Dressed Charges, Soft Sectors, and Infrared Information

Massless gauge fields tie charged hard excitations to soft radiation and dressing. Information quantities depend on whether soft quanta are resolved, traced over, or incorporated into dressed asymptotic states. Finite detector time and energy resolution are therefore part of the subsystem definition; ideal soft-sector labels need not be perfectly distinguishable in an operational experiment.

Required background. Superselection and accessible entanglement supplies sector-conditioned information; reference frames and asymmetry supplies relational charge resources.

Helpful background. Zero modes, boundaries, and infrared effects supplies the regulator boundary.

Hard, soft, and dressed algebras

In QED, a charged state cannot be separated from its long-range electromagnetic field. Bare Fock states lead to infrared-divergent amplitudes, while inclusive observables sum over unresolved radiation and dressed-state approaches attach coherent soft clouds to charged particles.

An information calculation must state the algebra. One possible regulated split is

$$\mathcal A_{\rm meas} =\mathcal A_{\rm hard}(E>\Delta E) \vee \mathcal A_{\rm soft,resolved}(1/T<E<\Delta E),$$

with softer modes treated as unresolved during measurement time $T$. The thresholds are detector parameters, not fundamental boundaries in the field.

Soft information is a sector problem with an additional detector-resolution choice. Dressing, inclusive tracing, and resolved soft measurements define different accessible algebras. Schematic and not to scale.

Inclusive decoherence

Suppose a hard scattering amplitude is entangled with soft radiation,

$$|\Psi_{\rm out}\rangle =\sum_\alpha c_\alpha |h_\alpha\rangle|s_\alpha\rangle.$$

If the detector ignores the soft sector, the hard density matrix contains overlaps

$$(\rho_{\rm hard})_{\alpha\beta} =c_\alpha c_\beta^*\langle s_\beta|s_\alpha\rangle.$$

Soft clouds associated with sufficiently different charged trajectories can suppress off-diagonal hard coherences in ideal asymptotic limits. At finite $T$ and $\Delta E$, only radiation distinguishable within the detector bandwidth contributes operationally. Resummation or dressed asymptotic states is needed before taking the infrared regulator to zero.

The inclusive hard-state decoherence factors and their infrared-finite resummation are Carney et al. 2017, Eqs. (5)–(11).

The entropy assigned to “soft photons” depends on the hard/soft split and dressing convention. Inclusive transition probabilities can be infrared finite even when an intermediate bare-state entropy is not.

Dressed-state comparison

Faddeev–Kulish-type dressings build asymptotic charged states with coherent soft fields determined by their momenta. Two dressings may differ by radiation or by a large-gauge sector. Their overlap and the observables that distinguish them depend on boundary conditions and finite resolution.

The asymptotic dressing construction is Kulish and Faddeev 1970, §§ 2–3, while its density-matrix and information-theoretic implementation is Carney et al. 2018, §§ II–III.

One should not infer that every formal asymptotic soft charge labels a perfectly distinguishable qubit. An operational sector requires a measurement coupling, a finite integration time, and control of memory or boundary observables. Conversely, tracing all soft modes can erase information that a suitably designed detector could retain.

Finite-time detector example

Compare two charged wavepackets that scatter into nearby momenta. Choose a detector response $f_T(\omega)$ and energy threshold $\Delta E$. Compute the inclusive hard-state fidelity after tracing modes weighted as unresolved. Then vary $T$, $\Delta E$, the photon-mass or dimensional infrared regulator, and the wavepacket width.

A robust conclusion is a resolution-dependent distinguishability curve with a stable regulator limit. Taking momentum eigenstates, infinite time, and zero infrared cutoff before wavepacket localization can produce exact orthogonality that no finite detector realizes.

Evidence boundary as of 10 August 2026

Infrared-safe information quantities are established in specific inclusive and dressed scattering frameworks. Their detailed entropies and coherences depend on dressings, asymptotic algebras, detector resolution, and the order of limits. This page does not replace the scattering-theory treatment of infrared resummation, and it does not claim one universal hard/soft tensor factorization.

Validity map for infrared information. Dressing and the measurable algebra must be fixed, while the soft cutoff and detector resolution are refined separately. Unresolved labels cannot be treated as perfectly accessible sectors. Schematic and not to scale.

Common pitfalls

Tracing “the soft sector” without defining it. State the energy response, time window, dressing, and regulator.

Using ideal asymptotic orthogonality as finite-time distinguishability. A real detector has wavepacket and bandwidth limits.

Calling an intermediate bare-state entropy infrared safe. Check the inclusive or dressed observable after the regulator is removed.

References

• Carney, Daniel, Laurent Chaurette, Dominik Neuenfeld, and Gordon Walter Semenoff. “Infrared Quantum Information.” Physical Review Letters 119 (2017): 180502. DOI.
• Carney, Daniel, Laurent Chaurette, Dominik Neuenfeld, and Gordon Walter Semenoff. “Dressed Infrared Quantum Information.” Physical Review D 97 (2018): 025007. DOI.
• Kulish, P. P., and L. D. Faddeev. “Asymptotic Conditions and Infrared Divergences in Quantum Electrodynamics.” Theoretical and Mathematical Physics 4 (1970): 745–757. DOI.