scalar field Q on the world sheet is introduced which takes the value 1 on lines and 0
away from them. The authors then build a Lagrangian density through the Lightcone
World Sheet formalism on a continuous world-sheet with a simple cutoff (see chapter 2 for
a thorough discussion of the formalism). A continuous field Q replaces the Ising like spin
system introducing a cutoff so that the world-sheet boundary conditions only exactly hold
when the cutoff is removed while maintaining it amounts to imposing an infrared cutoff.
The constraints Q = 0 and = 1 are implemented by a Lagrange multiplier 7r(, 7), so
that '],. v end up with two fields 0(j, 7) and 7r(a7,7) on the world-sheet. These two fields
are treated as a background on which the quantum fields live, the authors then compute
the ground state of the quantum system in the presence of the background and then solve
the classical equations of motion for the background fields thereby minimizing the total
energy. In later refinements [14] the mean-fields were taken to be scalar bilinears of the
target space world-sheet fields, which makes the mean-field approximation more clearly
applicable but obscures the interpretation of the mean-field as background, representing
the solid lines. A further extension to the mean-field approach was later published [8].
Here, instead of a uniform field 0(j, r) = on the world-sheet and representing the
"smeared-out" solid lines (condensation), two fields 0(a, 7r) and 0'(a, ,r) are introduced at
alternatingt sites on a discretized world-sheet lattice. These fields, although each is taken
to be homogenous on the world-sheet, allow for inhomogeneity in the distribution of solid
lines. Note in particular that = Q' corresponds to the previous work but = 1 and
' = 0 corresponds to an ordering of solid lines reminiscent of an anti-Ferromagnetic Ising
spin arrangement. Such an arrangement of solid lines yields a so-called fishnet diagram
Not to be confused with the quantum matrix field Q in the original scalar f3
Lagrangian density.
t Not exactly alternating sites is required to reproduce the fishnet diagrams, but rather
spin configurations of the type 1, T, T, 1, T, T, T, I, ....