# A microscopic semiclassical confining field equation for $U(1)$ lattice gauge theory in 2+1 dimensions

## TL;DR

This paper introduces a semiclassical nonlinear field equation derived from the microscopic Hamiltonian of 2+1D $U(1)$ lattice gauge theory, capturing magnetic monopole effects that lead to electric field confinement.

## Contribution

It presents a novel semiclassical field equation explicitly derived from the microscopic Hamiltonian, modeling magnetic monopole-induced confinement in 2+1D $U(1)$ lattice gauge theory.

## Key findings

- Numerical solutions show the equation models confinement via magnetic monopoles.
- The equation can be interpreted as a London relation in a dual superconductor.
- Demonstrates the nonlinear dynamics leading to electric field confinement.

## Abstract

We present a semiclassical nonlinear field equation for the confining field in 2+1--dimensional $U(1)$ lattice gauge theory (compact QED). The equation is derived directly from the underlying microscopic quantum Hamiltonian by means of truncation. Its nonlinearities express the dynamic creation of magnetic monopole currents leading to the confinement of the electric field between two static electric charges. We solve the equation numerically and show that it can be interpreted as a London relation in a dual superconductor.