  The present work continues the program of summing planar Feynman graphs on
the world sheet. Although it is based on the same classical action introduced
in the earlier work, there are important new features: Instead of the path
integral used in the earlier work, the model is quantized using the canonical
algebra and the Hamiltonian picture. The new approach has an important
advantage over the old one: The ultraviolet divergence that plagued the earlier
work is absent. Using a family of projection operators, we are able to give an
exact representation on the world sheet of the planar graphs of both the phi^3
theory, on which most of the previous work was based, and also of the phi^4
theory. We then apply the mean field approximation to determine the structure
of the ground state. In agreement with the earlier work, we find that the
graphs of phi^3 theory form a dense network (condensate) on the world sheet. In
the case of the phi^4 theory, graphs condense for the unphysical (attractive)
sign of the coupling, whereas there is no condensation for the physical
(repulsive) sign.
