Fluctuating string of lattice gauge theory: The heavy-quark potential, the restoration of rotational symmetry, and roughening

String dynamics, the heavy-quark potential, and the restoration of rotational symmetry in lattice gauge theory are studied using Hamiltonian methods. We find that off-axis strings experience unbounded transverse fluctuations at all finite couplings. A fermion formulation of lattice strings is used to do systematic calculations. The heavy-quark potential is seen to have power-law corrections to linear confinement. Rotational symmetry in the string sector of the theory is restored in two stages. There is a finite coupling $g_R$ ("roughening") where the equipotential surfaces of the heavy-quark potential become well-approximated by spheres, and there is the bulk critical point $g=0\mathrm{}$ where short-distance violations of rotational symmetry disappear. A new order parameter, the "kink" mass, is introduced to distinguish the smooth and rough phases of the string sector of any lattice gauge theory. Roughening coupling constants are determined using the kink mass for a variety of theories. These results are compared against more traditional width calculations. In the Hamiltonian formulation of SU(3) lattice gauge theory the roughening point lies inside the weak-coupling region.