Friday, January 6, 2017

Lorentz invariance of string theory in the light-cone gauge

On page 261 in his book [1] Zwiebach writes ''There is much at stake in this calculation. It is in fact, one of the most important calculations in string theory. [...] The calculation is long and uses many of our previously derived results''. Then Zwiebach states the result \begin{align} \left[ M^{- I}, M^{- J}\right] = &- \frac{1}{\alpha'\ { p^+ }^2} \sum_{m=1}^{\infty} \left(\alpha^I_{-m} \alpha^J_m - \alpha^J_{-m} \alpha^I_m \right)\nonumber\\ &\times \left\{ m \left[1 - \dfrac{1}{24} (D-2) \right] + \dfrac{1}{m} \left[ \dfrac{1}{24} (D-2) + a \right] \right\}\label{eq:20170103} \end{align} This is the commutator of two Lorentz transformations in the light-cone gauge. The commutator should be zero for string theory to be Lorentz invariant. The calculation of the commutator is indeed very tedious and after scribling too many pages I gave up. I googled for a quick way to obtain \eqref{eq:20170103}. Here are the more interesting results that I found. According to [2], the original paper is [3]. In this paper there is some detail but it is mostly stated that the calculation is tedious and it refers to another paper which I cannot access. In a recent paper [4] there is more information, but [4] is still very long. Also [4] uses a regulator and I think that in principle, depending on its exact use, the use of the regulator could lead to incorrect results. There is more detail in lecture notes by 't Hooft [5], but the bulk of the calculation is still left as an exercise. Anyway, I think Zwiebach is wise to not include the calculation in his book. Even if I finished the calculation of \eqref{eq:20170103}, I would not learn much, so I decided to stop trying to prove \eqref{eq:20170103}.
In the calculation of \eqref{eq:20170103} many terms will cancel and the right hand side of \eqref{eq:20170103} are some terms that survive the cancellations. What I would like to have is some way to calculate \eqref{eq:20170103} in which those cancellations do not ''miraculously'' happen. In general, I think that if one does a calculation and many terms cancel, so that the final result is surprisingly simple, that often means one does not have sufficient understanding about what is going on.
A similar situation occurs in checking that \begin{equation*} [p^{-} , p^I ] = [ p^{-} , p^{+} ] = 0 \end{equation*} In this case the calculation is short. The right hand side is indeed zero, so the quantum string theory is translation invariant. Although in this case the calculation is very short, I am not satisfied that there is no better way to understand this calculation in the light-cone gauge.

References

[1] A First Course in String Theory, Zwiebach, 2009

[2] A footnote in Lectures on String Theory by David Tong

[3] Goddard, Goldstone Rebbi and Thorn “Quantum Dynamics of a Massless Relativistic String”, Nucl. Phys. B56, (1973)

[4] A Note on Angular Momentum Commutators in Light-Cone Formulation of Open Bosonic String Theory, Bering, arXiv:1104.4446v2

[5] Introduction to String Theory, 't Hooft