Lectures on String Theory

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Finally, we can write the level-matching condition in terms of transversal oscillators.
We find
V = 1 ∑ (
)
ᾱ i
2T
nᾱ−n i − αnα i −n
i = 0 . (3.80)
n≠0
Thus, the level-matching condition tells that the left- and right-moving oscillators
contribute the same amount of energy.
3.4.2 Poisson structure of the light-cone theory
Using the Poisson brackets of physical fields we can now establish the Poisson relations
between all quantities of interest. We first summarize the basic Poisson relations
for the closed string case
{, } p + p − p j x j x − αm j αm
−
p + 0 0 0 0 1 0 0
p − 0 0 0 − pi
− p− 2πiT
mα j 2πiT
p + p + p + m mα − p + m
p i 0 0 0 − δ ij 0 0 0
x i 0
x − − 1
p i
δ ij δ
0 0 ij δ m
p + 4πT
p −
0 0 0 0
p +
α i n 0 − 2πiT
p + nα i n 0 − δij δ n
αn − 0 − 2πiT nα − p + n 0 − αi n
p +
4πT
0 − inδ ij δ n+m − i √ 4πT
− α− n
p +
α i m
p +
α − m
p +
nα i p + n+m
i √ 4πT
p + mα i n+m {α − n , α − m}
Tab. 1. Poisson brackets of the light-cone modes. The variable p − is
essentially the Hamiltonian: p − = 2πT H . The brackets involving ᾱ
p +
variables are the same.
These relations are easy to derive. For instance,
{p − , x − } = {2πT H p + , x− } = −2πT H 1
(p + ) 2 {p+ , x − } = −2πT H
(p + ) 2 = −p− p + .
Also, one has to remember that the variable α − n contains the zero mode α i 0
α − n =
√
πT
p +
2αi 0α i n + . . . = pi α i n
p + + . . .
and, therefore,
{x i , α − n } = 1
p + {xi , p j α j n} = 1
p + αi n .
The most complicated bracket is {α − m, α − n }.