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arXiv:2606.27498

Soft QED as Open Quantum System: Infrared Cancellation and Soft-Shell Coarse Graining

Soo-Jong Rey

hep-thcond-mat.stat-mechhep-phquant-phSubmitted Jun 25, 2026• Announced Jun 29, 2026
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Abstract

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 $f$ of multiple hard particles and photons has probability $P_{f}(i) =\sum_n|\langle f,n|S|i,0\rangle|^2=\langle i|F_f|i\rangle$, with Kraus operators $K_n= {}_{\rm soft} \langle n|S|0 \rangle_{\rm soft}$ and effect $F_f=\sum_nK_n^\daggerΠ_fK_n$. 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.

Comments
23 pages, 1 figure

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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.
The discard ZX-calculus, a diagrammatic language for mixed-state quantum mechanics, is used to give a nonperturbative, categorical proof of the Bloch-Nordsieck cancellation of infrared divergences in QED. Soft photons are treated as an open quantum system: the resolved charged particles and hard photons form the system, while photons below a detector resolution form the environment. The reduced hard channel is a completely positive trace-preserving (CPTP) map, and the soft-photon theorem replaces the full S-matrix by a controlled displacement operator whose Feynman-Vernon influence functional satisfies the equal-history normalization ${\cal F}[J,J]=1 $. In the ZX-calculus, this normalization is a single diagrammatic identity: the doubled displacement diagram collapses to the bare wire under the unitarity, cyclicity, and discard rules. The proof therefore serves as a categorical consistency check on the open-system treatment of soft QED given in a companion paper; it confirms that the physical derivation is logically complete and free of hidden assumptions about the infrared limit. For off-diagonal hard-state elements, the same diagram yields the coherent-state overlap, giving a first-principles account of soft-cloud decoherence. The soft-shell coarse graining is then constructed as a CPTP Schur channel whose infinitesimal limit produces the exact Lindblad generator with jump operators determined by the eikonal emission amplitudes. Finally, a local CPTP-certification pipeline is developed for non-Markovian process tensors, enabling constant-time verification of trace preservation in open quantum simulations. The framework bridges categorical quantum semantics, non-equilibrium field theory, and practical open-system compilation.
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.
I formulate black-hole-horizon-induced decoherence of charged branch codes as the leading-soft QED restricted to an exterior algebra, formulated as an open quantum system. The fixed-history Feynman--Vernon identity F[J,J]=1{\cal F}[J,J]=1F[J,J]=1 remains exact. Decoherence enters through the unequal-history influence factor that survives exterior monitoring and belongs to the complementary horizon output. In the coherent eikonal regime, I derive the completely positive Schur channel (EH(0)ρ)ab=⟨ΦbH,(0)∣ΦaH,(0)⟩ ρab({\cal E}_H^{(0)}ρ)_{ab}=\langleΦ_b^{H,(0)}|Φ_a^{H,(0)}\rangle \, ρ_{ab}(EH(0)​ρ)ab​=⟨ΦbH,(0)​∣ΦaH,(0)​⟩ρab​. The leading soft input is the eikonal factor, projected onto the horizon radiative algebra. The channel yields Gram-positivity constraints, an exterior quantum-eraser bound, finite-time non-Markovianity tests, soft/hard scaling criteria, and a charged-qutrit interferometer measuring a leading-soft Bargmann holonomy. The holonomy phase is the rephasing-invariant symplectic area of a triangle in horizon soft phase space. I show that its orientation, common-mode, triangulation, and completely positive determinant identities render falsifiable tests beyond pairwise two-path visibility.