  Real-time lattice quantum electrodynamics (QED) provides a unique tool for
simulating plasmas in the strong-field regime, where collective plasma scales
are not well-separated from relativistic-quantum scales. As a toy model, we
study scalar QED, which describes self-consistent interactions between charged
bosons and electromagnetic fields. To solve this model on a computer, we first
discretize the scalar-QED action on a lattice, in a way that respects geometric
structures of exterior calculus and U(1)-gauge symmetry. The lattice scalar QED
can then be solved, in the classical-statistics regime, by advancing an
ensemble of statistically equivalent initial conditions in time, using
classical field equations obtained by extremizing the discrete action. To
demonstrate the capability of our numerical scheme, we apply it to two example
problems. The first example is the propagation of linear waves, where we
recover analytic wave dispersion relations using numerical spectrum. The second
example is an intense laser interacting with a 1D plasma slab, where we
demonstrate natural transition from wakefield acceleration to pair production
when the wave amplitude exceeds the Schwinger threshold. Our real-time lattice
scheme is fully explicit and respects local conservation laws, making it
reliable for long-time dynamics. The algorithm is readily parallelized using
domain decomposition, and the ensemble may be computed using quantum
parallelism in the future.
