Weak Coupling Limit of U(1) Lattice Model in Fourier Basis

TL;DR

This paper analyzes the U(1) lattice model's transfer-matrix in the Fourier basis, revealing the nature of gauge invariant states and spectrum behavior in the weak coupling limit across various dimensions.

Contribution

It provides an analytical derivation of the spectrum at weak coupling for any lattice size and dimension, clarifying the role of link-currents and gauge invariance.

Findings

Gauge invariant states transition from closed loop currents to link-currents along periodic directions in weak coupling.

Spectrum at weak coupling matches continuum model predictions in the large lattice limit.

Proper handling of zero eigenvalues allows for accurate extraction of diverging group volume.

Abstract

The transfer-matrix of the U(1) lattice model is considered in the Fourier basis and in the weak coupling limit. The issues of Gauss law constraint and gauge invariant states are addressed in the Fourier basis. In particular, it is shown that in the strong coupling limit the gauge invariant Fourier states are effectively the finite size closed loop currents. In the weak coupling limit, however, the link-currents along periodic or infinite spatial directions find comparable roles as gauge invariant states. The subtleties related to the extreme weak coupling of the transfer-matrix in the Fourier basis are discussed. A careful analysis of the zero eigenvalues of the matrix in the quadratic action leads to a safe extraction of the diverging group volume in the limit $g\to 0$. By means of the very basic notions and tools of the lattice model, the spectrum at the weak coupling limit for any dimension and size of lattice is obtained analytically. The spectrum at the weak coupling limit corresponds to the expected one by the continuum model in the large lattice limit.