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Hamiltonian formulation of string field theory
Volume 195, number 4
PHYSICS LETTERS B
17 September 1987
H A M I L T O N I A N F O R M U L A T I O N O F S T R I N G F I E L D T H E O R Y ~r George SIOPSIS CaBforma Instttute of Technology, Pasadena, CA 91125, USA Receaved 15 May 1987
Watten's stnng field theory asquantazedm the hamdtoman formalism. The constraints are solved and the hamdtoman asexpressed m terms of only physaealdegrees of freedom. Thus, no Faddeev-Popov ghosts are introduced. Instead, the action contains terms of arbatrarlly hagh order m the string funchonals. Agreement wath the standard results is demonstrated by an exphclt calculation of the resadues of the first few poles of the four-tacbyon tree amphtude
Quantization o f string field theory has recently been achieved [ 1,2 ] in the case o f open bosonic strings [ 3 ]. The method employed is analogous to gauge theories where F a d d e e v - P o p o v ghosts are introduced when the gauge is fixed. The procedure was carried out in the lagrangian formalism. The drawback o f this approach is that it does not ensure unitarity o f the theory. To this end, a hamiltonian formalism is needed. Progress in this direction was made by the identification o f a symplectic structure in string phase space [ 4 ]. A subsequent study o f Poincar6 invariance made possible the construction o f a classical hamiltonian [ 5 ]. Here, we perform the quantization o f string field theory for open bosonic strings, using the hamiltonian formalism. Starting from Witten's lagrangian [ 3 ], 5eIA l, where A is a string functional, we define m o m e n t a P and thus we construct the hamiltonian as a function o f coordinates A and m o m e n t a P. The transformation from the lagrangian to the hamiltonian picture is singular in the same manner as in gauge theories. Thus, constraints emerge, analogous to Gauss' law. We then quantize this system by solving the constraints and integrating over the unphysical degrees o f freedom. It should be noted that the symplectic structure that emerges by our identification o f m o m e n t a differs from the one adopted by Witten [4], because o f a different choice o f time. Also, the resulting hamiltonian is the same as in ref. [ 6 ] in the case where interactions are ignored. An essential difference with the lagrangian approach is that the introduction o f F a d d e e v - P o p o v ghosts is not necessary. We start with a list o f definitions to fix our notation. We expand the coordinates x~'(a), the ghost c ( a ) and the anti-ghost b ( a ) (H=0, 1..... 25, a s [0, n]) in modes a~, c, and bn, respectively, satisfying (anti-)commutation relations ,u
~,
--
Jan, a m ] - m O m + , , o ~
¢tp
,
{bm, Cn}=C~m+n.0,
(la,b)
with all other (anti-)commutators vanishing, a~ is related to the center-of-mass m o m e n t u m by a~ = v/2p u. Thus, if qU is the position of the center o f mass, then [qU, p~]=iO ~. The v a c u u m is defined as an eigenstate of ag: a~ IP> =,~/~P~[P>- Modes with positive (negative) indices are creation (annihilation) operators. It is convenient to separate the zero modes o f the ghost (co) and anti-ghost (bo) from the rest of the string modes. They act on a two-dimensional space spanned by the kets I + > and I - ), where bo] + > = [ - >, Col - > = [ + >, bo I - > = Co] + > = 0. We also define bras < + I and < - [ that are annihilated by negahve-index modes. An innerproduct is defined by < + f - ) = ( - J + ) = 1, ( + I + ) = < - I - ) = 0. Thus, a general state IA > can be written as IA> = IA+ > + IA_ >, where IA+ > ( IA_ > ) is constructed from the I + > ( I - ) ) vacuum. We can also define ¢r Work supported in part by the US Department of Energy under contract DEAC 03-81-ER40050. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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Volume 195, number 4
PHYSICS LETTERS B
17 September 1987
a functional A [ z ( a ) ] , where z = (x u, b, c), by A [ z ( a ) ] = ( z l A ) . A Z2 grading is imposed on the string fields by assigning a n u m b e r ( - ) IAIto each functional A, where GA = ( IA I - ½)A, G being the ghost-number operator:
G=~:c_.b~:-½.
(2)
Integration is defined by f A= ( I I A ) , where the "identity" state I I ) is [7]
[I)=b+b_exp
~ ~ ( - ) ~ , ,~~- . a ~- n ]'~ exp n=l
n
~ (
-
)
_.c_~
I+)
(3)
n=l
where b ± = Z ( + i) "b.. We also define a multiplication operation by IA ~rB) 1= 1(A Iz (BI 1/3) i23, where the three-string vertex is given by I V3 ) 123=
d26pld26p2d26p3~(Pl+ P 2 + P 3 )
exp
~
~ -rs nC_n b_mNm,
r,s= 1 m,n=O
×exp ~ ~ k
r , s = l rr/, rt = 0
)
aU';.N~.aS% [Pl, + ) l i P 2 , "q-)21P3, + ) 3 -
(4)
The explicit form of the N e u m a n n coefficients Ngn and N,,,. -'~ can be found in ref. [7]. It is useful to define a two-string vertex by [ V2) 1z = 3(11 V3 ) 123. Explicitly, I V2)12 =
d26p,dZ6pz~(Plq-p2)exp ool ( -
x (Ipl, + ) 1 1 p 2 , - ) 2 +
[pl,-),lpz,
)"
/ exp,,,=,~ ( - , ,
+)2).
.........
j (5)
Is also follows that Y, A~B = 1(A12 ( B I V 2)12. Because o f the special form of the v a c u u m in I V2)12 (eq. (5)), we have Z A+-kB+ = ZA_'kB_ = 0 . The action is S = f ~, where the lagrangian density LZis defined by [3]
5g = ½A~QA + ½A~rA~A .
(6)
This is different from the usual density in space-time; it is defined an the space of string modes. To obtain the m o m e n t a that are conjugate to the "coordinates" A[z(cr)], we define the center-of-mass time-coordinate q ° = f ~ d a x ° ( a ) as the time of the system. Then the m o m e n t a are given by
P=6£e/8 ( a°oA) .
(7)
Although [113)~23 (eq. (3)) contains arbitrarily high orders of the time derivative, ao°, the interaction term m the action, S,,t--- ½fA~A'kA, does not contribute to the momenta, P. This can be seen by writing Smt in terms of functionals:
Smt = f DZl DzzDz3 V3[zl (o-), z2(o), z3( a) ]A[ zl ( ff ) ]A [z2( ~) ]A [ z3( ¢7)] ,
(8)
where V3 [ zl, z2, z3 ] = i ( z l 2 ( Z [ 3 ( Z [ V 3 ) 123 represents an interaction potential that has absorbed all possible derivative terms ~1. It is convenient to expand the BRS charge Q in the zero modes bo, Co, and the time derivative a °. We obtain
Q=coaOaO+ ZKaO+co~+boT+ + ~ .
(9)
~ Notice that, wath our chome of time, V3[za,z2, z3] ~snon-local m both space and t~me. For locahty, we have to choose the coordinates of the mad-point of the string as the space-ume coordinates [ 4]. However, for explicit calculatxons at is necessary that we express the vertex (eq. (4)) an terms of a more convenient basis We hope to report on progress m this dlrectmn shortly. Henceforth, we shall ignore the problem on non-loeahty. 542
Volume 195, number 4
PHYSICS LETTERSB
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The operators K, ,~, T+, and Q (anti-)commute with each other by virtue of the nilpotency of Q in 26 dimensions. Also, {K, K} = ½T+, and Q2+AT+ =0. Another important property of these operators is their "hermitlcity", i.e., f A * K B = - (-)IAIfKA*B, and sxmilarly for zi and Q. They follow from the properties of Q as a derivation,
Q(A*B) = QA*B + ( - )l a lA,QB '
(10)
and fQA=O. A straightforward computation gives 6 5° 6Lf P_-=bo 6(aOA )=a°A_+ 2boKA+ , P+ -Co g(aOA+~=0.
(lla,b)
The factors bo and Cohave been introduced for convenience. In deriving these two equations we have discarded surface terms. We have also made use of the "hermiticity" of the operators K, Z and Q. Thus, the phase space is endowed with a symplectic structure described by the two-form
Ja(Y~o)~A*~P=j a(Zo)rA_*(coa°rA_ + 2KrA+ ),
co_=
(12)
where ~o is the hypersurface t = 0. ~(~o) is a fi-functxon which vanishes unless the center of mass of the string lies on the surface Z0. Notice that this is different from the one adopted by Witten [4]. The difference lies in the choice of time. co is obtained by choosing the center-of-mass time coordinate as the time of the system, whereas Witten's choice corresponds to time being the time coordinate of the mid-point of the string, i.e., x°(zc/2 ). Since P_ vanishes, it follows that A+ is similar to A8 in Yang-Mills theories. It plays the role of a Lagrange multiplier, as can be seen by constructing the hamlltonian density
~f_=P+*coa°A_ - ~ = ~ V f o + A + * F ,
(13)
where
~o = ~P+ coP+ + ½A_ ~Co,~A_ - ½A ~A ~A_
(14)
I'_= 2KP_ + QA_ -boco(A_*A_ ) .
(15)
The operator boco acts as a projection onto the space generated from the [ - ) vacuum. To derive the above equation, we used the fact that X - A + * A + = 0. To prove this, we have to make use of the explicit form of the three-string vertex (eq. (4)). Since IX)~--2 (A +13 (A +[ V3)123and ( + I + ) = ( - [ - ) = 0, separating the zero modes of the ghosts in the vertex, we obtain IX) I = 2 ( A + 13(A+ [ exp(Nu) exp(Nf)(1 +boC i 1 )~{C 1 2 , C3}1+)11-)21-)3,
(16)
where Nb= ZaU_'LN~naS~, Nx= Zb2mN~nnc~_, and Cr= ZN~%nc~, ( m > 0). Since the c~ anti-commute, we have {C 2, C 3} = 0 and therefore [X) I=0, which shows that only terms linear in A+ can appear in the hamiltonian. In this formulation, there is no sense in setting A+ = 0, because we lose the constraint F_ --0. Thus we see that the gauge A + = 0 (known as the Siegel gauge) is similar to the Ao = 0 gauge in Yang-Mills theories. However, unlike in gauge theories, F_ generates a transformation
6A_=2K~_,
6P_=-Qe_+bo[A_, e_],
(17a,b)
where [A, B] = A - B - ( - ) l a e J B * A , which does not leave the hamiltonian invariant. (The gauge parameter e_ is of course independent of time (a°e_ = 0 ) , and 6A+ =0. Also, ( - ) i , J = _ 1.) We therefore have to impose an additional constraint, P ' _ = {/'_, ~o} = 0, where {A, B} is the Poisson bracket of A and B. We easily obtain /~_' = 2 / ( ~ _ - 2boK( A_ ~rA_ )+ QP_ -boco[ P_ , A_ ],
(18) 543
Volume 195, number4
PHYSICSLETTERSB
17 September1987
generating the transformation
~A_ =O_.E'_+boco[A_, e ' ] ,
~P_ =KAe" +bo[P_, e" ]-2boK[A_, e" ] .
(19a,b)
No further constraints need to be imposed, because
{if'L, ~o} =AF +B(QF_ + 2 K F " ),
(20)
where B=Zfl_nb, and the coefficients fl_, ( n > 0 ) solve the system of equations Y , N ~ n f l _ , = l and --11 2,N,,,nnfl_~ = 0 ( m > 0 ) . These constraints form a closed algebra, because {F_, P ' } = F . To quantaze the theory, we couple the system to an external current J+, by adding a term J+~rA_ to the lagrangian. Thus, the second constraint (eq. (18)) becomes
F ; -if"_ + 2boKJ+ = 0 .
(21)
We define the generating functional by e x p { - W[J+]}=
NA_ NP_ N2+ N2~_ exp
P+~xcoa°A_ - o ~ o + 2 + ~ F _ +2+
_+J+~xA_
.
(22)
To compute Green functions, we have to fix the gauge. Exploiting the invarlance under the transformation generated by F_, we impose the conditzon KA_ = 0. Notice that this implies T+A_ = 4KZA_ = 0, and GA_ = O, where G is the ghost number operator. This gauge (T+A_ = GA_ = 0 ) was introduced by Siegel and Zwiebach [ 6 ], in the case of a free theory. The inclusion of interactions does not change things, because the transformation properties of A_ (eq. (17a)) are not affected by the interactions. To eliminate the second gauge invariance (eq. (19)), we impose the constraint (~A =0. These two constraints are implemented by the Faddeev-Popov procedure. Inserting the two factors
I = A I [ A _ ] jf~ e _ ~ [ K ( A _ + 2 K ~ _ ) ]
,
1 =A2[A_ ] ~ j e ' _ a[(~(A_ +(O_.+boco[A , .])eL)] ,
(23a,b)
into the path-integral (eq. (22)), and performing two gauge transformations, we obtain
' [A_ ]A2 [A _ ]g[KA_lg[O.A_] exp{ - W[J+ ] } = fj NA_ ~P+ N2+ N2+A~ × e x p ( _ f p + 4tcoaOA__~o+2+ ~rF_+2L.kF,_+j+ ~rA ) .
(24)
The Faddeev-Popov determininants are A~= det' K 2 and d2 = det' Q ( Q + boco[ A , • ]), where the prime denotes omission of the zero modes of the operators K and (~. Integration over the Lagrange multipliers, 2+ and 2~_ in eq. (22) produces two 6-functionals that enforce the constraints F_ = 0 and F L = 0, respectively. We can use these two constraints to integrate over the redundant degrees of freedom. Splitting the momentum P_ as P_ = H _ + H ; + H " , where QH_ =KH_ = K H ; = 0, the constraints (eqs. (15) and (21)) become
F _ - 2KH" + boco(A_ ~A_ ) = 0 ,
(25a)
F 2 -(Q+boco[A_, • 1) bo(H_ +H'_ )+boco[H'_, A_ ] - 2 K ( b o ( A _ ~rA_) + J + ) = 0 ,
(25b)
Using eqs. (25a) and (25b), we can express H'_ and/7'_' in terms of H_. Explicitly,
H " = ½K-' boco(A_ *A_ ) + ( Q + boco[A_," 1) - 1bo([H_, A_ ]+ 2K((A_ ~rA_ ) + J + )),
(26a)
/7" = - ½K-~boco(A •A_).
(26b)
Therefore, integration over H'_ and HL' gives rise to two factors, (det' K ) - ~ and (det' ((~+ boco[A_, .]))-1. 544
Volume 195, number 4
PHYSICS LETTERSB
17 September 1987
These two factors, together with the two Faddeev-Popov determinants, give a factor of det' (KQ) which is a constant, and can therefore be absorbed into the overall normalization of the generating functional. Hence, the final form of the generating functional (eq. (24)) is
e x p { - W [ J + ]}= f ~A
~II_ exp(-~H_*coa°A_-~o)
,
where A_ a n d / 7 _ are annihilated by both K and Q. The hamiltonian ~o is a function of A , m o m e n t u m / - / _ and the external current J+:
~o=½1-I-~rcolL +½II"kCo H '- + ~ -~,.o~rt"a-½ _*coAA_ - ~A_'kA_"xA_ +J+"xA_ ,
(27) its conjugate
(28)
where H'_ and H " are given by eqs. (26a) and (26b). This form of the hamiltonian agrees with the minimal form of ref. [ 6 ], if interactions are ignored. To demonstrate this, consider the generating functional for the free theory, exp{ - Wo[J+ ]} =
J A_
To derive this, we used H " = 2boQ- 1KJ+, H 2 = 0, which follow from eqs. (26a) and (26b), respectively, when the interactions are switched off. Integrating over H_ by completing the square in the exponent, we obtain
exp{-Wo[J+]}=f~A_exp(-~lA_~rcoAA_+J+.A_+½J+.boA-'J+),
(30)
where A-a°oa°+A= Z:a~_.a~:- 1 + Zn :c_.bn:. Thus, ½fA_*coAA_ is the free gauge-fixed action, in agreement with the results of ref. [ 6 ]. The free propagator is therefore A- t. The first few eigenstates of A are ]p, - ) (with eigenvalue p 2 - 1), a~_l IP, -- ) (with eigenvalue p2), a~_la~_l [P, - ), a_alP, - ), C_lb_l [P, - ) (with eigenvalue p a + 1 ), etc. Expanding a general state IA_ ) in terms of these states, we obtain
IA_ ) = f d26p[~(p)+Au(p)aU_t + g~,,(p)a~_, a~_, + C~,(p)a~_2+ (J(p)c_l b , +...liP, - ) •
(31 )
The constraint KA_ = 0 implies Ao = Bou = Co = g/= 0. It is also easily seen that the second constraint, QA_ = 0 implies O,A, = OiB,j= 0,C,= 0 (t, j = 1.... , 2 5 ) . Notice that for the massless gauge field Au, the second constraint is the Coulomb gauge. We shall now compute the contribution of the lowest modes of IA_ ) to the four-tachyon scattering amplitude. Our discussion will follow closely the discussion in ref. [2]. Consider on-shell tachyons, [p~, - ) , IPa, - ) , IP3, - ) and [P4, - ), where p~z =pa=p3=P4 2 2 2 = 1. Define the Mandelstam variables s = -(p~ +p2) 2, t = - ( p z + p 3 ) z and u = - ( p l +p3) 2. We shall concentrate on the s-t dual diagram. The amplitude is
S=S_I+So+...,
(32)
where S_ l, So .... are the contributions of an intermedmte virtual tachyon, massless field, etc., respectively. The propagator for the tachyon is just 1
D_~(p,p')= (p, - J A -~ [p', - ) =-~--1-10(p+p').
(33)
It is a little harder to find the propagator D~ ~(p, p ' ) for a massless field. Writing 545
Volume 195, number 4
U~d +2 D~"(p,p')=(p,-la~ ~7-f-W[J+]
PHYSICS LETTERS B
aU_llp', - ) ,
J+
17 September 1987
(34)
=0
after a little algebra we obtain
This is just the p r o p a g a t o r o f a p h o t o n in the C o u l o m b gauge. The other ingredients that are n e e d e d are the t a c h y o n - t a c h y o n - t a c h y o n and t a c h y o n - t a c h y o n - p h o t o n interaction vertices. A straightforward calculation gives V 1(Pl, Pz, P3) = 1 ( P l , - 12(P2, - 13 (P3, - I V3 ) 123=(4/3x/~)P}+P2+P~O(Pl+P2+P3),
(36)
V~(pl, P2, P3) = i ( P l , - 12 (P2, - 13 (P3, - l a~' 3 ] V3 ) 123 =( 4/3x/~)PI +P2+p~+l (1/,,/2 )(p2 -Pl )~6(Pl + P 2 + P 3 ) , where we used N~o = ½1n(16/27 )d rs a n d i nr31 Y 10 ~ obtain
--~v
(37)
~r32 o = 0 . Thus, using eqs. (33) a n d (36), we 10 = ( 4 / 2 7 )1/2, N l33
S_1= V_I(pl,P2,p)D_I(p) g_l(p3,P4,p)oc(16/27)-s-l(s+ 1) - I
(38)
Similarly, eqs. (35) a n d ( 3 7 ) give So=
Vg(pl, Pz, P)DFd"(P)lPo(P3,/)4, p)oc(16/27) - s ( 4 + 2t+s)/2s.
(39)
Therefore, the ratio o f the residues o f the poles at s = 0 a n d s = - 1 is 2 + t. It coincides with the ratio o f the residues o f the corresponding poles in the function B ( - s - 1, - t - 1 ) [ 2 ]. This provides an i n d i c a t i o n that our results agree with the results o f the dual theory at tree level. It would be interesting to extend these calculations by c o m p u t i n g the poles o f a loop diagram. This will allow a better u n d e r s t a n d i n g o f how closed strings arise in an open-string field theory, a n d will resolve questions o f unitarity. W o r k in this direction is in progress. I wish to t h a n k T. Allen, R. R o h m a n d J. Preskill for discussions.
References" [ 1] [2] [3] [4]
M. Bochlcchlo, Rome University preprmt Rome-527 (1986), Princeton Umverslty preprmt PUPT-1028. C.B. Thorn, Princeton Umverslty preplant Print-86-1334. E Wltten, Nucl. Phys. B 268 (1986) 253 E. Watten, Nucl Pbys B 276 (1986) 291, C. Crnkovi6 and E. Wxtten, Princeton University preprlnt Print-86-1309 [ 5] C. Crnkovl6, Princeton Umversxty preprant PUPT-1033. [6] W Siegel and B Zwlebach, Nucl. Phys B 263 (1986) 105 [7] D.J Gross and A. Jevlckl, Nucl. Phys B 283 (1987) 1.
546
PHYSICS LETTERS B
17 September 1987
H A M I L T O N I A N F O R M U L A T I O N O F S T R I N G F I E L D T H E O R Y ~r George SIOPSIS CaBforma Instttute of Technology, Pasadena, CA 91125, USA Receaved 15 May 1987
Watten's stnng field theory asquantazedm the hamdtoman formalism. The constraints are solved and the hamdtoman asexpressed m terms of only physaealdegrees of freedom. Thus, no Faddeev-Popov ghosts are introduced. Instead, the action contains terms of arbatrarlly hagh order m the string funchonals. Agreement wath the standard results is demonstrated by an exphclt calculation of the resadues of the first few poles of the four-tacbyon tree amphtude
Quantization o f string field theory has recently been achieved [ 1,2 ] in the case o f open bosonic strings [ 3 ]. The method employed is analogous to gauge theories where F a d d e e v - P o p o v ghosts are introduced when the gauge is fixed. The procedure was carried out in the lagrangian formalism. The drawback o f this approach is that it does not ensure unitarity o f the theory. To this end, a hamiltonian formalism is needed. Progress in this direction was made by the identification o f a symplectic structure in string phase space [ 4 ]. A subsequent study o f Poincar6 invariance made possible the construction o f a classical hamiltonian [ 5 ]. Here, we perform the quantization o f string field theory for open bosonic strings, using the hamiltonian formalism. Starting from Witten's lagrangian [ 3 ], 5eIA l, where A is a string functional, we define m o m e n t a P and thus we construct the hamiltonian as a function o f coordinates A and m o m e n t a P. The transformation from the lagrangian to the hamiltonian picture is singular in the same manner as in gauge theories. Thus, constraints emerge, analogous to Gauss' law. We then quantize this system by solving the constraints and integrating over the unphysical degrees o f freedom. It should be noted that the symplectic structure that emerges by our identification o f m o m e n t a differs from the one adopted by Witten [4], because o f a different choice o f time. Also, the resulting hamiltonian is the same as in ref. [ 6 ] in the case where interactions are ignored. An essential difference with the lagrangian approach is that the introduction o f F a d d e e v - P o p o v ghosts is not necessary. We start with a list o f definitions to fix our notation. We expand the coordinates x~'(a), the ghost c ( a ) and the anti-ghost b ( a ) (H=0, 1..... 25, a s [0, n]) in modes a~, c, and bn, respectively, satisfying (anti-)commutation relations ,u
~,
--
Jan, a m ] - m O m + , , o ~
¢tp
,
{bm, Cn}=C~m+n.0,
(la,b)
with all other (anti-)commutators vanishing, a~ is related to the center-of-mass m o m e n t u m by a~ = v/2p u. Thus, if qU is the position of the center o f mass, then [qU, p~]=iO ~. The v a c u u m is defined as an eigenstate of ag: a~ IP> =,~/~P~[P>- Modes with positive (negative) indices are creation (annihilation) operators. It is convenient to separate the zero modes o f the ghost (co) and anti-ghost (bo) from the rest of the string modes. They act on a two-dimensional space spanned by the kets I + > and I - ), where bo] + > = [ - >, Col - > = [ + >, bo I - > = Co] + > = 0. We also define bras < + I and < - [ that are annihilated by negahve-index modes. An innerproduct is defined by < + f - ) = ( - J + ) = 1, ( + I + ) = < - I - ) = 0. Thus, a general state IA > can be written as IA> = IA+ > + IA_ >, where IA+ > ( IA_ > ) is constructed from the I + > ( I - ) ) vacuum. We can also define ¢r Work supported in part by the US Department of Energy under contract DEAC 03-81-ER40050. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
541
Volume 195, number 4
PHYSICS LETTERS B
17 September 1987
a functional A [ z ( a ) ] , where z = (x u, b, c), by A [ z ( a ) ] = ( z l A ) . A Z2 grading is imposed on the string fields by assigning a n u m b e r ( - ) IAIto each functional A, where GA = ( IA I - ½)A, G being the ghost-number operator:
G=~:c_.b~:-½.
(2)
Integration is defined by f A= ( I I A ) , where the "identity" state I I ) is [7]
[I)=b+b_exp
~ ~ ( - ) ~ , ,~~- . a ~- n ]'~ exp n=l
n
~ (
-
)
_.c_~
I+)
(3)
n=l
where b ± = Z ( + i) "b.. We also define a multiplication operation by IA ~rB) 1= 1(A Iz (BI 1/3) i23, where the three-string vertex is given by I V3 ) 123=
d26pld26p2d26p3~(Pl+ P 2 + P 3 )
exp
~
~ -rs nC_n b_mNm,
r,s= 1 m,n=O
×exp ~ ~ k
r , s = l rr/, rt = 0
)
aU';.N~.aS% [Pl, + ) l i P 2 , "q-)21P3, + ) 3 -
(4)
The explicit form of the N e u m a n n coefficients Ngn and N,,,. -'~ can be found in ref. [7]. It is useful to define a two-string vertex by [ V2) 1z = 3(11 V3 ) 123. Explicitly, I V2)12 =
d26p,dZ6pz~(Plq-p2)exp ool ( -
x (Ipl, + ) 1 1 p 2 , - ) 2 +
[pl,-),lpz,
)"
/ exp,,,=,~ ( - , ,
+)2).
.........
j (5)
Is also follows that Y, A~B = 1(A12 ( B I V 2)12. Because o f the special form of the v a c u u m in I V2)12 (eq. (5)), we have Z A+-kB+ = ZA_'kB_ = 0 . The action is S = f ~, where the lagrangian density LZis defined by [3]
5g = ½A~QA + ½A~rA~A .
(6)
This is different from the usual density in space-time; it is defined an the space of string modes. To obtain the m o m e n t a that are conjugate to the "coordinates" A[z(cr)], we define the center-of-mass time-coordinate q ° = f ~ d a x ° ( a ) as the time of the system. Then the m o m e n t a are given by
P=6£e/8 ( a°oA) .
(7)
Although [113)~23 (eq. (3)) contains arbitrarily high orders of the time derivative, ao°, the interaction term m the action, S,,t--- ½fA~A'kA, does not contribute to the momenta, P. This can be seen by writing Smt in terms of functionals:
Smt = f DZl DzzDz3 V3[zl (o-), z2(o), z3( a) ]A[ zl ( ff ) ]A [z2( ~) ]A [ z3( ¢7)] ,
(8)
where V3 [ zl, z2, z3 ] = i ( z l 2 ( Z [ 3 ( Z [ V 3 ) 123 represents an interaction potential that has absorbed all possible derivative terms ~1. It is convenient to expand the BRS charge Q in the zero modes bo, Co, and the time derivative a °. We obtain
Q=coaOaO+ ZKaO+co~+boT+ + ~ .
(9)
~ Notice that, wath our chome of time, V3[za,z2, z3] ~snon-local m both space and t~me. For locahty, we have to choose the coordinates of the mad-point of the string as the space-ume coordinates [ 4]. However, for explicit calculatxons at is necessary that we express the vertex (eq. (4)) an terms of a more convenient basis We hope to report on progress m this dlrectmn shortly. Henceforth, we shall ignore the problem on non-loeahty. 542
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The operators K, ,~, T+, and Q (anti-)commute with each other by virtue of the nilpotency of Q in 26 dimensions. Also, {K, K} = ½T+, and Q2+AT+ =0. Another important property of these operators is their "hermitlcity", i.e., f A * K B = - (-)IAIfKA*B, and sxmilarly for zi and Q. They follow from the properties of Q as a derivation,
Q(A*B) = QA*B + ( - )l a lA,QB '
(10)
and fQA=O. A straightforward computation gives 6 5° 6Lf P_-=bo 6(aOA )=a°A_+ 2boKA+ , P+ -Co g(aOA+~=0.
(lla,b)
The factors bo and Cohave been introduced for convenience. In deriving these two equations we have discarded surface terms. We have also made use of the "hermiticity" of the operators K, Z and Q. Thus, the phase space is endowed with a symplectic structure described by the two-form
Ja(Y~o)~A*~P=j a(Zo)rA_*(coa°rA_ + 2KrA+ ),
co_=
(12)
where ~o is the hypersurface t = 0. ~(~o) is a fi-functxon which vanishes unless the center of mass of the string lies on the surface Z0. Notice that this is different from the one adopted by Witten [4]. The difference lies in the choice of time. co is obtained by choosing the center-of-mass time coordinate as the time of the system, whereas Witten's choice corresponds to time being the time coordinate of the mid-point of the string, i.e., x°(zc/2 ). Since P_ vanishes, it follows that A+ is similar to A8 in Yang-Mills theories. It plays the role of a Lagrange multiplier, as can be seen by constructing the hamlltonian density
~f_=P+*coa°A_ - ~ = ~ V f o + A + * F ,
(13)
where
~o = ~P+ coP+ + ½A_ ~Co,~A_ - ½A ~A ~A_
(14)
I'_= 2KP_ + QA_ -boco(A_*A_ ) .
(15)
The operator boco acts as a projection onto the space generated from the [ - ) vacuum. To derive the above equation, we used the fact that X - A + * A + = 0. To prove this, we have to make use of the explicit form of the three-string vertex (eq. (4)). Since IX)~--2 (A +13 (A +[ V3)123and ( + I + ) = ( - [ - ) = 0, separating the zero modes of the ghosts in the vertex, we obtain IX) I = 2 ( A + 13(A+ [ exp(Nu) exp(Nf)(1 +boC i 1 )~{C 1 2 , C3}1+)11-)21-)3,
(16)
where Nb= ZaU_'LN~naS~, Nx= Zb2mN~nnc~_, and Cr= ZN~%nc~, ( m > 0). Since the c~ anti-commute, we have {C 2, C 3} = 0 and therefore [X) I=0, which shows that only terms linear in A+ can appear in the hamiltonian. In this formulation, there is no sense in setting A+ = 0, because we lose the constraint F_ --0. Thus we see that the gauge A + = 0 (known as the Siegel gauge) is similar to the Ao = 0 gauge in Yang-Mills theories. However, unlike in gauge theories, F_ generates a transformation
6A_=2K~_,
6P_=-Qe_+bo[A_, e_],
(17a,b)
where [A, B] = A - B - ( - ) l a e J B * A , which does not leave the hamiltonian invariant. (The gauge parameter e_ is of course independent of time (a°e_ = 0 ) , and 6A+ =0. Also, ( - ) i , J = _ 1.) We therefore have to impose an additional constraint, P ' _ = {/'_, ~o} = 0, where {A, B} is the Poisson bracket of A and B. We easily obtain /~_' = 2 / ( ~ _ - 2boK( A_ ~rA_ )+ QP_ -boco[ P_ , A_ ],
(18) 543
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generating the transformation
~A_ =O_.E'_+boco[A_, e ' ] ,
~P_ =KAe" +bo[P_, e" ]-2boK[A_, e" ] .
(19a,b)
No further constraints need to be imposed, because
{if'L, ~o} =AF +B(QF_ + 2 K F " ),
(20)
where B=Zfl_nb, and the coefficients fl_, ( n > 0 ) solve the system of equations Y , N ~ n f l _ , = l and --11 2,N,,,nnfl_~ = 0 ( m > 0 ) . These constraints form a closed algebra, because {F_, P ' } = F . To quantaze the theory, we couple the system to an external current J+, by adding a term J+~rA_ to the lagrangian. Thus, the second constraint (eq. (18)) becomes
F ; -if"_ + 2boKJ+ = 0 .
(21)
We define the generating functional by e x p { - W[J+]}=
NA_ NP_ N2+ N2~_ exp
P+~xcoa°A_ - o ~ o + 2 + ~ F _ +2+
_+J+~xA_
.
(22)
To compute Green functions, we have to fix the gauge. Exploiting the invarlance under the transformation generated by F_, we impose the conditzon KA_ = 0. Notice that this implies T+A_ = 4KZA_ = 0, and GA_ = O, where G is the ghost number operator. This gauge (T+A_ = GA_ = 0 ) was introduced by Siegel and Zwiebach [ 6 ], in the case of a free theory. The inclusion of interactions does not change things, because the transformation properties of A_ (eq. (17a)) are not affected by the interactions. To eliminate the second gauge invariance (eq. (19)), we impose the constraint (~A =0. These two constraints are implemented by the Faddeev-Popov procedure. Inserting the two factors
I = A I [ A _ ] jf~ e _ ~ [ K ( A _ + 2 K ~ _ ) ]
,
1 =A2[A_ ] ~ j e ' _ a[(~(A_ +(O_.+boco[A , .])eL)] ,
(23a,b)
into the path-integral (eq. (22)), and performing two gauge transformations, we obtain
' [A_ ]A2 [A _ ]g[KA_lg[O.A_] exp{ - W[J+ ] } = fj NA_ ~P+ N2+ N2+A~ × e x p ( _ f p + 4tcoaOA__~o+2+ ~rF_+2L.kF,_+j+ ~rA ) .
(24)
The Faddeev-Popov determininants are A~= det' K 2 and d2 = det' Q ( Q + boco[ A , • ]), where the prime denotes omission of the zero modes of the operators K and (~. Integration over the Lagrange multipliers, 2+ and 2~_ in eq. (22) produces two 6-functionals that enforce the constraints F_ = 0 and F L = 0, respectively. We can use these two constraints to integrate over the redundant degrees of freedom. Splitting the momentum P_ as P_ = H _ + H ; + H " , where QH_ =KH_ = K H ; = 0, the constraints (eqs. (15) and (21)) become
F _ - 2KH" + boco(A_ ~A_ ) = 0 ,
(25a)
F 2 -(Q+boco[A_, • 1) bo(H_ +H'_ )+boco[H'_, A_ ] - 2 K ( b o ( A _ ~rA_) + J + ) = 0 ,
(25b)
Using eqs. (25a) and (25b), we can express H'_ and/7'_' in terms of H_. Explicitly,
H " = ½K-' boco(A_ *A_ ) + ( Q + boco[A_," 1) - 1bo([H_, A_ ]+ 2K((A_ ~rA_ ) + J + )),
(26a)
/7" = - ½K-~boco(A •A_).
(26b)
Therefore, integration over H'_ and HL' gives rise to two factors, (det' K ) - ~ and (det' ((~+ boco[A_, .]))-1. 544
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These two factors, together with the two Faddeev-Popov determinants, give a factor of det' (KQ) which is a constant, and can therefore be absorbed into the overall normalization of the generating functional. Hence, the final form of the generating functional (eq. (24)) is
e x p { - W [ J + ]}= f ~A
~II_ exp(-~H_*coa°A_-~o)
,
where A_ a n d / 7 _ are annihilated by both K and Q. The hamiltonian ~o is a function of A , m o m e n t u m / - / _ and the external current J+:
~o=½1-I-~rcolL +½II"kCo H '- + ~ -~,.o~rt"a-½ _*coAA_ - ~A_'kA_"xA_ +J+"xA_ ,
(27) its conjugate
(28)
where H'_ and H " are given by eqs. (26a) and (26b). This form of the hamiltonian agrees with the minimal form of ref. [ 6 ], if interactions are ignored. To demonstrate this, consider the generating functional for the free theory, exp{ - Wo[J+ ]} =
J A_
To derive this, we used H " = 2boQ- 1KJ+, H 2 = 0, which follow from eqs. (26a) and (26b), respectively, when the interactions are switched off. Integrating over H_ by completing the square in the exponent, we obtain
exp{-Wo[J+]}=f~A_exp(-~lA_~rcoAA_+J+.A_+½J+.boA-'J+),
(30)
where A-a°oa°+A= Z:a~_.a~:- 1 + Zn :c_.bn:. Thus, ½fA_*coAA_ is the free gauge-fixed action, in agreement with the results of ref. [ 6 ]. The free propagator is therefore A- t. The first few eigenstates of A are ]p, - ) (with eigenvalue p 2 - 1), a~_l IP, -- ) (with eigenvalue p2), a~_la~_l [P, - ), a_alP, - ), C_lb_l [P, - ) (with eigenvalue p a + 1 ), etc. Expanding a general state IA_ ) in terms of these states, we obtain
IA_ ) = f d26p[~(p)+Au(p)aU_t + g~,,(p)a~_, a~_, + C~,(p)a~_2+ (J(p)c_l b , +...liP, - ) •
(31 )
The constraint KA_ = 0 implies Ao = Bou = Co = g/= 0. It is also easily seen that the second constraint, QA_ = 0 implies O,A, = OiB,j= 0,C,= 0 (t, j = 1.... , 2 5 ) . Notice that for the massless gauge field Au, the second constraint is the Coulomb gauge. We shall now compute the contribution of the lowest modes of IA_ ) to the four-tachyon scattering amplitude. Our discussion will follow closely the discussion in ref. [2]. Consider on-shell tachyons, [p~, - ) , IPa, - ) , IP3, - ) and [P4, - ), where p~z =pa=p3=P4 2 2 2 = 1. Define the Mandelstam variables s = -(p~ +p2) 2, t = - ( p z + p 3 ) z and u = - ( p l +p3) 2. We shall concentrate on the s-t dual diagram. The amplitude is
S=S_I+So+...,
(32)
where S_ l, So .... are the contributions of an intermedmte virtual tachyon, massless field, etc., respectively. The propagator for the tachyon is just 1
D_~(p,p')= (p, - J A -~ [p', - ) =-~--1-10(p+p').
(33)
It is a little harder to find the propagator D~ ~(p, p ' ) for a massless field. Writing 545
Volume 195, number 4
U~d +2 D~"(p,p')=(p,-la~ ~7-f-W[J+]
PHYSICS LETTERS B
aU_llp', - ) ,
J+
17 September 1987
(34)
=0
after a little algebra we obtain
This is just the p r o p a g a t o r o f a p h o t o n in the C o u l o m b gauge. The other ingredients that are n e e d e d are the t a c h y o n - t a c h y o n - t a c h y o n and t a c h y o n - t a c h y o n - p h o t o n interaction vertices. A straightforward calculation gives V 1(Pl, Pz, P3) = 1 ( P l , - 12(P2, - 13 (P3, - I V3 ) 123=(4/3x/~)P}+P2+P~O(Pl+P2+P3),
(36)
V~(pl, P2, P3) = i ( P l , - 12 (P2, - 13 (P3, - l a~' 3 ] V3 ) 123 =( 4/3x/~)PI +P2+p~+l (1/,,/2 )(p2 -Pl )~6(Pl + P 2 + P 3 ) , where we used N~o = ½1n(16/27 )d rs a n d i nr31 Y 10 ~ obtain
--~v
(37)
~r32 o = 0 . Thus, using eqs. (33) a n d (36), we 10 = ( 4 / 2 7 )1/2, N l33
S_1= V_I(pl,P2,p)D_I(p) g_l(p3,P4,p)oc(16/27)-s-l(s+ 1) - I
(38)
Similarly, eqs. (35) a n d ( 3 7 ) give So=
Vg(pl, Pz, P)DFd"(P)lPo(P3,/)4, p)oc(16/27) - s ( 4 + 2t+s)/2s.
(39)
Therefore, the ratio o f the residues o f the poles at s = 0 a n d s = - 1 is 2 + t. It coincides with the ratio o f the residues o f the corresponding poles in the function B ( - s - 1, - t - 1 ) [ 2 ]. This provides an i n d i c a t i o n that our results agree with the results o f the dual theory at tree level. It would be interesting to extend these calculations by c o m p u t i n g the poles o f a loop diagram. This will allow a better u n d e r s t a n d i n g o f how closed strings arise in an open-string field theory, a n d will resolve questions o f unitarity. W o r k in this direction is in progress. I wish to t h a n k T. Allen, R. R o h m a n d J. Preskill for discussions.
References" [ 1] [2] [3] [4]
M. Bochlcchlo, Rome University preprmt Rome-527 (1986), Princeton Umverslty preprmt PUPT-1028. C.B. Thorn, Princeton Umverslty preplant Print-86-1334. E Wltten, Nucl. Phys. B 268 (1986) 253 E. Watten, Nucl Pbys B 276 (1986) 291, C. Crnkovi6 and E. Wxtten, Princeton University preprlnt Print-86-1309 [ 5] C. Crnkovl6, Princeton Umversxty preprant PUPT-1033. [6] W Siegel and B Zwlebach, Nucl. Phys B 263 (1986) 105 [7] D.J Gross and A. Jevlckl, Nucl. Phys B 283 (1987) 1.
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