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