Hamiltonian approach toZ(N)lattice gauge theories

We develop a Hamiltonian formalism for $Z\left(N\right)\mathrm{}$ lattice gauge theories. Duality is expressed by algebraic operator relations which are the analog of the interchange of electric and magnetic fields in $D=3\mathrm{}$ space dimensions. In $D=2\mathrm{}$ duality is used to solve the gauge condition. This leads to a generalized Ising Hamiltonian. In $D=3\mathrm{}$ our theory is self-dual. For $N\to \infty$ the theory turns into "periodic QED" in appropriate limits. This leads us to propose the existence of three phases for $N>\mathrm{}{N}_c\simeq 6$. Their physical properties can be classified as electric-confining, nonconfining, and magnetic-confining.