The Taming of the Group: On the Weak Coupling Limit of U(1) Lattice Model in Fourier Basis

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 the gauge invariant Fourier states along the spatial directions the continuum part of the spectrum in the extreme weak coupling limit is calculated in any dimension.