1/M and couplings for vertices, are then multiplied with appropriate combinations of six's
so that they are present where ',., v should be. For example, at endpoints of solid lines we
have s = -sj+1 so a factor of (1 + s')( + s )(1 s-)/8 is 1 at the beginning of a solid
line but 0 otherwise, whereas the factor (1 s+)(1 + st)(1 + s~- )/8 is 1 at the end of a
solid line but 0 otherwise. We then put an overall sum over all s' configurations in front of
the whole path integral and the resulting expression then represents the sum of all planar
Feynman diagrams:
/N M-1 d7 d d 7
Tf, N d exTp -S e+ bi [v' + vP']}
SexP{2-q 4 })
s 1 j 1 i 1 i -
exp iY (q qi Pi + ( -- PbP ) In 2 bPI[P-1[P j bc
I i3
+ (bic -b _) (1 P ,i_ I
where P7 = (1 + s')/2 and P (P) is a combination of s's which is 1 at beginnings (ends)
of solid lines that are at least 2 time steps long, and 0 otherwise.
The Ising-like spin system sj is clearly introduced simply to manipulate the presence
or absence of terms which have to do with boundaries of the world sheet. The spin system
is completely new and has no counterpart in the original field theory. In a way it is
precisely what makes the strong coupling regime reachable by this world sheet approach
as compared with Feynman diagram perturbation theory or lattice quantum field theory.
The various schemes to tackle the world sheet, by Monte Carlo simulation as will be done
here, or by introducing a mean-field [11, 14] or the anti-ferromagnetic-like configuration
considered by Thorn and Tran [8], all involve a particular choice for the treatment of the
spin system, and of course, the treatment of renormalization. What happens to the spin