D(x+,p+,p) = e--A(p)
J 271
0(x+) _ix+p /2p
2p+
where we have assumed p+ > 0. Next, discretize "imaginary time" and "momentum"
(p+ = Im, ix = ka with I 1,2,..., M and k = 1,2,..., N) and define To = m/a. The
expression becomes
D(x+,p+,p) (k) kp2/2lT
21mrn
So far the steps may admittedly seem ad-hoc and reminiscent of a cook-book recipe, but
notice that this propagator can be associated with a rectangular grid with width M and
length N. Furthermore, the imaginary time transcription has been shown to be analogous
to the analytic extension of the Schwinger representation to a real exponential [15],
which corresponds to the normal Wick-rotation. We wish to associate the mathematical
expression for the propagator with a path integral of local variables on this grid. To do so,
notice that a propagator always connects two vertices so as long as some final expression
for an amplitude or other Feynman diagram calculation contains all terms and factors,
we are free to redefine the rules for the diagrammatic construction of the expressions. In
this line, we can assign the factor of 1/1 present in the propagator, to one of the vertices it
connects. Since Lightcone parametrization only allows for propagation forward in time, it
is meaningful to assign the factor to the earlier vertex connected by the propagator. This
creates an ..-i-,iin. Iry between fission and fusion vertices which now require independent
treatment. Instead, propagation becomes a simple exponential which was the goal. The
resulting re-prescribed Feynman rules are summarized in Table 2-1.