Open string operator quantization

Transcription

1 Open strng operator quantzaton Requred readng: Zwebach -4 Suggested readng: Polchnsk 3 Green, Schwarz, & Wtten 3 upto eq 33 The lght-cone strng as a feld theory: Today we wll dscuss the quantzaton of an open strng; we wll just be consderng frst quantzaton, because we know the acton for a sngle strng Incdentally, t sn t known n general how to carry out second quantzaton for strngs, and t s a pretty techncal research subject However, to do frst quantzaton of strngs, we have to do second quantzaton on the worldsheet, snce the lght-cone acton s a feld theory! For now, let s work wth all Neumann boundary condtons The lght-cone gauge acton for an open strng s S = π 4πα dτ dσ 4α p + τ X + τ X σ X Recall that X s not really a degree of freedom, snce t s fxed by the constrants, and t won t enter nto the Hamltonan due to the form of ts acton Each of the remanng transverse coordnates X has the acton S = π 4πα dτ dσ α X α X Ths s just the acton for a massless feld n two dmensons, so we can just combne the results we have for scalar feld theory wth what we know about the mode expanson of X For an open strng, we had mode expanson X τ, σ = x + α α τ + α n n α n cosnσe nτ et s compare to the smlar expanson for a scalar feld n dmensons φx = p Ep ap e E pt e px + a pe E pt e px To compare, we should really dentfy t = τ, x = σ, = π, and φ = X/ πα

2 Apparently, there s a slght dfference due to the Neumann boundary condtons on the strng vs the perodc bc for the feld Ths sn t consequental Also, because the feld X for the strng s massless from the worldsheet pont of vew, the zero momentum part looks dfferent It s pretty clear, though, that α n s an annhlaton operator on the worldsheet for n > Also, α n = α n for n > We can set α n = na n For a massless φ and nonzero momentum, we requre p = n for n Z That means the energy s n We can then relate the commutators an, a ] ] m = δn,m αn, α j m = nδ n,m δ j n > For the strng, though, n can be any nteger To understand the zero-momentum part, we can t just use the feld theory analogy However, f we set the zero momentum pece to q τ, we fnd that the acton for q s just S q = 4α dτ q, whch s just that for a free partcle Snce the Hesenberg equaton says that q should be lnear n tme, we fnd q τ = x + α p τ, x, p j ] = δ j, ncludng the standard commutator We also fnd α = α p You can also start wth the equal-tme commutator between each X and ts canoncal momentum P Xτ, σ, Pτ, σ ] = δσ σ to work out the oscllator commutators The text descrbes ths approach Hamltonan: et s move on to dscuss the Hamltonan From the acton above, we can wrte the Hamltonan as π H = dσ πα P P + 4πα σx σ X, P = πα τ X

3 For the zero-modes, the Hamltonan just comes out to be α p p, the center of mass momentum-squared for the strng n the transverse drectons From our results for scalar felds, then, we fnd H = α p p + na na n + C = α p p + α nα n + C We ve combned all the zero pont energes nto one term et s remember that we defned the X oscllators n terms of Vrasoro modes now Vrasoro operators: α αn = p + n = p + αn p α p In partcular, = α α + α nα n + αnα n p = α α + α nα n + α n, α n] = α p p + α nα n + D n The last term s an nfnte-lookng zero-pont term As long as we dentfy ths constant wth the constant C, we have H = n= From the relaton α = α p, we have H = = α p + p If we remember from the lght-cone gauge pont partcle, ths s approprate for a Hamltonan that satsfes the Schrödnger equaton τ = H The mass spectrum s now modfed from the classcal formula! We have here m = p + p p p = α α nα n + C Due to the normalzaton of the α n, each exctaton at level n adds n/α to the mass-squared Note that the zero-pont energy has real meanng here, as t contrbutes to the mass of the strng state! That fact s related to the fact that the worldsheet theory ncludes gravty before we gauge-fx 3

4 Vrasoro operator algebra: Snce t s farly mportant f you progress n strng theory, let s brefly dscuss the commutator algebra of the Vrasoro operators Frst note that the conjugate operators are n = n We can make use of the commutator relaton AB, C] = AB, C]+A, C]B to fnd m, αn] = nα m+n, m, x ] = α αm Iteratng, we fnd m, n ] = p= p nα m+n p αp pαn pα m+p Iff n = m, we have nontrval commutators; n the end we fnd m, ] n = m n m+n + A m δ m, n wth A m = D p= p D m p= pp m = D + D m 3 m Gettng ths last requres fgurng out whch terms have nonzero commutators and dstrbutng the sums a dcey proposton wth dvergent sums, but t works ok here D s formally nfnte, and we ll dscuss t below et s look at the acton of the Vrasoro operators on the strng coordnates Wth some algebra, we fnd m, X τ, σ ] = α n= e m nτ σ + e m nτ+σ α n = e mτ cosmσ τ X + e mτ snmσ σ X Ths second lne shows that m generates a change of coordnates Zeta functons: τ, σ τ, σ + ɛe mτ cosmσ, snmσ Here we wll use the magc of analytcal contnuaton, and you wll justfy the formulae below on the homework Frst note that the Remann zeta functon s defned as ζs = n s, n Fortunately, ζs has a unque analytc contnuaton for all complex s, so we can evaluate what we want 4

5 The frst sum we need corresponds to ζ = / odd that a sum of postve ntegers s a negatve fracton, sn t t? We fnd that the normal orderng constant n = H s C = D 4 The other nfnte sum we have s ζ = That means that D = n A m 5