchio, and Franco Strocchi. 1979. “Charged Sectors and Scattering States in Quantum Electrodynamics.” Annals of Physics 119: 241–284. DOI.