and therefore a seemingly dramatic slowdown of the Monte Carlo simulation there are
shortcuts that can be used as will be explained.
In this simple case a world sheet configuration consists only of NM spins labelled by
st, where i = 1, 2,. M and j 1, 2,. N label the space and time lattice coordinates
respectively. An up spin could be represented by st = 1 and a down spin by st = 0.
We employ periodic spacial and temporal boundary conditions so that the field, in this
case only st, lives on a torus. In principle the Metropolis algorithm now just proceeds
analogously to the Ising spin system, where each site in the lattice is visited and a spin is
flipped with a corresponding probability to complete a full sweep of the lattice which is
then repeated. Lattice configurations Ck, where k labels the sweeps are generated and then
used to calculate observables of the physical system.
4.2.1 Generating the Lattice Configurations
The basic ingredient for the Metropolis algorithm of generating lattice configurations
is to determine the change in the action under the possible local spin flips that may occur.
When a spin is flipped as to make a new solid line the mass term of the propagator goes
from e6p2/ (Mi+M2) to being e62/,mM, Ca2/mM2 where the M, and I.[. denote the lattice
steps to the nearest up spin to the left and right respectively. Further when a spin flip
results in a solid line being lengthened upwards (downwards) the factor for the fusion
(fission) vertex will be moved upwards (downwards) possibly resulting in a change in
which M1, 3 [_-, to use. Then there is the appearance or disappearance of fission and fusion
vertices as solid lines split or join. These basic ingredients are summarized in Table 4-3.
However the table (4-3) does not tell the whole story because there are a number of
subtleties that need to be addressed. These can be summarized as follows:
1. Particle Interpretation.
2. No Four Vertex.
3. Ergodicity.