Modified $U(1)$ lattice gauge theory: towards realistic lattice QED

We study properties of the compact $4D$ $U(1)$ lattice gauge theory with monopoles {\it removed}. Employing Monte Carlo simulations we calculate correlators of scalar, vector and tensor operators at zero and nonzero momenta $\vec{p}$. We confirm that the theory without monopoles has no phase transition, at least, in the interval $0<\beta \le 2$. There the photon becomes massless and fits the lattice free field theory dispersion relation very well. The energies of the $0^{++}$, $1^{+-}$ and $2^{++}$ states show a rather weak dependence on the coupling in the interval of $\beta$ investigated, and their ratios are practically constant. We show also a further modification of the theory suppressing the negative plaquettes to improve drastically the overlap with the lowest states (at least, for $J=1$).