The Infraparticle Edge

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 $\mathrm{d}{N}_h(\ddot{I}‰)=\hat{I}{\cdot }_h\mathrm{d}\ddot{I}‰/\ddot{I}‰+\mathrm{d}{N}_{h,\mathrm{reg}}(\ddot{I}‰)$, I obtain $\ddot{I}{}_{\mathrm{inc}}(s)\sim C\hat{I}¸(s-m^2)(s-m^2{)}^{-1+\hat{I}{\cdot }_h}$. I retain the coherence kernel and derive hard-sector dephasing and the spectral edge from two contractions of one soft environment. The diagonal coefficient $\hat{I}{º}_{aa}$ fixes the endpoint exponent, while $\frac{1}{2}(\hat{I}{º}_{aa}+\hat{I}{º}_{bb}-2\mathrm{Re}\hat{I}{º}_{ba})$ fixes the dephasing exponent between hard alternatives. I then classify infrared energy marginals, derive the finite-resolution residue $Z(\hat{I}¼)=(\hat{I}¼/\hat{I}›{)}^{\hat{I}{\cdot }_h}$, 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.

18 pages; 1 figure

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