Electric Field Decay Without Pair Production: Lattice, Bosonization and Novel Worldline Instantons

Electric fields can spontaneously decay via the Schwinger effect, the nucleation of a charged particle-anti particle pair separated by a critical distance \(d\). What happens if the available distance is smaller than \(d\)? Previous work on this question has produced contradictory results. Here, we study the quantum evolution of electric fields when the field points in a compact direction with circumference \(L < d\) using the massive Schwinger model, quantum electrodynamics in one space dimension with massive charged fermions. We uncover a new and previously unknown set of instantons that result in novel physics that disagrees with all previous estimates. In parameter regimes where the field value can be well-defined in the quantum theory, generic initial fields \(E\) are in fact stable and do not decay, while initial values that are quantized in half-integer units of the charge \(E = (k/2) g\) with \(k\in \mathbb Z\) oscillate in time from \(+(k/2) g\) to \(-(k/2) g\), with exponentially small probability of ever taking any other value. We verify our results with four distinct techniques: numerically by measuring the decay directly in Lorentzian time on the lattice, numerically using the spectrum of the Hamiltonian, numerically and semi-analytically using the bosonized description of the Schwinger model, and analytically via our instanton estimate.