U(1) Lattice Gauge Theory

U(1) gauge theory is a model of the electromagnetic force that incorporates relativistic and quantum effects. Simulations are performed using U(1) gauge theory to study the photon field and the interactions between photons and charged particles. Starting from Maxwell’s equations and the relativistic Lagrangians for electron fields and photon fields, forcing a local invariance on the Lagrangians provides a gauge symmetry in Maxwell’s equations. A gauge symmetry leaves Maxwell’s equations invariant under specific transformations to the potential four-vector that underlies Maxwell’s equations. Simulations of the photon field can be achieved through a discretization of space and time on a lattice. The lattice contains sites, which represent positions in space and time, and links that connect the sites. The links carry information about the potential four-vector and the discretization of the electromagnetic field strength tensor. The propagation of the photon field through the lattice relies on Metropolis updates that select field states from a distribution given by the photon field Lagrangian. Measurements can be made on the simulated lattice to derive relationships between charged particles such as the Coulomb potential. The critical “temperature” at which charged particles become decoupled within the simulation can also be found by analyzing the data produced in the simulations. The computational implementation of U(1) lattice gauge theory shares many concepts in common with the implementation of lattice quantum chromodynamics, the study of quarks and gluons on the lattice