DBI Analysis of Open String Bound States on Non-compact D-branes

CHAPTER 2. BOSONIC STRINGS 28This algebra can be derived by using the commutation relations for the expansion modesexpressed in Eqs. 2.49 - 2.51. Quantum mechanically, things get a bit messier because<strong>of</strong> the earlier noted ordering ambiguity, and the algebra becomes[L m , L n ] = (m − n)L m+n + c12 m( m 2 − 1 ) δ m+n . (2.96)The derivation <strong>of</strong> this result is somewhat tricky, and can be found in full glory in e.g.the Appendix to Chapter 3, p. 54-55 in [14], or via an alternative approach in [9], p.80-81.As is usual in quantum theories, we will order our operators using the normalorderingconvention <strong>of</strong> “lowering operators to the right and raising operators to theleft,” resulting inL m = 1 2+∞∑n=−∞: α m−n · α n : . (2.97)We should be careful though. Since we find ourselves in the beloved and charmingpresence <strong>of</strong> gravity (on the world sheet), we can no longer ignore the extra constant thatresults from the normal ordering prescription as we did in the case <strong>of</strong> QFT. However,since all operators commute except those constituting L 0 , this ordering ambiguity isonly relevant for L 0 , and we take this into account by replacing L 0 by (L 0 − a) wheneverneeded, where a represents the as-<strong>of</strong>-yet undetermined ordering constant. 6This constant also has repercussions for the mass <strong>of</strong> states. We demand that for aphysical state |φ〉L m |φ〉 = ˜L m |φ〉 = 0 m > 0, (2.98))(L 0 − a) |φ〉 =(˜L0 − a |φ〉 = 0. (2.99)The second condition is simply the mass-shell condition. The first condition can be understoodas the quantum translation <strong>of</strong> the classical vanishing <strong>of</strong> the energy-momentumtensor T αβ . Quantum mechanically, our first guess would be to impose thatT αβ |φ〉 = 0, (2.100)however this would imply that all operators annihilate physical states. This would leaveus a little embarassed, as little physics would survive. The next best thing we can do isdemand that〈φ|T αβ |φ〉 = 0. (2.101)Remembering that in the mean time we normal-ordered our Virasoro generators, we seethat this time around the lowering operators can destroy |φ〉, while the raising operatorscan destroy 〈φ|, which leaves a fully consistent spectrum <strong>of</strong> physical states. This isanalogous to the statement expressed in Eq. 2.98.6 Basically, we are simply rewriting L 0 = `12P: α −n · α n : +a´ as L 0 − a = 1 2P: α −n · α n :.