Date: 2024-10-19 | Article ID: 51376 |

This article introduces one of the simplest examples of a quantum model with a gauge field, treating D-dimensional space as a lattice so that the math is straightforward. The model is a special case of compact quantum electrodynamics (**compact QED**), namely the case with no electrically charged matter, so the quantum electromagnetic field is the only physical entity.

The adjective *compact* in the name refers to the fact that the model uses the compact group U(1) as its gauged group, in contrast to traditional electrodynamics in which the gauged group is the noncompact group R. The choice U(1) is motivated by the fact that the electric charges of all known elementary particles appear to be precisely integer multiples of a single elementary unit of charge. The model constructed here does not include charged matter, but it uses U(1) as the gauged group to prepare for models that do.

**2024-10-19**(simplified a paragraph about continuum limits when space is two-dimensional and cited a new article for more detail)**2024-06-23**(added a footnote about normal operators and observables, replaced "gauge invariant" with a more specific term as a reminder that we only require invariance under gauge transformations that act trivially on the boundary, modified the hamiltonian to include plaquettes that touch the boundary when the lattice is truncated, and updated the wording in several sections to support that modification)**2024-06-01**(replaced a confused paragraph about two different continuum limits in two-dimensional space and added a reference that compares those different continuum limits)**2024-05-19**(added a reference)**2024-03-24**(corrected a typo)**2024-03-07**(fixed statements involving Stokes's theorem so that they remain valid when space is not contractible)**2024-03-04**(first version)

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