Lorentz invariance of lightcone string field theory

Nuclear Physics B306 (1988) 282-304 North-Holland, Amsterdam

LORENTZ INVARIANCE OF LIGHTCONE STRING FIELD THEORY t (I). Bosonic dosed string Sang-Jin SIN

Department of Physics and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720, USA Received 23 July 1987 (Revised 9 November 1987)

We construct the Lorentz generators for bosonic closed lightcone string field theory with cubic interactions, and prove that the Lorentz algebra closes if the spacetime dimension is 26.

1. Introduction Recently much interest has been focused on string field theory. Although people are interested in covariant formulations of interacting string theory, it seems that having a closed string, without which the theory is known to be inconsistent, is not an easy problem in Witten's approach [1]. On the other hand, if the Feynman rules of the theory to be constructed are given by Polyakov's prescription [2], it has been proved [3] that they give the same S-matrix elements given by the lightcone approach [4]. From this and also from the history of superstring theory, we expect that lightcone string field theory will be helpful to formulate and to choose the correct covariant string field theory. In the lightcone frame, we believe that we have the correct interacting string theory. But the Lorentz invariance of the lightcone approach has been proved only at the S-matrix level [M]. In this paper we will prove that the theory is a correct nonlinear realization of the Lorentz symmetry by constructing the Poincar6 generators and showing that the algebra closes if the space time dimension d is 26. Since we are mainly interested in the closed string theory, we shall work out only for this case. Lightcone string field theory in the second quantization formalism was written * This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the US Department of Energy under Contract DE-AC03-76SF00098, and the National Science Foundation under Research Grant No. PHY-8515857. 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Sang-Jin Sin / Lightcone string theo~' (I)

283

down by Kaku and Kikawa [K] and its covariant analogue was given by Hata et al. [5] recently. We shall follow the notation of [M] and [K]. The outline of this paper is as follows. In sect. 2 we review the Lorentz generators for free strings in X += 1- gauge. The lightcone gauge condition changes under the Lorentz transformation, which can be restored by using the conformal invariance of the theory. In sect. 3, we define the three-string vertex and construct the cubic part of the Lorentz generators by calculating the commutators between the free part of the Lorentz generators and the interaction part of the total hamiltonian. In sect. 4, we prove that the Poincar4 algebra closes. The proof of sects. 3 and 4 has three parts according to the order in the coupling constant g. For zeroth order, the work was done by G G R T [6]. The method of proving gl order is the closed string version of Mandelstam's technique [M]. We prove the g2 order by using the fact that different types of lightcone string diagrams for four strings have the same integrand. They differ only by their domains of integration [7]. This method was also used by those seeking the covariant generalization of the lightcone string theory to prove the gauge invariance of their theory [8]. Sect. 5 gives a summary and some comments.

2. Lorentz generators in the free string theory We begin our discussion by fixing our notation. The action of a free string is given by

1

. . . .J~

det(0~X O~X,))

.

(2.1a)

Orthogonality: O, X " OoX~, = O,

(2.2a)

(aoX,)2= 0.

(2.2b)

Lightcone gauge: X + ~ 1-,

(2.3a)

P + = 1/47r.

(2.3b)

With these choices, the action and the momentum become 1

S= -8-~Trf[( o,X')2- ( GXi)-] dod1-, 1

(2.1b)

(2.4)

Sang-Jin Sin / Lightconestring theory (I)

284

Using the lightcone gauge condition (2.3), we can express X- and P - in terms of the transverse variables X i. Since X + is time, P - is the hamiltonian. Due to the gauge choice (2.3), p + = fd¢"~thdoP+(o)= length/4~r. If we define a = 2p +, the parameter length of the string is equal to 2~ra, and all the transverse variables X ~'s depend on p + through the range of their parameters. Since X - and P - are nonlinear in X ~, the Lorentz generators containing them, i.e. M i-, are less trivial than the others. Under the Lorentz transformation 1 i e M i-, X + goes to X++ e X ( To keep the identification of X + as time, we should reparametrize ~"by ~ = r + e X i. This change in r induces the change in o to preserve the orthogonality conditions (2.2) by 6 = o + Y*(o), where y ; = fg d o' a, Xi(o'). These reparametrizations give extra contributions to the change in XJ under the Lorentz transformation M i-, so that a x J = 5:J(o( <

- x J ( a,

= e S u X - + 8oOoXJ + 8 r O , X j = e[6iJx - - X i O T X j - YiOoXJ] .

(2.5)

In this way, the rigid spacetime Lorentz transformation is related to local two dimensional conformal transformation on the world sheet. So the conformal invariance in the quantum level implies the Lorentz invariance in the quantum level and vice versa. Another consequence of the reparametrization is the change of the parameter length by eY;(o = 2rra)= 4"n'ep i, which is exactly the change of 2~ra under the Lorentz transformation. Fortunately we can construct an M i- that has the required properties from the canonical expression M i-= [2"~'~do(xie-- PiX-), ao

by expressing P From (2.2),

(2.6)

and X- in terms of the transverse variables.

X - ( o , r) = Q - ( r ) - f Z ~ a d a 0 ( o ' - o) O , x i o , , X i, a0 1

r

l

2

7)= El(
(2.7)

where ),

Q-=q-+2a-tfdo(oT+rP 1

T~

- -

4~r

O , X i OoX i ,

q

=

1 fo2~radoX (O). 2~ra

(2.8)

Sang-Jin Sin / Lightcone string theory (I)

285

Using these, M i-=

fo2~da(p-xi+ TY;) _piQ ,

(2.9)

where pi= fZ~dopi(o). Since M ~- gives the change in parameter length, a subtle question arises about the number of degrees of freedom; if we let each o define an operator Xi(o, ~) at the fixed time ,r, observers in different Lorentz frames would describe the same string with different degrees of freedom. To avoid this difficulty, we expand X;(o, ~') in Fourier series and take the oscillators as basic dynamical variables.

Xi(°,~') =xi+ 2a-lP iz+i E ±(a.e-i"('+°)/"+a.e-i"('-°)/") • (2.10) n~O n The basic commutation relations are

[X i, p J] = iSiJ, [OLn, Olrn] = [ ~n, ~m] = n~m+n,O, [ q-, a] = - 2i,

others are zero.

(2.11)

We shall take symmetric ordering in (2.6) rather than the normal ordering to make M ; - an integral of a local expression, so that Lorentz invariance is more evident. (We should treat the infinities explicitly, however.) To see that (2.9) indeed generates the expected change in X j, eq. (2.5), it is crucial to notice the following identity. [Q-, x J ( o ) ] = 0.

(2.12)

Once (2.12) is proved, deriving (2.5) is an easy exercise. It is also interesting to notice that Q - = X-(o = 2~ra,~). Eq. (2.12) can be seen from the fact that XJ depends on o and ~- only through o/a and ~'/a. (Look at the mode expansion (2.10).) Using (2.11), we can verify

[ x i ( o , T), PJ(o',

T)] =

iSiJS(o - o'),

so that

[q-, X'(o, r)] = i 2 a - t ( o 0 o X i + rO,X;), 2a-l f do(oT + TP-), xi(o, ~')1 = - i Z a - l ( ° O o Xi + TO,xi) • J

With these results, (2.12) follows immediately. In words, the action of the q- on

286

Sang-Jin Sin / Lightcone string theory (I)

Xi(o, r) cancels that of 2 a - l f d o ( o T + rP-). This fact is very important in sects. 3 and 4 where we calculate the commutator between M t- and the vertex. Next we want to introduce some notation in second quantization. The string field ~/i is a function(al) of X*(o, r) and a. We use the following notation. +1 =

=

(2.13)

4)*. q ' = fo°°da f D X ( o ) Of(X, a ) q ' ( X , a ) . We also impose the commutation relation [K],

[~1, ~2*] =8(1,2) ~-=~(O~I--R2) H ~[Xl(°)-X2(°)] o~
•

(2.14)

With these notations, we can write the lagrangian and hamiltonian for free strings in second quantization language. aeo=

(ia

~ 0 = ~*" Hob,

(2.15)

where (~2

1

2 (2.16)

The Lorentz generators for free strings can be written as Jg~)~ = ~*. M ~ , where

M ~'~= fo2'~"do(X~'P ~ - X ~P~')s"

(2.17)

The subscript s means symmetric ordering. Now using (2.14) we can easily prove that ~ 0 ~ satisfies the algebra of M ~ [K]. When we introduce interactions, only P - changes and other components of X or P do not change. Hence M ij and M i+ are not modified by the interaction. Furthermore M ÷ - = X ( p + q - + q - p + ) does not contain P - either. So the only component that is modified by interaction is ,/g'-. This finishes our discussion of the Lorentz generators for free strings.

Sang-Jin Sin / Lightcone string theory (I)

287

3. Construction o| the Lorentz generators We begin by describing the 3-string vertex for closed strings. Define the joining operation • by X1(Ol)

when 0 ~
(3.1)

when 2'trot I ~< 0" ~< 2 ~ ( o t I + or2) , where 01

-~-

O,

0"2

=

0"

-

-

2~ra 1

.

The open string vertex was given by the connection condition [K]; ( X 3 ( o ) - X 1 • X2(o)l V) = 0,

(3.2a)

with cq + a 2 - a 3 = 0, or equivalently as the overlapping 3 functional V (0) = g ~ ( o / 1 -4- OL2 -- o ~ 3 ) ~ ( X 3 - X 1 ~ : ) / 2 ) ,

(3.2b)

where 3 ( X ) --1-I,=03 2 ~ ( (Xo ) ) . It is worthwhile to emphasize that X[ = X ~ i _- X~, i at the interaction points so that we can denote them by X ] unambiguously. To obtain the vertex for closed strings, we should implement the periodic boundary condition. Actually we demand the invariance under the origin shifting of o. This can be done most easily by applying the projection operator

f dOexp(iOG) ,

G= fa' do O,X ' OoX,'

(3.3)

"0

to the naive vertex V (°) given by (3.2b), so that the three string vertex for closed string is V3,12 =: .~V(,°l)~l~2 ,

or symbolically V = ~ t V O ) . ~ .

(3.4)

Notice that G is the origin shifting operator and also notice that the vertex given by (3.4) is not hermitian by itself. We shall use the notation (Vt)12, 3 = (V3,~2)t. The role of ~ is two-fold. First, it projects the initial or final string states into closed string states which satisfy the periodic boundary condition. Second, it generates the twist for intermediate strings. For example, consider the four-string tree amplitude with external string states [i) = 11,2) and If) = 13,4) •

a , = fd( 2 - ~t)(fl v3*4,6('r2) v5,12 ('q)l i)

= fd(

2-

lh)(3,4]~at~]Va(°)6(~'2)~6~stVs(,%(~'l)~1~211,2)

•

(3.5a)

Sang-Jin Sin / Lightcone string theory (I)

288

Since ~r's project the external states into the closed string states, a 4 =

f d~-d0<3,4;"r21V3~°?6(~2)e-i(ar+~mVs!°½(zl)ll, 2; Tx> ,

(3.5b)

where "r = r2 - ~1, 0 = 06 - 05. Eq. (3.5b) clearly shows that G generates twist for the intermediate string just as H does time evolution. Generally for each intermediate string there is one variable that measures its twist. Hence V generates lightcone string diagrams for closed strings in which intermediate strings can have twist. It is well known that the lightcone diagrams can be mapped into the complex plane (possibly with some disks a n d / o r points taken out) by the map known as Mandelstam mapping. The interaction lagrangian .L~al is "~1 = -- g 0 3 j'" V 3 , 1 2 0 1 0 2 -

h.c.,

(3.6a)

where g is a coupling constant and summation (functional integration) over repeated indices is understood. By introducing the * product (O * 0)3 = V 3 , 1 2 0 1 0 2 , we can write the interaction hamiltonian as ~'1 = g Or" (O * O) + h.c. =: 3~'+ y/-t

(3.6b)

Now we can construct the Lorentz generators for the closed string with a cubic interaction. Recall that ~¢t'~- is the only one that is modified by the interaction. ~/,= ~ 0 + ~ + ~t , jCi-= ~-

+ ( ~t'~- + h.c.).

(3.7)

Since ..¢¢6- was given in the previous section, only J[~- is to be constructed. We require

0= [ ~ ' - , ~ ] = [ ~ - , ~ 0 ]

+ ( [ ~ - , ~'1 + [~i-, ~0] } -h.c. + {[~-,~]

+ [~-,~q

} - h.c.

(3.8)

where we used the following identities;

[At, B l =

- [A, B t ] t ,

[At, Bt] = -[A, BI*.

(3.9)

We start with I =-" [..¢t'6-, ze']. By repeated use of [O1, O2~ ] = ~(1,2), [01, 02] = 0, I= [Ot.Mi-O, or. (0.0)1

=OtMi-.(O*O)-Ot.(Mi-O*O+O*Mi-O).

(3.10)

289

Sang-Jin Sin / Lightcone string theory (I)

We can evaluate (3.10) by a technique similar to that used by Mandelstam [M]. The result of the calculation is [ dPt . M i - dP, dPt . ( dP * dP) ] = [ dPt . H dP, dPt . X ] ( Cb * eb ) ] ,

if d = 26, (3.11)

where XJ = X i ( O l ) and H = f o 2 " ~ d o P - ( a ) . This result suggests that we define (3.12)

j [ ~ - = : ~ t . X] ( ~ * ~ )

to get +

e0] = 0,

which is precisely the O(g 1) part of (3.8). The fact that ( 3 . 1 2 ) is the correct definition of ,,l¢{- so that j g i -

is given by

J [ , - = ~ t . M ' - q~ + ¢bt " X ] ( q~ * dp ) + ( ~ , ep ) t • X ] eb

(3.13)

is the main result of this paper.

The rest of this section will be devoted to the proof of (3.11). As the first step, we take the matrix element of I between ] i ) = [1,2,) and If) = 13), (31 [Jg~-, f ] l l , 2 ) =

(31M'- V3,x2- V 3 , 1 2 ( M [ - + M d - ) I 1 , 2 ) .

(3.14a)

Equivalently, (1,2, 3 IMd-+ M [ - + M d - I V ) .

(3.14b)

We remark that there are differences in sign conventions between (3.14a) and (3.14b). In (3.14a) all the p+ momenta are positive while in (3.14b) the outgoing string has negative p +. Furthermore, in (3.14b), o runs from the bottom to the top of the string diagram for incoming strings and the other way around for the out going strings, while in (3.14a) o runs from the bottom to the top in both cases. Now recall that Mi-=

f2*ado(P-Xi

+ Tyi) _piQ-.

Jo So the object we should calculate is

[fc

(P-Xi

+ TYi) -P~Q{

= C3 - Ct - Cz

where the contours Cr's are shown in fig. 1.

- P 2 iQ 2- _ p ~ Q 3 ] l V )

(3.15a)

290

Sang-Jin Sin / Lightcone string theory (I)

l

2

C2 1

1

:t, C1 !

C3

3

Fig. 1. Contours in (3.15). In all string diagrams in this paper, identifications at the boundaries are assumed since we are considering the closed string.

_i+'~- do not contribute to (3.15a). This can As the second step, we prove that ~3r=llJr~,:~r be shown by proving that the vertex is annihilated by Q-. We first scale both a and o with the same small factor I + e (this does not change any thing since X's depend on a and o only through o/a). Then we expand the vertex function(al) by a Taylor series in e and equate the result with the original expression. Vanishing of the first order says that the X dependent part of the vertex is killed by each QT. The 8 ( a . . . ) part of the vertex is annihilated by Ep~Q7 due to the pi momentum conservation. This completes our second step. From now on to the end of this section, we refer to M i- without its last term p'Q-. What we have to calculate is reduced to

fc(P-Xi + TY+)IV).

(3.15b)

Notice that (3.15b) is not a well defined quantity, since it contains a product of many operators defined at the same point. Our prescription of regularization is "point-splitting" in the time direction of the string diagram followed by replacing the integrand of (3.15b) with its time ordered product. That is

f J { J,-x, +

}1v>,

(3.15c)

(1,2,31fc,~( P- g} }lV).

(3.16)

where J means time ordered product. Similarly, we express the matrix element of

xj(e, e)]

between 11,2) and 13) as

Sang-Jin Sin / Lightconestring theory (I)

291

Let A be the difference between (3.15c) and (3.16). That is,

z~=: fc,Y - { P - ( X ~ ( o ) - X ] ) +

T(Y~(o) - IT/))+ fc,Y-(TY/}.

(3.17)

Notice that the last term in (3.17) acting on the vertex vanishes since the vertex is invariant under the origin shifting. Finally we prove that A vanishes when the spacetime dimension is 26. We do a Wick rotation r ~ ~ = ir to evaluate the integral using Cauchy's theorem. Define a complex variable O --- ~ + io, and ~ = ~ - io. Then X + Ygoes to X+ iY which we call 2 X. (Let's suppress the Lorentz indices unless it is ambiguous.) X is independent of ~ and X= x + ~. With these preparations, we can express the Wick-rotated A as a sum of integral of an analytic function of p and ~ so that (3.17) becomes 1 2~'i fc dp T{(

OoX)2(X - X,)} + h.c.

(3.18)

The singularity due to the interaction can be singled out most easily by mapping the string diagram to z-plane, by

p(z)=aaln(z-1)+aalnz

(z3= ~).

The contour C is mapped into three disjoint closed curve in z-plane. (See fig. 2a.) Since the integrand is meromorphic, we can deform the contour into a small circle around the interaction point z I. Call this circle C z. (See fig. 2; b, c.) On C z,

p(Z)--OI=~OI(Z--ZI)2nL l b ( z - - z I ) 3 q - I c ( z - - z I ) 4 - I

[s]

-

"''.

(3.19)

[el

Fig. 2. (a) Contours of fig. 1 mapped into z-plane. Dotted line is the image of the line r = "rI (interaction time) in the string diagram; (b), (c) Deformation of the contour.

Sang-Jin Sin / Lightconestring theory (I)

292

Expressing (3.18) in terms of z, we get 1

1

2~rif~czdz(dPl \dz] T{(OzX)2(X-XI)}+h'c"

(3.20)

Let's call this integral F. Now we expand the time ordered product by Wick's theorem using the two-point function ( x i ( z ) x J ( w ) ) = - 8iJln(z - w). Then,

V=~dz

(d )l -~z

(d )l

;(Ozx)2(x-xI):-~-~ dZ -~z

(3.21)

(OzX'Ozx)(x-xI)

The first term gives zero due to the (X - X~) factor. The second and third terms are regularized by "point-splitting" in the p-plane [M]. That is, do/-1

(-~z] (SzX'OzX)~[~z(Zl)]

-1

(OzX(Zx)'O~X(Z))'

(3.23a)

( OzX" X(z)> ~ (OzX(Z) " X(Z1)>,

(3.23b)

p(zl)= p(z)+ e, e

where z 1 is defined by is small compared with [ z - ziI. One might think that (3.23b) is not a unique choice because we could also regularize it by dp/-1

-1

It turns out that they give the same answer. Hence there is no ambiguity. It is a straightforward calculation to show

[do

] -1 (z,) (Z)

10'"(z)] {o'(z) }

1 [1 (P"(z)) 2 (zl- z) 2 (0'(z)) 3 (Z -- ZI~---- -- 2 ---~2 (p'(z))

6 (p,(z)) 2 +

+

-e + O(e) .

--~

+ o(e)

, (3.24a)

(3.25a)

Notice that terms in { ) do not contribute to the contour integral. They either have no poles or vanish in e ~ 0 limit. To evaluate the second integral in (3.22), we need

293

Sang-Jin Sin / Lightcone string theory (I)

up to (z - zi) -2 in (3.24), which is 1

1

b

4a ( z - zl) 3

1

(3.24b)

12a2 ( z - zi) 2"

To evaluate the third integral in (3.22) we need all multiple pole in (3.25a), which is just 1

1

2a ( z - z i ) 2"

(3.25b)

We also need [~z [dP(z)]-I ]

1

1

1 a (z-zi) 2

(z-zi)

b 1 a 2 (z-zi)

(3.26)

"

Using (3.24), (3.25), (3.29) and (3.21), 1 1 b ] second term in (3.22) = 2¢ri. ( - 1 ) ( d - 2) -~. ~ a OzOzXI - ------70zxI 12a ' (3.27)

[

1

b

third term in (3.22) = 2-2~ri. ( - 1) ( - ½ - 1) a OzOzXl + ~-70zXI

] •

(3.28)

Finally adding (3.27) and (3.28) we get the result of (3.22) F=2~ri-(-1)

8

)1

3 aOzOzXit-~-~

2--

t ]

1----~ OzXI "

(3.29)

Both terms in the bracket of (3.29) vanish at d = 26. This finishes the proof of (3.11).

4. Proof of the Poincar~ algebra

In this section we will prove that the j ~ i - constructed in the previous section satisfies the Poincar6 algebra commutation relations so that the theory is a good representation of the Poincar6 symmetry. We begin with the Lorentz algebra. Here again, we need to work out only [M/i-, , g J - ] = 0. [ Jr'i-, ¢¢t' j - ] = [Jr%-, ~g~-] + { [..¢t'~-, jglJ- ] - [ JC0J-,~'xi- ] } - h.c. + ([~g~-, ..¢t'¢-] + [ ~ ¢ ~ _ , ( . . g ~ _ ) t ] } - h . c .

(4.1)

294

Sang-Jin Sin / Lightconestring theory (I)

Notice that terms symmetric in i and j are cancelled automatically. We will work out (4.1) order by order in g. The proof of the zeroth order was done by G G R T [6]. For O(gl), we need to do similar calculations to those done in sect. 3. We just write down the result here, mentioning the differences between this case and the previous one. As in sect. 3,

[.~iO- ' ...~j-] = gift. M i - X [ ( t i . tip) _ titxJ. ( M i - t i , (71_)-'k ti * M i - ti).

(4.2)

By taking the matrix element, this can be shown to be

( 1 , 2 , 3 1 f J - { P-X~X[ )I V ) ,

(4.3)

which is symmetric in i and j (notice that only one interaction point is involved in O(gl)); therefore [..¢t'6-, .A((-] and [Jr'6-, a/t'{-] in the first brace of (4.1) cancel against one another. The ground state's fluctuation vanishes in 26 dimension as before. Hence O(g l) of (4.1) vanishes. Next we prove (4.1) in g2 order, i.e. [ Jt'~-, J / ( - + (..¢/(-) t ] - h.c. = O.

(4.4)

We start with the first term in the second brace of (4.1), I11 =: [,A[~ , ..//J- ]

= [ti*-xi(ti,ti),

ti*. x¢(ti, ti)].

(4.5)

After some algebra we get II 1 = t i t . X ; X i [ ( t i ,

t i ) , ,ti] + tit. X;X{[ti *'(ti * ti)]

- tit • X~XI [(ti *'ti) * ti ] - tit • X~X/[ ti * (ti * ' t i ) ] ,

(4.6)

where *' means the vertex associated with XI, and * with X[. The readers should notice that XI and X[ are defined at different values of o. If we take the matrix element of II between I i ) = 11,2,3) and I f ) = 14), each term in (4.6) gives three kinds of diagrams A, B, C which are Az (=.- $x - r2) ~ 0 case of fig. 3. The string diagrams were calculated explicitly by Cremmer and Gervais for open string [7]. Their result is that for the given values of a 1, a 2, a3, ot4, various types of string diagrams are integrals of the same integrand with different domains. Different string diagrams correspond to different values of Az. Recently, Hata et al. [8] performed the calculation for closed strings and confirmed the result. From this fact we get the following identity. (ti • t i ) . ti = ti • ( t i . t i ) ,

(4.7)

Sang-Jin Sin / Lightcone string theory (1)

295

1

3 2

.i

1

J"

-.i ' J

4

2

A-1

I

1 4

4

j. 3

B-1

C-1

AT i. i

j~

B-2

C-2

- - i A-2

1

_ _ :

-J J

4

2

J

3

" I.

2

"

A-3

--il - - j

B-4

4

1

:I

3

B-3

'

C-3

~ i l

~ i

- - j

~ j

C-4

.'j

I

A-4

Fig. 3. Contents of (4.6). The limit AT ~ 0 should be taken. The indices i, j at the interaction points indicate that X~, X{ are sitting there.

Sang-Jin Sin / Lightcone string theory (I)

296 3

3 "

2

•

1

""

4

Ar.-.~0 ~

2 1

Fig. 4. Associativity of the * product.

which in diagrammatic language implies fig. 4. The existence of the extra factors X1, X[ does not change the result. From these considerations we see that the diagrams A-l, B-l, C-1 cancel A-4, B4, C-4 respectively. And A-2, B-2, C-2 cancel A-3, B-3, C-3 respectively. This proves [~¢/~-, ~¢~-] = 0 and its hermitian conjugate. The calculation of 112 =." [¢¢(-, ( j / ~ - ) t ] is a bit more involved. I I 2 = [ ~ t • XJ((~* ~), ( ~ * ~ ) t • X~@]

= __( (~ ,t~)+ X~X[( (~ * (~) .4- (~;. X [ [ ( (~* (~)3, ( (~ * '(~))~'] X~(~3', = -1121 + 1122.

(4.8)

Here again we use *' to indicate the vertex associated with X[, and * with X{. The second term of (4.8), which we denoted by 1122, is I122 = ~ t3I z J12,3 t V3',1'2'[~'1'~2~1,2 i [ ~ r]~ R ' -~ (~)11~)2'32,I' q- 1~)~,1~)231,1, q_ ~11~1'32,2 '] ~3' (4.9a) =

+

+ ~3~2,V~2,3V3,,12,~2~3, t t Jt i + ~3~1,V~2,3V3, t ~" J'~ i 1,2~1~3, ,

(4.9b)

where V i =, VX[. We normal ordered the last two terms in (4.9a) to avoid the tadpole-type terms. Now, to see the contents of (4.8) and (4.9), we take the matrix element of them between 1 i ) = [1,2), I f ) = [3,4). (The indices 1,2,3 etc. are dummy, so they have nothing to do with the string states [i) = [1,2), [f) = [3,4).) We assume p f > p~ > O, p~ > p~ > 0 with p f + p~= p~ + p~. In terms of the string diagram, (3, 4[(1st term in (4.8))[1, 2) generates diagrams denoted by A 1, A 2 in fig. 5 with multiplicity 2. After some consideration, it is easy to see that the first, second, third, and fourth terms in (4.9) generate B1, C1, C2, and B2 in fig. 5, respectively with multiplicity 2. Moreover A 1 and A 2 are related by the twist angle ~r. The same relations are true for BI, B2 and for C1,C 2. So it is enough to consider A1,Bt, C 1 only.

297

Sang-Jin Sin / Lightcone string theory (I)

2

:i--

1

I

~ i

"

AI

] m A2

j

j i

i

Bt

C!

j__i [

iwi

Cz

Bz

Fig. 5. Contents of (4.8) (4.9)• II21 = A 1 + A2, II22 = B1 + B2 + C 1 + C2. Only the 0 = 0 case is drawn.

N o w we again use the fact that different types of lightcone diagrams have the same integrand with different integration region. The relevant diagrams in our case are AT ~ 0 configurations. Looking at (4.8) we expect that any diagram in II21 can find its partner in II22 to cancel each other. Actually we claim that a part of A1 cancels B 1 and C 1, and that the remaining part of A 1 is cancelled by its hermitian conjugate. T o see this, we transform the lightcone diagram into the Koba-Nielsen plane b y p = p(z) = alln(z-

1) + a a l n ( z -

x ) + a41nz

(4 .lo)

a n d draw fig. 6 which shows how string diagrams are parametrized b y x as we vary twist angle 0 with AT fixed*. T h e solid contours in fig. 6 correspond to AT = 0 trajectories in x-plane. (Each value of x represents a string diagram.) They are boundaries of the region A, B, C; D i a g r a m s of type A 1 correspond to the sum of three boundaries (AB), (AC), and ( A A ) where the b o u n d a r y (AA) consists of two lines joining x+ and x above and below the real axis. (x + are the two zeros of the discriminant of d p / d z = 0. See the

* This figure appears in the work of Hata et al. [8].

298

Sang-Jin Sin / Lightcone string theory (I)

( .-:" O=0 o-t,-.""

B

";

/

Fig. 6. Zlr, 0 versus x. x is defined by P = 0(z) = aaln(z - 1) + a31n(z - x) + a41n z(a3, 4 < 0); x o corresponds to Al(0=0 ), CI(0= 0); xo corresponds to Ba(0 = 0), A2(0=0); x_ corresponds to C2(0 = 0), Al(_+0_); x+ corresponds to B2(0 = 0), Aa(_+0+).

appendix.) Let's call the diagrams corresponding to these two lines A t _ , A t +; i.e. A t + = {At(0)10_-<<0~< 0 + } , A I _ = {AI(0)I2cr - 0+~< 0 ~< 21r - 0_ }.

(4.11)

O n the other hand, B x corresponds to (AB), C a to (AC) boundary. We express what we have observed by A 1- Bt + C x +At++

At_.

(4.12)

Since A 1 contribute to (4.6) with opposite sign relative to B1, C 1, we have proved half of our claim; a part of A t cancels B t and C a. We shall show that A t + + A t_ is hermitian, thus completing the proof of our claim; hence the p r o o f of [ J C J - , J t '~-] = 0. L o o k at

(4.13) which says that taking the hermitian conjugate is equivalent to interchanging i and j together with changing the sign of the 0. Fig. 7 shows that A t _ ( - 0 ) and A t + ( 0 ) are related b y interchanging the interaction points if 0 is between 0_ and 0+; i.e. [AI+(0)] t = At_ (-0)

= At_(2~r- 0),

which implies that A I + + A 1_ is hermitian. Q E D

Sang-Jin Sin / Lightcone string theory (I)

299

Fig. 7. AI+ and A 1 - are related by interchanging the vertices. The figure demonstrates this by constructing A l + ( 8 ) and A 1 (8) from the B = 0 configuration.

To complete the Poincar6 algebra, we need to prove [ ~ , ~ i ]= 0 in O(g2). (Recall that O(g 1) was proved in the last section.) The proof is nearly the same as the proof of the Lorentz algebra in O(g 2) which was performed in this section. They differ just by the extra factor X{, hence there is no essential difference. So we have finished the proof of the Poincar6 algebra. In the appendix of this paper, we will discuss the difference between the open and closed string theory.

5. Summary We have proved that the generators constructed in Sect. 3 satisfies the Poincar6 algebra so that the lightcone string field theory respects special relativity. The generators of the Lorentz algebra are as follows. ~ij = ~'. MiJ~,

M ij = f02~ra d a ( x i p j _ X J p i ) ,

...~i+ = ¢ * . Mi+cTp

m i + = _ _ f2"~do(1 _ ,rS,)X i, 4,r "o

1

'

.///+- = ~ , . M + - ~ , . ~ i - - = ¢'~. M ' - ¢

M+-= l(p+q-+

q-p+),

+ g e t . X~(~ * q~) + g ( ~ * ff))t. X ~ ,

300

Sang-Jin Sin / Lightcone string theory (I)

where

M'-= fo2='~do(P-Xi + TY')

_piQ-

,

P-: =~I { ( 8~X)2 + ( 8~X)2 } 1

r,(o)

= fo°dO'2'(o').

The critical dimension is the same as for the free string case. There are two kinds of infinities which need much care: one arises from the operator product's short distance behavior which is basically the quantum fluctuation due to the uncertainty principle; the other arises from singular geometry of the interaction. The regularization of these infinities and the imposition of the Lorentz symmetries are the origin of the critical dimension. Our method of the proof is quite general and geometric. It can be extended to the NSR superstring in the supersheet formalism [10]. I would like to thank Prof. Stanely Mandelstam for suggesting this problem and guidance throughout this work. I would also like to thank Nathan Berkovits and Dae-Sung Hwang for useful discussions, and Hidenori Sonoda for discussion about various aspects of the string theory.

Note added to the proof After this work was finished, the author noticed that some other authors considered the same subject in different approaches [11,12].

Appendix In this appendix we want to point out where our proof of the Poincar6 algebra breaks down for the open string case. First we recall that string diagrams can be mapped into a complex plane by the mapping p = ~2arln(z- Zr), with E a r = 0. Using the projective invariance, we fix Z 1 = 1, Z 2 = ~ , Z 3 = x, Z 4 = 0; here x is real for open strings, and complex for closed strings. Then p = a l l n ( z - 1) + a 3 1 n ( z - x ) + a41n z.

(A.1)

As is well known, the single variable x covers all possible lightcone diagrams, and the open string diagrams corresponding to x_ ~
Sang-Jin Sin / Lightcone string theory (I)

301

Now look at fig. 5, regarded as open string diagrams. (Hence the identifications at the boundaries are not assumed.) The diagrams AI(A2) cancel just Ba(C1) and diagrams B2 and C 2 are singular by themselves (we will show this later). To avoid these singularities, we should look at small but nonzero AT (regularization). Nonzero A~" means that two interaction points are separated so that B2 does not become C 2 when we interchange the two vertices. Since interchanging the vertices and taking the hermitian conjugate is equivalent, this in turn means that B2 + C 2 is not hermitian. In short, the singularity breaks down the self-adjointness of B2 + C 2, which is essential for the proof to work. The difference between B2 + C 2 and its hermitian conjugate is expected to be cancelled by the term involving the 4-string vertex. For this point, relevant calculations indicating the affirmative result were performed by Itoh et al. [9] when they proved the nilpotency of the BRS generators in the covariant approach. Once we recognized that we should worry about these singularities, we must ask why they do not cause any problem in our proof of the closed string algebra. The essence of the answer is that integration over the twist angle 0 gives a finite quantity even though there are singular diagrams for some values of 0, as will be shown below. We calculate the singular behavior of the string diagrams using Mandelstam's mapping. To clarify the singularities we consider the four string scattering amplitude since the relevant quantities are limitting cases of them. At the interaction points, p is stationary. do

dz

0/1 -

-

0/3

-

z- 1

+

z- x

0/4 +

--

z

f~(z) =-"

z ( z - 1 ) ( z - x)

.

(A.2)

Then

L ( z ) = (0/1 + 0/~ + 0 / . ) ( z - z+(x))(~ - z_(~)), (0/~ + 0 / . ) x + 0/3 + 0/4 + ~ ( - x ) Z_+(X) --

2(0/1+0/3+0/4)

a(~)

= ( , ~ + 0/4):(~ -- x + ) ( x

-- ~ _ ) ,

-- 0/10/3-- 0/20/4 -IX_+=

(0/1 + 0/4)2

(A.3) (A.4) (A.5) (A.6)

Let's consider the closed string first. The relation between the lightcone variables and x is given by

(to=~A~=Re[p+(x)-p_(x)],

wherep+(x)=p(z+_(x))

( I m [ p + ( x ) - p _ ( x ) ] + const)/aint~rm~diate.

(A.7)

Sang-Jin Sin / Lightcone string theory (I)

302

The scattering amplitude [4] sums over 0 and t. amp - f dt d0(det 0 2)-(d- 2)/2. (momentum dependent factor),

(A.8)

where 0 2 means the scalar laplacian defined on the string diagram. The integrand of (A.8) corresponds to a string diagram with specified 0 and t. The relevant quantity in our analysis (matrix element of a commutator) is c o m m - ~ = o d t d O ( d e t 0 2) (d-2)/2(...).

(A.9)

The momentum dependent factor is nonsingular, while the determinant factor gives a singular contribution ~z 2

"~z2 :

,

(A.lO)

where h = ( d - 2)/24 for closed strings, and h = ( d - 2)/48 for open strings [4]. From eqs. (A.4), (A.5), if x = x~, z + ( x ) = z ( x ) . These values of x correspond to diagrams B 2 or C 2 (with t = 0 = 0 in closed string theory) in fig. 5, i.e. the singular diagrams. Let x 0 be one of these x values and look at a nearby point x on the AT = 0 trajectory (see fig, 6). We show that the singular behavior of those diagrams at spacetime dimension 26 is like (x - x0) -1, and dO a , = o - ( X - X o ) 1/2, dx so that the value of the string diagram integrated over 0 is

fa

d x ( x - X o ) - t / 2 (something nonsingular at x = Xo),

(A.11)

r~0

which would complete the proof of finiteness in the closed string case. To show these we evaluate (A.10).

p+-p_=p(z+)-p(z_) 1 d2p ~_ -" 2 d z 2 " ( z + ( x ) - z d2p z_= f'(z ) dz 2 z (z - 1 ) ( z _ - x ) "

(x))Z[l+O(z+-

z )],

since p'( z +) = O,

Sang-Jin Sin / Lightconestring theory (I)

303

It is easy to see that / x t ( Z + ) = "~-~ ( ~

,

Z+(X)--Z(X)=~X-)/(Oll"~Ol3Jt-Ol4).

(A.12)

Notice that z is "far" from 1,0, oo, because they correspond t = + oo, while we are looking at t = 0. Putting all these together p + - t o = constA (x) 3/2

near x = x o

= const(x - Xo) 3/2, d2p z+ = + const(x - Xo) 1/2.

dz 2

(A.13) (A.14)

Finally

dO ( dZp c°mm-f=0dX~x-x/~z2

d2p -1( +~z2z ) -..)

(A.15)

- (x - Xo) 1/2,

(A.16)

and dO

--

dx

- (x - x0) 1/2,

dt

--

dx

from which dx

comm - Jt/'=o(x - Xo "1/2) ( ' ' " ) follows immediately. In other words, the singularities of the integrand ( - (x - x0)-1) is smoothed out by the dO measure ( - (x - x0) 1/2) to produce finite integral in closed string cases. In open string cases, there is no dO integration in (A.15) and t is fixed to be zero. Hence c o m m - det 3 2 ( . . .

) - ( x - Xo) 1/2,

which is singular; so that we have to consider the t v~ 0 case.

References [M] [K] [1] [2]

S. Mandelstam, Nucl. Phys. B83 (1974) 413 M. Kaku and K. Kikawa, Phys. Rev. D10 (1974) 1110 E. Witten, Nucl. Phys. B268 (1986) 253 A.M. Polyakov, Phys. Lett. 103B (1981) 207, 211

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Sang-Jin Sin / Lightcone string theory (I)

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

E. D'Hoker and S.B. Giddings, preprint PUPT-1046 S. Mandelstam, in Unified string theory, ed. M. Green and D. Gross (World Scientific, 1986) H. Hata, K. Itho, T. Kugo, H. Kunimoto and K. Ogawa, Phys. Lett. 172B (1986) 186, 195 P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Nucl. Phys. B56 (1973) 109 E. Cremmer and J.L. Gervais, Nucl. Phys. B90 (1975) 410 H. Hata, K. Itho, T. Kugo, H. Kunimoto and K. Ogawa, Phys. Rev. D35 (1987) 1318 K. Itho, T. Kugo, H. Kunimoto and K. Ogawa, Phys. Rev. D34 (1987) 2360 S-J. Sin, in preparation T. Kugo, KUNS 868 HE(TH) 87/09, Prog. Theor. Phys. to appear A.K.H. Bengtsson and N. Linden, Phys. Lett. B187 (1987) 289