Quantum Origin of Classical Background Fields from Coherent States: A First-Principles Formulation in QED

TL;DR

This work provides a first-principles derivation of classical background electromagnetic fields in QED from coherent states, showing that the background is the expectation value $\mathcal{A}^\mu(x)=\langle\alpha_H|\hat{A}_H^\mu(x)|\alpha_H\rangle$ and that a displaced Hamiltonian naturally connects Schrödinger- and Heisenberg-picture formalisms. By constructing a coherent-field QED framework, the authors unify operator-based, Furry-picture, and path-integral descriptions, and show how depletion/backreaction emerge as genuine quantum effects through transitions between coherent states. They demonstrate that the conventional fixed-background generating functional $Z[\mathcal{A}]$ is a limiting case of their more general coherent-state amplitude, clarifying the conceptual origin and domain of validity of background fields. The approach also relates to BRST quantization via the Gupta–Bleuler condition and provides a practical route to incorporate dynamical background fields in strong-field QED, with potential impact on laser-matter interaction theory and related high-intensity phenomena.

Abstract

Classical background electromagnetic fields are routinely employed in quantum electrodynamics to describe a wide range of physical situations, from laser-matter interactions to strong-field phenomena. In this work, we present a first-principles formulation that clarifies the quantum origin of such classical background fields in QED by systematically deriving them from coherent states of the electromagnetic field. Abstract Starting from the operator formulation of QED, we show how scattering amplitudes between coherent states naturally lead to an effective description in terms of background fields, while maintaining a clear separation between the coherent laser mode and other quantized photon degrees of freedom. This framework allows one to consistently incorporate effects beyond the fixed background approximation, such as depletion and backreaction, without assuming any particular field strength or intensity regime. Abstract We further demonstrate how the conventional generating functional with a prescribed background field emerges as a limiting case, corresponding to fixed coherent state boundary conditions. The path integral representation is then obtained as a reformulation of the same underlying Heisenberg picture amplitudes, providing a unified view of operator-based and functional approaches. Abstract Our results establish a general and intensity-independent foundation for QED with coherent background fields, within which the standard formulations of strong-field QED arise as well-defined special cases.