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arXiv:2607.13001

The Infraparticle Edge

Soo-Jong Rey

hep-thcond-mat.str-elgr-qchep-phquant-phSubmitted Jul 14, 2026• Announced Jul 15, 2026
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Abstract

I derive the charged-particle spectral edge from the quantum instrument of soft QED. I use two projections of that instrument. Tracing over unresolved photons gives the reduced hard-sector channel. Pushing the outcomes to total energy gives the inclusive energy distribution. Its Laplace exponent is fixed by the diagonal soft intensity. For ${\rm d} N_h(ω)=η_h{\rm d}ω/ω+{\rm d} N_{h,\mathrm{reg}}(ω)$, I obtain $ρ_{\mathrm{inc}}(s)\sim Cθ(s-m^2)(s-m^2)^{-1+η_h}$. I retain the coherence kernel and derive hard-sector dephasing and the spectral edge from two contractions of one soft environment. The diagonal coefficient $κ_{aa}$ fixes the endpoint exponent, while $\frac12(κ_{aa}+κ_{bb}-2\operatorname{Re}κ_{ba})$ fixes the dephasing exponent between hard alternatives. I then classify infrared energy marginals, derive the finite-resolution residue $Z(μ)=(μ/Λ)^{η_h}$, prove stability under infrared-integrable perturbations, and separate the bath exponent from a hard threshold exponent. For the one-electron spectral measure, the hard threshold factor is regular. The resulting edge has the local power law of a gapped unparticle spectrum, while its exponent remains a response coefficient of the unresolved photon sector.

Comments
18 pages; 1 figure

Related papers

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I derive the charged-particle spectral edge from the quantum instrument of soft QED. I use two projections of that instrument. Tracing over unresolved photons gives the reduced hard-sector channel. Pushing the outcomes to total energy gives the inclusive energy distribution. Its Laplace exponent is fixed by the diagonal soft intensity. For dNh(ω)=ηhdω/ω+dNh,reg(ω){\rm d} N_h(ω)=η_h{\rm d}ω/ω+{\rm d} N_{h,\mathrm{reg}}(ω)dNh​(ω)=ηh​dω/ω+dNh,reg​(ω), I obtain ρinc(s)∼Cθ(s−m2)(s−m2)−1+ηhρ_{\mathrm{inc}}(s)\sim Cθ(s-m^2)(s-m^2)^{-1+η_h}ρinc​(s)∼Cθ(s−m2)(s−m2)−1+ηh​. I retain the coherence kernel and derive hard-sector dephasing and the spectral edge from two contractions of one soft environment. The diagonal coefficient κaaκ_{aa}κaa​ fixes the endpoint exponent, while 12(κaa+κbb−2Re⁡κba)\frac12(κ_{aa}+κ_{bb}-2\operatorname{Re}κ_{ba})21​(κaa​+κbb​−2Reκba​) fixes the dephasing exponent between hard alternatives. I then classify infrared energy marginals, derive the finite-resolution residue Z(μ)=(μ/Λ)ηhZ(μ)=(μ/Λ)^{η_h}Z(μ)=(μ/Λ)ηh​, prove stability under infrared-integrable perturbations, and separate the bath exponent from a hard threshold exponent. For the one-electron spectral measure, the hard threshold factor is regular. The resulting edge has the local power law of a gapped unparticle spectrum, while its exponent remains a response coefficient of the unresolved photon sector.
I formulate unresolved soft-photon sector of QED as open quantum system. Resolved charged particles and hard photons form the system, unresolved soft photons form the environment, and basic object is the reduced density matrix. A resolved outcome fff of multiple hard particles and photons has probability Pf(i)=∑n∣⟨f,n∣S∣i,0⟩∣2=⟨i∣Ff∣i⟩P_{f}(i) =\sum_n|\langle f,n|S|i,0\rangle|^2=\langle i|F_f|i\ranglePf​(i)=∑n​∣⟨f,n∣S∣i,0⟩∣2=⟨i∣Ff​∣i⟩, with Kraus operators Kn=soft⟨n∣S∣0⟩softK_n= {}_{\rm soft} \langle n|S|0 \rangle_{\rm soft}Kn​=soft​⟨n∣S∣0⟩soft​ and effect Ff=∑nKn†ΠfKnF_f=\sum_nK_n^\daggerΠ_fK_nFf​=∑n​Kn†​Πf​Kn​. The SK formulation places unresolved virtual and real terms in one doubled-contour expansion. At one loop they carry the same on-shell eikonal kernel with opposite signs. This elegantly organizes the QED probability: for the same observable, perturbative order, diagrams, and phase space, the OQS gives the same infrared-finite terms as the full-QED. The soft-photon evolution is a unitary coherent-state displacement driven by the scattering current. The equal-history identity of influence functional exactly normalizes this soft evolution; together with the soft-photon theorem it removes the IR divergent leading-soft factor from inclusive probability. I also derive explicit leading-soft QED realization of scale-parametrized Lindblad evolution on a fixed hard-branch space. Tracing an infinitesimal soft-photon shell produces diagonal jump operators whose entries are fixed by the corresponding eikonal emission amplitudes. The finite-shell map is a completely positive unital Schur channel and, in the sharp scale-invariant leading-soft regime, a dephasing semigroup of a completely-positive-divisible scale flow. The resulting logarithmic visibility slope and monotonic purity loss are off-diagonal predictions of the reduced-state description. The same controlled-displacement dilation gives the Sudakov probability, Poisson soft-photon multiplicities, and the bremsstrahlung number spectrum.
Exact Solution for Bulk-Edge Coupling in the Non-Abelian ν=5/2\nu=5/2ν=5/2 Quantum Hall Interferometer
Built upon infrared QED formulated as an open quantum system, I use the complementary soft photons as a quantum instrument. The experiment selects the hard-state code, resolved process, trajectory, accepted modes, and photon readout. QED fixes the interaction. On an eikonal-degenerate reducing code, the momentum-flow jump acts as a scalar. A Lindbladian displacement removes this common term from the dissipator. Its interference with the Low--Burnett--Kroll operator enters a finite Hamiltonian. The residual dissipator Jλ(q^){\cal J}_\lambda(\hat q)Jλ​(q^​) measures direction- and helicity-dependent projections of orbital and spin angular momentum. I isolate the same operator in a selective photon measurement by coherently nulling the branch-independent eikonal output. The nulled click effects and the logical dissipator scale as Ed2E_d^2Ed2​. Both saturate at the lower infrared endpoint. In the semiclassical branch limit, the decoherence functionalΓab(1)(Ed)=e22∑λ=±∫0Edd3q(2π)32ω∣Jλ,a(q^)−Jλ,b(q^)∣2 \Gamma^{(1)}_{ab}(E_d)= \frac{e^2}{2}\sum_{\lambda=\pm}\int_0^{E_d} \frac{{\rm d}^3q}{(2\pi)^3 2\omega} \left|{\cal J}_{\lambda,a}(\hat q)-{\cal J}_{\lambda,b}(\hat q)\right|^2Γab(1)​(Ed​)=2e2​∑λ=±​∫0Ed​​(2π)32ωd3q​∣Jλ,a​(q^​)−Jλ,b​(q^​)∣2sets the soft-output overlap. Coherent prescribed currents exponentiate this functional. Repeated outputs give the exact sample complexity for hard-branch identification. General mixed outputs obey the quantum Chernoff bound. I also specialize privacy--correctability duality to the vacuum-plus-one-photon instrument. The resulting criterion identifies hard codes that hide their logical state from the retained soft output.I extend the instrument to orthogonal mode restrictions and zero-mean Gaussian soft states. I prove covariance under unitary mode transformations. I derive the equivalent positive noise-kernel form. The result promotes the finite two-path record to an operator-valued QED instrument. Soft radiation becomes a controlled probe of angular momentum.
We develop a manifestly gauge and Lorentz invariant perturbative calculus to study the finite time and distance evolution of the electrically charged states created by Wilson-line-dressed scalar field operators in scalar quantum electrodynamics. By an Osterwalder-Schrader reflection and Euclidean gluing, the overlap of two dressed states is expressed as a Euclidean 2 point function of dressed operators, which is then evaluated in renormalized perturbation theory with a Stueckelberg photon mass as a gauge, BRST and Lorentz invariant infrared cutoff. This renders explicit and computable in closed form several structural features of the charged sector that have so far been accessible mainly through non-perturbative or algebraic arguments. We find that states dressed by non-parallel Wilson lines are ``cloud orthogonal". Their overlap vanishes as a power of the infrared regulator, with an exponent that depends only on the angle between the two dressings and is an infrared analogue of the cusp anomalous dimension with the difference that it is one-loop exact whenever the charged matter is mass-gapped. Charged states are thereby superselected by the orientation of their dressing. Within a superselection sector the 2 point function is infrared finite but exhibits the infraparticle. The mass shell pole is replaced by the edge of a branch cut. As a result, the asymptotic spread of a dressed charge is a power law in proper time with a one-loop-exact computable exponent depending on the dressing and on the direction of motion. We show that although cloud orthogonality and the infraparticle are each extremely sensitive to fundamental infrared cutoffs, the infraparticle scaling law persists across a wide window of proper time.
Strong coupling to a reservoir can do more than shift, broaden, or dress the work peaks of a driven quantum system. When the reservoir is infrared singular, a sudden change of a local control parameter can alter the boundary condition seen by infinitely many low-energy modes, converting a quasiparticle-like threshold line into a many-body edge. We demonstrate this mechanism for the inclusive work distribution of the biased spin-boson model under a sudden bias inversion. In the independent-boson limit, the problem is exactly solvable and gives a sharp infrared classification: a super-Ohmic bath can retain a finite elastic threshold weight, whereas Ohmic and sub-Ohmic baths extinguish the elastic line through boundary orthogonality. At the Ohmic fixed point, the same exponent controls both the vanishing elastic residue and the low-work continuum. We then ask how this edge is resolved away from the static-boundary limit. Using displaced-basis exact diagonalization of logarithmically discretized baths, we find that finite tunnelling leaves an edge-like continuum over the accessible energy window, while separating two operational diagnostics of the threshold: the cumulative-continuum exponent extracted from zzz-interleaved spectra lies above the elastic-overlap exponent extracted from zzz-averaged overlaps, θC>θZθ_C>θ_ZθC​>θZ​. We interpret this separation as a finite-energy crossover away from the static-boundary fixed point, not as evidence for a new asymptotic fixed point. The separation survives fitting-window variation, oscillator-cutoff checks, spectrum-size checks, and leave-one-zzz-out tests, while time-domain characteristic functions provide a compatible but non-decisive diagnostic. Finally, the same threshold edge controls the sampling cost of Jarzynski-type exponential averages, making rare low-work events increasingly important at low temperature.