I will present our ongoing program at Princeton [1–7], which aims to develop a structure-preserving theoretical framework for modeling the dynamics of charge fields coupled to gauge fields. The focus is on capturing physical phenomena beyond the scope of perturbation theory in the radiation–matter coupling, and on enabling accurate derivation and numerical simulation of the resulting equations, both in the quantum regime and relevant semiclassical limits, using simplicial mesh-based numerical methods. Applications will be presented to some problems in scalar quantum electrodynamics, capturing the dynamics of low-lying collective excitations in superconducting quantum devices, and to the massive Schwinger Model, which describes the relativistic dynamics of charged leptons interacting with an electromagnetic gauge field in 1+1 dimensions. Within this framework, a DEC-inspired approach is used to derive an effective discretized field theory that describes the dynamics of spatially and/or temporally coarse-grained fields within a finite space-time volume, tailored to accurately capture the features resolvable by a physical measurement apparatus.

[1] D. Pham et al., “Flux-Based Three-Dimensional Electrodynamic Modeling Approach to Superconducting Circuits and Materials.” Phys. Rev. A 107, 053704 (2023).[2] K. Sinha et al., “Radiative Properties of an Artificial Atom Coupled to a Josephson Junction Array.” Phys. Rev. A 106, 033714 (2022).[3] T. Maldonado et al., “Negative Electrohydrostatic Pressure Between Superconducting Bodies.” Physical Review B 110.1 (2024).[4] W. Fan et al., “Model Order Reduction for Open Quantum Systems Based on Measurement-Adapted Time-Coarse Graining.” arxiv:2410.23116[5] D. Pham et al., “Spectral Theory for Non-Linear Superconducting Microwave Systems: Extracting Relaxation Rates and Mode Hybridization.”[6] D. Pham et al., “Long-Time Soliton Dynamics via a Coarse-Grained Space-Time Method.", arxiv:2504.12286[7] T. Maldonado et al., “Mesoscopic Theory of the Josephson Junction.” Physical Review B 111, L140505 (2025).