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

CHAPTER 2. BOSONIC STRINGS 22which are the usual equal time classical Poisson brackets. This can be translated to theP.B.’s <strong>of</strong> the modes by inserting the expansions found earlier (Eqs. 2.29, 2.30, 2.39 and2.41), and results in{α µ m, α ν n} P.B. = imη µν δ m+n,0 , (2.49){˜α µ m, ˜α ν n} P.B. = imη µν δ m+n,0 , (2.50){α µ n, ˜α ν n} P.B. = 0. (2.51)Of course, the two last equations are only applicable to the case <strong>of</strong> the closed string.The next step would be to replace these classical P.B.’s with quantum mechanicalcommutation relations usig the prescriptionThus we obtain{. . .,...} P.B. −→ i [. . . , . . .] . (2.52)[α µ m, α ν n] = [˜α µ n, ˜α ν n] = mη µν δ m+n,0 ; [α µ m, ˜α ν n] = 0. (2.53)Just as in quantum mechanics, these operators are normalized following the definitiona µ m = 1 √ mα µ m ; a µ†m = 1 √ mα µ −m with m > 0. (2.54)Essentialy, this leaves us with the algebra <strong>of</strong> the harmonic oscillator operators,[ ] [ ]a µ m, a ν†n = ã µ m, ã ν†n = η µν δ m,n with m, n > 0. (2.55)This immediatly alerts us to a serious problem, namely[ ]a 0 m, a 0†m = −1, (2.56)which demonstrates that the timelike mode (µ = 0) is a very naughty boy because itgives rise to negatively normed states, e.g.〈0|a 0 ma 0†m|0〉 = −1,in which we assumed that 〈0|0〉 = 1, and in which |0〉 represents the ground state, i.e.the state that is annihilated by all lowering operators,α µ m|0〉 = 0 ∀ m > 0. (2.57)This, and other negative norm states, are called ghosts, and we will need to find a wayto get rid <strong>of</strong> them if we want our theory to make any sense.