formalism was put forth in [11]. In their paper, the authors review the Lightcone World
Sheet formalism as a means of realizing the sum over planar graphs by coupling the world
sheet fields to a two dimensional Ising spin system. They then regard the resulting two
dimensional system as a noninteracting string moving in a background described by the
Ising spin system. The mean-field approximation is therefore applied to the spin system
so that a qualitative understanding of the physics of the sums of planar graphs could be
achieved. This mean-field work was done in stages, where refinements and improvements
were constantly being published, and it is of course an ongoing effort since the more
complicated theories, such as QCD and Supersymmetric Gauge-theory, have yet to be
tackled. There are more recent views of this project [12, 13]. We will concentrate on the
early developments for sake of simplicity; the refinements and especially the string theory
implications are beyond the scope of this thesis.
As will be clear after the introduction to the Lightcone World Sheet formalism,
the Ising spin system describes the Feynman diagram topology of each graph. For
now it is sufficient to note that the Ising system represents the lines making up the
smaller rectangles in the center graph of Figure 1-2. Notice that as we go to higher
order perturbative Feynman diagrams, the solid lines become more numerous and in the
asymptotic regime we can imagine a limit where the lines acquire a finite density on the
world sheet. The authors of the mean-field papers [11, 14] refer to this mechanism as the
condensation of boundaries, and depending on the dynamics investigations on whether
string formation occurs through this condensation was considered in some limiting cases.
Further work on string formation and the implication is also found in later articles [12, 13].
To give the reader a taste of the mean-field idea, we will elaborate briefly on the
original attempts [11]. To keep track of the solid lines which in turn keep track of the
splitting of momentum between propagators, i.e., the topology of Feynman diagrams, a