S = Tr { pa v+! m2& } (2-1)
Here 0 is a NA-by-NA matrix of scalar fields and the derivative is taken element-wise in the
matrix. The propagator is given by
where a, 3, 7, 6 E {1, 2,..., N}. Using the double line notation due to t'Hooft as explained
earlier the Greek indices correspond to the "color" of the lines. As t'Hooft prescribed, we
deal with the colors diagrammatically and since we will be taking the large N, limit, thus
ignoring all but planar diagrams, we suppress the color factors S6 &3 altogether in what
follows. Introducing next Lightcone coordinates defined for a D-dimensional Minkowski
vector xP as
x (X= 0 D-l) /2. (2-2)
There is no transformation of the remaining components, and we distinguish them
instead by Latin indices, or as vector bold-face type. The coordinates are (x+, x-, xk) or
(x+, x-, x) and the Lorentz invariant scalar product becomes x y x= x y x+y- x-y+
. By now choosing x+ to be the quantum evolution operator, or "time", its Hamiltonian
(, ,iii: p- = p2/2p+ becomes the massless on-shell "energy" of a particle. We choose
the variables (x+,p+,p) to represent the Feynman rules and arrive at the following
expression for the propagator:
This is called "mixed representation", i.e., Fourier transforming back the p- variable
but retaining the momentum representation in the other components.