# Lorentz transformation in light cone coordinates in string theory

What is the explicit form of the Lorentz transformation changing the light cone coordinates in the light cone gauge in string theory? The extended nature of the strings complicate matters, especially with interactions turned on. String modes tend to be mixed up.

The Poincaré generators in the light cone gauge are divided to $P^{i}\sim \int d\sigma p^i(\sigma)$, $P^+$ (which is proportional to the length $\sigma_{\rm max}$ of the string in the light cone gauge), $P^-\sim \int d\sigma (\dot x^2+ p^2)$; the latter is the real dynamical generator, related to the world sheet Hamiltonian.
So far, I mentioned the momenta. The Lorentz generators are the rotations $J^{ij}\sim \int d\sigma (x^i p^j - x^j p^i)$, $J^{+i}$, $J^{+-}$, and $J^{i-}$. All of them may be written as particular integrals over $\sigma$; see e.g. Chapter 4, 5, 6, 11 of Green-Schwarz-Witten or similar portions of Polchinski's or other basic string theory books. Sorry, I don't think it makes sense to copy the formulae.
One may verify that the commutators are what they should be; the generators span a copy of the Poincaré algebra. The only truly nontrivial commutator whose calculation is tough is $[J^{i-},J^{j-}]$ which has to vanish because $g^{--}=0$. The calculation of the commutator in general deviates from the classical Poisson bracket computations – by "double commutator" terms – and in order to show that it vanishes, you also need to use the critical dimension, $D-2=24$ or $D-2=8$ for the superstring.