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

CHAPTER 2. BOSONIC STRINGS 34<strong>Open</strong> string spectrumAs always, we construct the spectrum <strong>of</strong> the theory by acting on the ground state withthe raising operators. Before doing this, we remind ourselves that the groundstate hadM 2 equal toα ′ M 2 |0〉 = −a|0〉 = −1|0〉. (2.127)Hence, we see that our groundstate is in fact a tachyon. This, to say the least, is badnews, but there is not much we can do about it. 9Next, we turn to the first excited state, generated by acting once upon the groundstatewith a raising operator,α i −1|0〉, (2.128)and as we saw earlier, this should be a massless particle. Since there are 24 raisingoperators, we find ourselves with 24 massless states.Going one step further, we find several ways to create the next state, namelyα−2|0〉 i or α−1α i j −1 |0〉. (2.129)Both ways give rise to a massive particle with α ′ M 2 = 1. The first possibility clearlyleads to 24 different states. Recalling that the “non-zero mode” operators commute, thesecond one amounts to24 2 − 242+ 24 =24 (24 − 1) + 482=24 · 252= 300 states, (2.130)so in total, we find ourselves with 324 states with α ′ M 2 = 1.To construct the rest <strong>of</strong> the spectrum, which is infinite, one just has to carry onthis game as long as one wishes. This is however a nice time to introduce a nice littletool that allows one to easily compute the number <strong>of</strong> states at each and every masslevel, which responds to the name <strong>of</strong> generating function. As an appetizer, consider theidentity1+∞∑1 − x = 1 + x + x2 + x 3 + · · · = x n . (2.131)n=0In this case, we see that expanding 11−xgenerates all the terms 1 + x + . . .. We can putthis knowledge to our advantage by considering how we construct open string states.The ground state is unique. Then we act on the ground state with raising operators, andwe have 24 sets <strong>of</strong> these. For the first excited level, we have 24 choices correspondingto the 24 α−1 i operators. Going one step further, we also need to consider the αi −2operators, even one step further also α−3 i and so on. But if we consider all the modesseparately for a second, and considering only one direction, say i = 1, we see that theα−1 1 operator will give rise to one level one state, one level two state ((α1 −1 )2 ), one levelthree state, etc. Assuming x represents a state, and that the exponent <strong>of</strong> x corresponds9 An interpretation for this has been found as the decay <strong>of</strong> D25-branes into closed strings, but we willcome back to this in due time.