The field K is then given a Lightcone quantization on each slice of x+ and x is made
compact leading to the discretization of the .i, i-' i,.! k+ = nP+/K. By writing out the
(now discrete) oscillator mode expansion for KP the authors arrive at the following normal
ordered expression for the Lightcone Hamiltonian:
P- DPK (V xT) (4-41)
2P+
where V and T are given as series in the (discrete) creation and annihilation matrix
operators Aj(n) and Ab (n), and x )= DK/21PDK i. Out of all states the authors
concentrate on the global SU(N) singlets:
1
Tr[At(ni).. At(nB)] |0). (4-42)
The states are defined by ordered partitions of K into B positive integers, modulo
cyclic permutations. Here s denotes the multiplicity of the state with respect to cyclic
permutations of the sequence of n-s into themselves, and Z i nk = K is the total number
of momentum units of the states. These states correspond to oriented, closed strings
with total momentum P+ 2wK/L where L is radius of x--space. For a fixed value
of the cutoff K there are only a finite number of states of the -1 i,.-I and the authors
were able to write out the action of the operator P- from Eq. (4-41) on these states as
a (finite-size) matrix, which was subsequently diagonalized. The authors arrive at the
energy spectrum and show that there is an indication that the separation between levels
gets denser as K is increased. The problem with their results is the very limited size of K
which is tractable using their methods. The size of the matrix being diagonalized grows as
exp K, so even if one would choose an extremely effective diagonalization technique, then
the problem is what is referred to in computer-science as numerically intractable. We shall
see that the method developed here to obtain the energy gap is in fact a computationally
tractable problem at the expense of being stochastic. Even though the results are not as