Gauge invariant canonical symplectic algorithms for real-time lattice strong-field quantum electrodynamics

TL;DR

This paper develops high-order, gauge-invariant symplectic algorithms for simulating strong-field quantum electrodynamics and relativistic quantum plasmas, ensuring preservation of geometric structures and stability for accurate long-term simulations.

Contribution

The paper introduces a novel class of high-order geometric algorithms based on discrete exterior calculus for simulating quantized Dirac-Maxwell theory with gauge invariance and structure preservation.

Findings

Algorithms accurately simulate nonlinear Schwinger mechanism.

Numerical schemes preserve gauge symmetry and energy conservation.

Successful real-time simulations of vacuum effects and pair creation.

Abstract

A class of high-order canonical symplectic structure-preserving geometric algorithms are developed for high-quality simulations of the quantized Dirac-Maxwell theory based strong-field quantum electrodynamics (SFQED) and relativistic quantum plasmas (RQP) phenomena. The Lagrangian density of an interacting bispinor-gauge fields theory is constructed in a conjugate real fields form. The canonical symplectic form and canonical equations of this field theory are obtained by the general Hamilton's principle on cotangent bundle. Based on discrete exterior calculus, the gauge field components are discreted to form a cochain complex, and the bispinor components are naturally discreted on a staggered dual lattice as combinations of differential forms. With pull-back and push-forward gauge covariant derivatives, the discrete action is gauge invariant. A well-defined discrete canonical Poisson bracket generates a semi-discrete lattice canonical field theory (LCFT), which admits the canonical symplectic form, unitary property, gauge symmetry and discrete Poincar\'e subgroup. The Hamiltonian splitting method, Cayley transformation and symmetric composition technique are introduced to construct a class of high-order numerical schemes. These schemes involve two degenerate fermion flavors and are locally unconditional stable, which also preserve the geometric structures. Equipped with statistically quantization-equivalent ensemble models of the Dirac vacuum and non-trivial plasma backgrounds, the schemes are expected to have excellent performance in secular simulations of relativistic quantum effects. The algorithms are verified in detail by numerical energy spectra. Real-time LCFT simulations are successfully implemented for the nonlinear Schwinger mechanism induced $e$-$e^+$ pairs creation and vacuum Kerr effect, which open a new door toward high-quality simulations in SFQED and RQP.