Table 2-1: Lightcone Feynman rules. The arrows denote the flow of "time" ix+.
9 1
87T3/2To MI+M2
1 2
8\3/2TO MIM2
1 2
S -kp2/21To
The local world sheet variables alluded to above, are now within reach. Write the
total momentum as a difference p = qM q and define:
S = Sg + Sq
M-1
Sq 20- (qI l q)2 (2-3)
j i=0
M-2
S, -T I c + M b+E I bj) (1 I c() (2-4)
f 1 i= 1_
Then
ex1{ /2 / (- 1 7 77
S-(qM- 2 )2 1 H I dq e-Sg-Sq (25)
SN j=1 i= 1
where bk, c are a pair of Grassmann fields for each point (i, k) on the grid (or lattice).
We implement qkM qM and q k q0 for all k by putting in J-functions. Proof of the
above identity and discussion of useful intermediate results are found in the original paper
[5], but neither is necessary to appreciate the fact that we have here a world sheet local
representation of the free field theory. To put it less dramatically, we have spread out the
very simple exponential momentum propagation over the width of a world sheet. We can
even write the above mentioned 6-functions as discretized path integrals of an exponential
emphasizing the interpretation of (2-5) as a path integral over an N x M discretized world
sheet grid. Even though the construction is a bit cumbersome considering that we still
only have the free field theory, the important point to notice is that here is a completely