The transfer-matrix of the U(1) lattice gauge theory is investigated in the field Fourier space, the basis of which consists of the quantized currents on lattice links. In the strong coupling limit, a series expansion is developed for the elements of the transfer matrix, to which a diagrammatic representation based on the occurrence of virtual link and loop currents can be associated. Either by interpretation or through their role in fixing the relevant terms, the diagrams are reminiscent of the Feynman ones of the perturbative small coupling expansions. 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. 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->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.
