U(1) gauge theory on a spatial lattice: duality, photons, and shadow states

We present Hamiltonian approach to compact and noncompact (pure) U(1) gauge theory Oil a regular cubic spatial lattice in (2 + 1) and (3+1) dimensions. The diagonalization of the kinetic part of the Hamiltonian via Fourier transformation of the wave functionals induces an electromagnetic duality transformation. The dual variables are naturally associated with the dual lattice. The notation we borrow from algebraic topology suggests a straightforward generalization to irregular spatial lattices. We determine the states of the theory in the different representations in the strong- and weak-coupling limits, and compare the vacuum and the coherent states in the weak-coupling limit with the (shadow) states obtained some years ago by Varadarajan and Ashtekar and Lewandowski in an ultraviolet-regularized version of loop-quantized continuum U(1) gauge theory. Possible implications for the formulation of a nonperturbative renormalization group in loopquantized theories and the description of confinement in non-abelian gauge theories are discussed.