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Vacuum photon emission and mean electromagnetic field in pair-creating external backgrounds

I. A. Aleksandrov, E. V. Perelygin, and D. V. Chubukov

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Phys. Rev. D 114, 056009 – Published 8 September, 2026

DOI: https://doi.org/10.1103/6r3d-w1tl

Abstract

We develop a perturbative description of vacuum radiative processes in quantum electrodynamics with a prescribed external electromagnetic background capable of producing electron-positron pairs. Since the initial vacuum is then unstable and the in- and out-vacua are inequivalent, radiative observables require a real-time formulation beyond the ordinary in-out approach of vacuum-stable QED. Using the Keldysh-Schwinger-Fradkin nonequilibrium technique, we derive the mean number density of emitted photons through the second nonvanishing order in the fine-structure constant. The leading term, of order α, reproduces the known vertex and tadpole mechanisms, while the complete order-α2 correction contains interference, loop, and induced-current contributions. We also give an independent derivation based on the spectral decomposition of the identity operator in the in-Fock space, where the photon number density is represented as a sum of squared transition amplitudes and vacuum-disconnected terms are canceled by the optical theorem generalized to an unstable vacuum. In addition, we compute the mean electromagnetic field through order e3, including the electromagnetic dressing of the induced vacuum current, and verify it using the corresponding Schwinger-Dyson equations. The final formulas are expressed in terms of exact solutions and propagators of the Dirac equation in the external background and apply to general spacetime-dependent field configurations.

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Phys. Rev. D 114, 056009– Published 8 September, 2026
  • Received 16 June 2026
  • Accepted 5 August 2026
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