Coherent-state boundary conditions as the first-principles origin of background fields in QED

TL;DR

This paper rigorously derives the background field approximation in QED from full quantum electrodynamics using coherent-state boundary conditions, clarifying its foundational origin and relation to backreaction effects.

Contribution

It provides the first operator-level proof that background fields in QED emerge as a boundary-condition limit of full QED, linking classical backgrounds to quantum coherent states.

Findings

Background fields arise from coherent-state boundary conditions in full QED.

The background-field approximation is shown to be a limit of full QED, not a separate theory.

Time dependence of backgrounds is a choice of quantum picture, not an explicit Hamiltonian property.

Abstract

QED formulated in prescribed classical background electromagnetic fields is a standard framework for strong-field and laser\textendash matter interactions. It is usually treated as a theory modified by externally imposed fields, obscuring its precise relation to full QED and, in particular, the role of asymptotic boundary conditions for the gauge field. Starting from an operator-based formulation, we show that QED with background fields is not a distinct theory but arises as a well-defined boundary-condition limit of full QED, in which the classical background field emerges from coherent-state boundary conditions on the quantized electromagnetic field. In this limit, the conventional generating functional used in calculations with prescribed background fields is recovered naturally, while relaxing the boundary conditions allows depletion and backreaction effects to be incorporated within the same framework. The central new result is a rigorous operator-level proof of the equivalence between the fixed background-field approximation and coherent-state asymptotic boundary conditions\textemdash a formulation that, to our knowledge, has not been made explicit in previous approaches. We further demonstrate that the apparent time dependence of background fields does not originate from an explicitly time-dependent Hamiltonian, but instead reflects the choice of picture\textemdash Schr\"{o}dinger versus Heisenberg\textemdash in the underlying quantum theory. Rather than introducing new properties of coherent states, our analysis provides a first-principles reinterpretation of the fixed background-field approximation as a controlled and picture-dependent limit of full QED.