can mean k = 1, 2 but also k = A, V. Here the Lorentz invariant product can be written
XpYj, X^ + XvY^ X+Y- X-Y+.
To clarify the meaning of these Lightcone coordinates we look a little closer at a
simple example. Consider a scalar field theory with Lagrangian density:
haa 1 (
= 02m2 V(0). (1-3)
We wish to study this theory in the canonical formalism, with x+ representing time, i.e.,
the evolution parameter. We write 40 =0+) so that
9,ap = (Vr)2 2 Q a9_ and then
In H _1 (1-4)
where II is the conjugate momentum to 0. We shall sometimes want to work with the
Lightcone Hamiltonian density, written as
P- = (VO)2 + 2 + v(W) (1 -5)
2 2
This procedure goes through analogously for Gauge theory, although the algebra is a
bit more cumbersome, as we shall see in section 2.2.1. Let us consider a pure U(NA) Gauge
Theory with Lagrangian:
1
S= -Tr (GG) (1-6)
where
G/,, ,A, ,A, + i g [A, A,] (1-7)
A Z A T2 (1-8)
and the r-s are the generators of U(NA) (the quantum operator nature of A. resides in the
coefficients A2). Now the index a labels the matrices in the chosen representation of the