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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.

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All these (infinitesimal) transformations act locally on the world sheet; the strings are extended but physics is (and symmetry transformations and compensations needed to restore a gauge-fixing condition are) still local on the world sheet when interpreted properly. The transformation of individual fields may be computed as the commutators (or supercommutators, if we include supersymmetry generators) of the generators of the symmetries with the field operators.

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.

All the generators may be rewritten in terms of the stringy oscillators (and the zero modes of the coordinates).

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Lubos' answer doesn't give the whole story. His answer is right for free strings, but the OP specifically asked about the interacting case. First, let me point out the Poincare group only acts on-shell in the light cone gauge after eliminating a few auxiliary worldsheet fields and gauge fixing. With interactions, the light cone Hamiltonian picks up interaction terms changing the number of strings. To close the algebra on-shell, interaction terms also have to be added to the boost generators. This is analogous to SUSY where we have to add interaction terms to the SUSY generators in component notation after getting rid of the auxiliary fields because the Hamiltonian generator of time translations also has interaction terms. A boost will change the number of strings. It has to.

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