Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Lorentz transformation in light cone coordinates in string theory

+ 4 like - 0 dislike
1100 views

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.

thanks a million

This post has been migrated from (A51.SE)
asked Nov 4, 2011 in Theoretical Physics by kuchi (20 points) [ no revision ]

2 Answers

+ 3 like - 0 dislike

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).

This post has been migrated from (A51.SE)
answered Nov 4, 2011 by Luboš Motl (10,278 points) [ no revision ]
+ 2 like - 0 dislike

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.

This post has been migrated from (A51.SE)
answered Nov 14, 2011 by kompre (20 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOv$\varnothing$rflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...