compromise this argument and render the method unusable in that case. Fortunately this
case corresponds to small coupling where perturbation theory is a better method anyway.
The above arguments show how we can obtain the energy gaps of the field theory
from the correlations of spins on the Lightcone World Sheet. Even if contributions from
non-cyclically symmetric states may act as noise in our data there is hope that when the
coupling is large, the energy gaps can be read off from the exp behavior of the correlation
as a function of time (j e [1, N]). In Figure (4-6) an example of the correlation obtained
from the system simulation. The figure shows a number of interesting and representative
facts about the methods employed in this work. It shows the fit (solid line) plotted
together with the data points (with error bars). The fit is non-linear and obtained using
specialized methods which we developed for this purpose, in order to capture the specifics
of equation (4-20). The points on the graph are not really the correlation between spins,
but rather, the value of R(j) = (sl ,), with (i', j') (1,500) a fixed up-spin. The
correlation is equal to R (s) (sf,), which, since (i',j') is fixed, is just equal to R less
the average of spins on the lattice. This average has no lattice dependence (no (i,j)
dependance), which is why we work with R rather than the correlation itself. From the
graph we can see that even though correlations are longer ranged in the weak coupling
(that is, long lines predominant) the overall average of spins is much lower, there is much
less happening in the low coupling regime, which is what we could see qualitatively from
the world sheet figures (4-5).
Before data fits such as the one above could be tried, we had to relax the system as
explained earlier. When we relax the system we are essentially finding a state in which
to start the simulation off in, i.e., a state which is representative for the states near the
action minima. Of course, we cannot ever rigorously prove that we are in a truly relaxed
state of the system, but in practice we achieve relaxation by observing system variables,
preferably true physical observables such as the correlations or the total magnetization
(the average value of the spins) of the system. We observe these variables starting off from