An off-shell propagator for string theory

Nuclcar Physics B267 {1986) 143-157 ' North-llolland Publishing Company

AN O F F - S H E L L P R O P A G A T O R FOR STRING T H E O R Y Andrcv, COHEN, Gregory, MOORE and Philip NEI.SON

I.vman l.ahoratorv of Physics. Ilart'ard Umt'er~itv, ('amhrtdge, AIA O21.'¢8, USA Joseph POI.('HINSKI 77u,om (;roup, Pl{l'sics Department, I,'mver~t(v of Tcxa.s..4ttsttn, Tx 78712, I,'N.4 Received 27 August 1985

The l'ol',akov integral with boundaries can be used to provide a propagator for the secondquantiTed bosonic string. This propagator makes sense off-shell. For special initial and final states ~c c~mlpute it cxplicitl.', in the tree approximation, where it reduces to the correct ~,um of field thcorctic particle propagators.

1. Introduction Superstring theory is now a strong candidate for a fundamental theory of nature. However. the formulation of string theories is in many respects incomplete. Questions such as the origin of general covariance, the derivation of an effective quantum field theory, and the understanding of nontrivial vacuum structure remain to be answered. Attempts at a more complete formulation are usually based on the idea of a field theory of strings [1]. Despite encouraging recent progress [2- 4] this program is not yet complete. We would like, as an alternative, to consider Green functions in a covariant second-quantized string theory without making use of string fields. The method is the Polyakov functional integral [5]. which was first considered for this purpose by O. Alvarez'[6]. The present paper is an extension of [6] and also of [7,8]. In the next section, as an exercise, we evaluate the "Polyakov" path integral for the relativistic point particle propagator. In sect. 3 we examine carefully the boundary conditions for the sum over all surfaces with given initial and final boundary curves. In sect. 4 we give a general expression for the propagator and evaluate it explicitly, in sect. 5 for the special case of point-like initial and final curves. Such states were considered in the old dual model literature in attempts to couple strings to pointlike currents [9]. In sect. 6 we compare this with the result in the usual canonical approach. Finally, we discuss general questions and directions for future work. 143

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.4.G. ("~,henet aL / ,Strtnt~theory

2. The point particle r h e analog of Polyakov's prescription for a point particle of mass m includes an "'cinbcin'" auxiliary field e(t):

S p = ' 2 f~'[e

'(t)x'(t) z +e(t)m2]dt

The einbcin is related to a metric on the parameter interval / = [0, 1] by' g( t ) = e(t )-'. The amplitude for a particle localized at x,~ to be observed at xf' is

=f[de][dx"lexp(-Sp[e, x]).

(2.1)

"1o c o m p u t e (2.1), we need an intelligent representation of the space of metrics. Following [5, 6], we first note that every metric on 1 can be brought to a constant by a coordinate transformation. The only coordinate invariant property of an einbein e ( t ) is the total length c - f ~ ] e ( t ) d t . To put e ( t ) into standard form, we simply integrate the einbein transformation equation

e(f(t))f'(t)

= c.

(2.2)

w.ith f(0) = 0. f ( l ) = 1, to find the transformation f: 1--+ I. Thus every e ( t ) can be described by the number c and the map f ( t ) ; in the language of [6]. c is analogous to the Teichmiiller parameters. We will transform [de] to these variables and then perform (2.1) Since coordinate indices take on only one value t. we will omit them. bearing in mind that x ~', e and g transform as a scalar, 1-form, and tensor respectively. That is. under the map h we have

x~'(t ) --* x ~ ' ( h ( t ) ) - x" o h,

(2.3a)

e ( t ) --* h'e o h,

(2.3b)

g(t ) --, ( h')'-g o j,.

(2.3c)

a,x~'(t) - , h ' ( 0 , x " ) o h.

(2.3d)

S~ is invariant under these reparameterizations. We will also need the analog of (2.3b) in terms of the allernate coordinates c, f ( t ) . Applying (2.3b) to (2.2) we find c-*c.

f--+h

tof.

(2.4)

O u r strategy to compute (2.1) is the one used in [7.8]: we define measures [de] and [dx] using locality, change variables to physical coordinates and gauge degrees

145

.4.(*'. Cohen et al. / String theory

of freedom, then integrate. The job is simpler than in the string case because c and f ( t ) specify e ( t ) completely; there is no analog of the Weyl factor to integrate over. As in [7, 8]. we note that it suffices to work on the tangent space to the space of all einbcins. Hence we define [d(6e)] by requiring

= f Id( Be)] exp( - llaell2),

{2.5)

118eli 2 = Z l d t e - l ( t ) ( a e ( t )) 2

(2.6)

where

is the coordinate invariant metric on the tangent. As in [7]. the principle of locality determines the measure (2.5) up to a factor of exp k[~l d t e(t) for some k. This can be absorbed into the definition of m 2. We wish to express 8e in terms of 8c and 8f(t). near c and f ( t ) corresponding to some e. Then (2.2) says

(e + & , ) o ( f + 8 f ) . ( f ' + 8f') = c + &', d

8e of.f'+ ~t(e

o f ' 8 f ) = 3c.

Hence I dr

Ilaell' = £ ,--[~) (&'( r ) )-

=£1

dt

[Seof(t)f'(t)]2

,, o / { t ) / ' ( , )

v,herc r = f(t). Tile denominator is just c, so

-

c

+ c--

dtt ( e ° f ' a f )

dt

(2.7)

(the cross term vanishes by integration by parts). It will be convenient to refer 8f to the origin of the reparameterization group f + 6 f = (1 + 8./'o f ' ) o f . The infinitesimal map 1 + 6 f o f i pushes each point of 1 a small distance: thatis. 8 f o f i isa rectm'./ield ~ on I. The invariant norm on the space of small changes in f is thus

II~ll: = Z/[dte

(t)] g ( t ) [ ~ ( t )]2

=~ldte(t)3((t)2.

.4.G. Cohen et al. / String theory

146

We now have d

d

c d

dZ [e° f ~ f ] = dt [e° f " ~° f ] = e ~-~( e~)o f

=c(v.~)of, since ~ has one upper coordinate index. Thus,

ac 2

ll6ell2=--+c (

8C2

~1

(~7.~5

)2

ofdt

~le(w.~)2dr

C

(2.8) where we have introduced the laplacian A - g lV. V = g l ( d / d r ) e l ( d / d r ) e=

c 2e l(d2/dt2)e. We can nov,' define the jacobian by [d(Se)] = J ( e ) d(6c)[d~]. The LHS is defined by' (2.5). d,~ is defined analogously, and d(8c) is the ordinary Lebesgue measure times a constant such that fd(Sc)exp[-(&.)2/c] = c 1/2. Inserting into (2.5) we thus get , d 2 '}

l=J(g)cl/2det

1/'2 - c

dt 2 .

Hence . / = J I c ) only'" it is independent of f(t). In (2.1) we can perform f[dx*'] by letting x ~ ' ( t ) = x l ' + t A x ~ ' + y " ( t ) , A.x-~' - x~' - x~. The result is

f [dx;,]e

s. = e x p

2c

2

where

det'(-,.

lqserting into (2.1) and gauge fixing f = 1. we get

;

(~x)2

tH 2C

Next we must compute the regularized determinant. Since y ( 0 ) = y ( l ) = 0. the eigenfunctions are sin(heft) with eigenvalues n e v 2 / c 2. Thus, using ~'-function

.4.(;. Cohen et al. / String theory

147

regularization

//2,/}.2 ,)

d

log det' -

d s .,.= o

E 7.

= - 2~'(0)log c + const = log c + const,

SO

~. ~:P(xr" xi) = c°nst f~}

=

(.1x )2

"

d{'c-a/2exp{

consCf dap

= const'f d'lp

2c

,

"~'f~/~dc exp(

(2'/7)': p2e_'"-~' + m2

m2c') 2

i

_ 12c( p2 +,,12))

"

which is indeed the field-theoretic propagator. Note that, in contrast to [7], we are neglecting the overall normalization of the result, since this can be absorbed into " w a v e function" renormalization. While this is admittedly a longwinded way to get the propagator, we will see that the above steps all have simple generalizations to the string case. with Sp now the Polyakov action [5].

3. The string case We wish to consider the amplitude ~ ( £ f , ~e) for a closed oriented string localized at a loop J', in spacetime to be detected as coinciding with some other loop ,el. We represent this in terms of a sum over all world sheets whose boundaries coincide with the two given loops. We now discuss the appropriate b o u n d a r y conditions on such sheets. We will use the notation of [7, 8]: our treatment, although motivated by [6], will differ somewhat. For now we restrict attention to the tree level, so that our parameter space has the t o p o l o g y of a cylinder: M = [0,1] x [0.1] where the coordinate o I is periodic and o 2 = 0, 1 are the boundary curves. Thus the boundary aM of M is two circles. While physics tells us to integrate over world-sheets with given b o u n d a r y loops, in practice we must instead use parameterized surfaces x " ( o ) and later fix the gauge red u n d a n c y thus introduced. In particular we want . law to be any parameterization of the loops gi, "gr" l.et x','do 1) be representatives of J'i.f and v f any reparameterizations of the circle. Then we wish to integrate over the space 8; = u~,,,-,~" where b~,. =

14:";

.4. (;. ("ohep~c'l al. / .Slrm.gIheorv

{.X"(O)'X"],;M=Xi.r°Vi.f}. We will actually integrate over the spaces ~,;,- with Dirichlet b o u n d a r y conditions, saving for last an integral over ",.2:indeed the case we will do explicitly will not need the 2,' integral at all. A n o t h e r way to see that w:e must in general integrate over "-" is to note that even if wc consider all metrics and embeddings ( g, M') with one particular parametcrization of the boundary, in general when we take these to a specific (e.g. conformal) gauge v,e perform diffeomorphisms of M which are nontrivial on the boundary. The gauge-transformed configuration thus lives in some general b;,.. Turning this around. in order for a gauge-fixed integral to include all physical world sheets it must include all reparameterizations ~2 of the boundary data. Following [6] we should demand that the metric and embedding spaces be such that classical extrema of the action always exist, i.e. that (nv. 0 x ; ' ) f x ; ' - 0 for tangents &x', to b',. ltere n~ is the normal vector to the boundary aM of M, in the metric g. Since we are integrating over b:,. with Dirichlet conditions, this condition is always satisfied*. Wc therefore want to perform the integral [7]

[dgl /: [dale f dZ f v,,.Vw

%

,.,

(3.;)

or the equivalent gauge-fixed version. As pointed out in [6], however, we still need more b o u n d a r y conditions before ,g(o) has a good mode expansion defining its measure. We will choose an inward pointing vector field, n " ( o l). defined on aM, and require n"¢"yg,,,-- 0

on a m ,

(3.2}

where t;' is the tangent to the boundary. The resulting expression is in fact independent of the choice n". Make a change of variables corresponding to a reparameterization f "( o )**. g,o,--+ ( g , a ° f

)t,J.,J.t, r,; - ( f * g ),,,,

M' + M' o f-= f * x ' . The action and measures are invariant under this change. Only (3.2) changes, to (n")'t;'g,,;, = 0, with n " = ( n ' f " ) o f l Since our choice of n" is immaterial by coordinate invariance, we can. if we like. average over it. In any event n" does not really amount to specifying extra * Nmc that if ,.~.e had not split .~.; into ,,arious ,~.-,. then this condition ,aould ha,,c become a linear boundar,, condition on ,4. This v,ould ha,.'..: made the region of integration for [dg i &'petal , , ~. prc,.cnting us from integrating [d.x] first. "* The reparameterizatiom, ,.~.¢consider al'aa,.~, take i . i and f ~ f, that i:, /2(a;.()) = (L I 2( % . [ ) =. 1.

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•4.(;. Cohen et al. / String theory

geometrical data to define the integral (3.1). Similarly, exact conformal invariance in 26 dimensions means that we may put a boundary condition on the conformal part of g,a, or not, as we choose. Again any such choice is immaterial and does not specify anything about the physical initial and final states. It is important to verify that the quantity we are considering is actually Weyl invariant, since off-shell quantities computed with vertex operators are not. Comparing the functional integral evaluated on a given gauge slice, and that evaluated on a conformally related gauge slice g't, = (1 + 6+)g,,t,, we have

8 1 o g W = f~ d2ov/g(A + BR)Sep+ f~ • 1

ds . h.i) h 80] do-d-~o[(C+ Dn,,t t, V',,t ,, ) 8 0 + k,n M

(3.3)

R is the scalar curvature. Eq. (3.3) is the most general kx:al invariant with at most two derivatives that can be constructed out of g,,j,, the induced metric on ,3M. d s / d o , 6~, and the vector fields t", n". The latter are normalized to unit length since only their directions enter. In [8] it is shown that nonlocal factors do not appear. We have the freedom to add to the original string action a local term L,,

d s

,.,=f

. ) .

1341

We know [5] that B = ( d - 2 6 ) / 9 6 ~ ' . Further, an application of the Wess-Zumino consistency condition, insisting that the second variation of log W be symmetric, gives B = D. Comparing (3.3) with the variation of (3.4),

6 ( l n W + L,,) =f,~ d'-o frg-(A + a ) 6 0 .1

+

m

96~"

+

f,,

t

d2o vrg R 60 +

M

do-

do n

,,l ~'~ ,~,,,'" 80)

.

.

M

Thus. at d = 26, be appropriate choice of a, c, and h - ~e,l we obtain a Weyl-invariant quantity. This agreees with the statement in [6] that local scale-dependent factors can all be renormalized away.

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,1.G. Cohen et aL / 5,'trm,~fdu'on

4. The propagator We follow the analysis of [7.8] now including boundaries. The cylinder has just one Teichm'uller parameter X [10]. so that we can choose for our slice g'X = ( dOl )2 + XX(do2 )2. Thus ~, is the metric length of the cylinder. Clearly no two of these are eonformallv or diffemorphically related. The normal vector is n oc///b~o-'] aM. The vector field ~9/0o l, defined on all of M, generates a reparameterization which takes ,fI,~ to itself for every X. By the Riemann-Roch theorem, this is the onl', such "'conformal killing transformation" [6]. It generates a circle subgroup of diffemorphisms. Not only is this group's volunle finite, it is also constant since we choose O / 0 o I independent of ~,. It thus contributes only an overall constant to d' [8]. which we ignore. Changing variables, fixing the gauge, and performing the ,x" integral (see [7.8] and sect. 2) we get ~,'=

f d Z f dX ( d e t ' A " , ) ' / 2 ( d e t -

,.1) ,C-'

( det ll( P") ) '/2 ×

detH(P)

e .s,.(~

(4.1)

All factors in (4.1) are to be evaluated at g = ,{% Any small variation of ,g,,;, can be resolved into 8g,,;, = 8q, g,,;, + ~,:;,

~,.... + g,,;,.xaX.

+

with ,~'JaM = 0.

0,,~=laM = 0.

(4.2)

The latter condition follows from (3.2). while the former says that the diffemorphism generated by ~ leaves the region fixed. Eq. (4.2) gives the boundary condition on the vcctor laplacian J " . A is the scalar laplacian with boundary condition y],,,,~ = 0 (see sect. 2). II(P), H(P +) are the inner product matrices of the kernels of P. t '+ (see [6])..i:(o) is the solution to the classical equation of motion for x ( o ) with given .~;, and b o u n d a r y value x,.f o 2. We begin evaluating (4.1) by considering J " = - 8 " , ( 0 1 0 ~ +X "-0202). From (4.2) we get nonzero eigenvalues

I(,,":) 4,,,r 2

2 + ~4X 5

['4rr 2 n 2+ ,

I

. m > 0,

from ,~2 = 0.

m>lO.m.n not b o t h 0 , I'

(4.3a)

from 02~'=0

(4.3b) "

.4.(/. Cohen This gives (replacing

m ~

151

et al. / String theory

in (4.3b))

-m

•

,

~ 2

•

where the prime excludes only m = n = 0. This was evaluated in [7]: thus [det'k"]

'/2=2he

4~,,x)2.

~a/3H(I_e

(4.4)

n ,0

A similar discussion shows that d e t l / 2 ( - A ) is the square root of (4.4), times a k-i n d e p e n d e n t constant. We now turn to the one-dimensional determinants of H ( P ) , H( P " ). First

H(,°)

=

~O 1 2 = f d - ' o ¢ f f g , ,

= X.

For t l ( P ' ) w e must find the basis vector '4' for kernel P" corresponding to our choice of slice ~'x [8]. This is the traceless part of

d

,

aX,¢~ = i °

) 2h."

i.e.

,t,=

( -x

) X '

(Recall we contract indices with ~'x). Thus

tl( P

) = II,/.'112 =

fd2o v@Tr,gx

2 l~/'gx 'g ' = ~ -

('ollecting results. (4.1) is now d)t 4~x l-I (1 - e 4~.,,x) 24e s,,l~l d' = const f d"" £ ~ ~T~-e

(4.5)

tt - (I

Solving for x,

i

L

Sp ( .x" ) - 4'n'og ,, ..

2~?1 :,: sinh2wnX [(17%'12+ Ix,,d-' )cosh 2,a',,X - 2 Re(x,,,- x,*f)],

where

x,,. r =

LI do I e2.~,,,~,.~ ,.f (ol)

•

152

,,I.(;. ('ohen et al. / .gtrmg theory

We can re-express the product factor in (4.5) by recalling that the partition function for open strings is

y" do(N)q,,'.~4~= ~. qV l=q ', :-0

{ %,,}

,I-I(l_q,,

) -'4

n -I)

Hcre. N--[£,,.,neV,,, and d,,(N) is the degeneracy of an open string. While it may seem mysterious that the open-string degeneracy should enter our closed-string calculation, this will in fact be just what we need. At this point we would like to extract the spectrum contained in the propagator. In the appendix we reexpress the full propagator as the matrix element of the exponential of the string hamiltonian H, between parameterized string states: 1

E (-])"

'TT

m. n c Z

1

ll+213(m2,-n z j + m + n ]

]xi(°))"

(4.6)

Thus the integrand has poles at h =

-6(m2+1;

,2

-2(re+n).

m,n~T7,

where h = e/pC+ 4 ( N - 1 ) . h = 0 corresponds to the physical string modes p'~= - - ( l / ~ x ' ) 4 ( N - 1 ) = - m ~ . . The remaining poles arc necessary to reproduce the non-local effects of determinants in the path integral. Although these poles do not correspond to physical states in the Hilbert space, they produce necessary effects as intermediate states in graphs contributing to N-point functions. They presumably correspond to the Sti~ckelberg fields which must be included ill other approaches to string field theory [4.3].

5. P o i n t states

Formula (4.5) holds quite generally', but integration over reparameterizations ~' is not easily carried out. For nov,' we go to the special case where , a g~ are points: that is .v,.r( o ) = x,. r are constant maps. Then every ~.,';_,.is the same and we can drop the integration over ~. In this case. our final expression is

c '=constfxv f

d,,(N)e

4,,.x,,',,, ', ,a, "/4,-,,a

'V=O

1 , e ' p j ' . .deep .. = const x~=,, d"(N)fp:+4mk'. ( 2"tr,)>'

(5.1)

Here. m~, is the spectrum of the Nth open string level, so that 4m~. is the spectrum

A.G. ('ohen et al. / S t r t n g theon"

153

of the .¥th closed string level. The state Ix~'(a)= x~) contains closed string modes of all possible masses, but weighted by the degeneracy of the open string. The quantity we are calculating bears a certain resemblance to an off-shell tachyon vertex function, that is. the amplitude for a closed string world surface to pass through certain points. In both cases the embedded world surface is a closed surface passing through two specified points. However. the surface is defined not onlv by its embedding, but also by its internal metric structure, and here the two differ. The vertex operator instructs us to consider boundarylcss M. with x~'(o) constrained to touch given points x i of spacetime at fixed a~ which are later integrated. The present approach on the othcr hand uses an M with finite-sized boundaries which happen to map to points of spaeetime in the case in question. The off-shell vertex function is known to lose its Weyl invariancc, whilc the present one does not*. Note, also, that the point-like boundary couples not just to the tachyon, b u t tO states of all masses. The propagator (5.1) is highly singular at short distance, because of the point-like initial and final configurations. For loops of finite size, it would presumably be soft at short distance. In any case for any given amplitude computed with our prescription we can extract finite contributions from the particle species on the external legs just as wc can in the 2-point function (5.1).

6. Comparison with canonical result "lhc propagators in (5.1) look reasonable, but we must check the weights. These should be thc sum ovcr each state of mass m v of the squarcd amplitude for Ix,) to contain the state. To find these, we construct the state [x,). in canonical light-cone gauge formalism. The position operator is, in the Heisenberg picture.

x(a.r):q+2edpr+i~F2a'S.

1 n

[ % e '2''''~ "'+~t,,e : " " " ° ' l

We will usually drop the vector indices, which run from 1. . . . . 24. Thus [ a , , . a,,,] = n S ( n + m ). N = ~'. . a ,,a,,. and similarly with tildes. Also. the hamiltonian t l = a ' p e + 2( N + ,g,; ) - 4. o&n 2 = 2(N + N) - 4, with the constraint N - .,~' = 0. To dcal with * On->hell it ~,eetm. likely, that the two pre.',cnptions '.'.'ill p r o ; e to be equivalent es~,entialb, becau>,c v+e onl'. V,ant the re~.idue of the poles '.,.hen all external legs go on-shell. It turns out that the i m p o r t a n t metric c t m f i g u r a t i o n s in thb, case have v a n i s h i n g metric circumference for the boundaries,, and ~.o the latter bout, me essentially points on a sphere.

154

4.(,.

( . h e n ct al

.'~'trmgtlw,.'~

thc c o n s t r a i n t we introduce a new oscillator basis. [.ct

O! -"

1

-

[(,,,,--a

)+(,~

2¢2 ,,

-a,,)].

. . . .

Q,~-2~/2,,[(a,,-6

,,)-(a

,,-6,,)],

?,,-'= ~[(<,+a ,,)+(. ,,+a,,)]. Then

[ Q;:,. ,,,(:,] = ia,,,,,ao,a',. ,xhere (t. fl run from one to two. and

tl = a'p e +

(

- 4 8 n -4-.,i

'f

i

-4

E x p r e s s i n g x i ( o . r = 0) in terms of P , Q . we see that the point state satisfies

Qi]'lx, > =- 0. Such a state indeed satisfies N - /~' = 0. It has a definite value of x " as ~vcll. since x" cc r (the time), and one can verify that x t o ) I x , ) is also a constant. T h u s ~ e wish to c o m p u t e

(xf,qr ie "JTIxi,q, ).

At>0.

(6.1)

F o u r i e r t r a n s f o r m i n g we get 2 r r 3 ( p r- _ p ( ) 3 ( p f _ p i ) e 1-] ( Q " - 0]exp[ - i

,,,'p:a, 4,a,

..kr(-

48n + H,, )]1Q" = 0 ) .

It, I

where tt,, is the harmonic oscillator h a m i l t o n i a n of frequenc',' 2n. Using the 0 - 0 a m p l i t u d e of II. this equals

a(p(-p,)~(?,-p,)e

'°*"~'lq(1-e ~'"J~) -'~ n :" ()

=3(Ap')8(Ap)e

,,,'p'a~ ~.. d , , ( N ) c •V

0

4,a~(., 1,

A.G. Cohen et al. / 5,'trm,g them'v

155

Supplementing this equation with the corresponding one for antiparticle propagation (to get the Feynman propagator) and Fourier transforming J'r, we recover the same weighted sum of propagators as (5.1). again with the open-string degeneracy, d,,( N ). We can also see the appearance of d 0 ( N ) level bx' level. For instance, at the m 2 -- 0 level we have states ~' ~ ! 110). Most of these have wave functions vanishing at Q]' - 0, however. Only the states i = j survive: these are clearly in 1 1 correspondence ~ilh open string states ¢~' ~]()), and each makes unit contribution to (6.1).

7. Conclusions We have shown that the Polyakov path integral does define an off-shell string propagator. The same technique applies to n-point functions. Although the analysis in this case is harder, it is not so hard to apply it approximately to the case where all external lines are nearly on-shell. The resulting residues should then reproduce the Koba-Nielsen anaplitudes, and extend them in a sensible way off-shell. Work on this problem is in progress. It is important to go beyond the case of point strings. The average over "2," effectively projects onto states annihilated by 1.,,- l. ,, for all n, rather than onto transverse states. Even if we cannot perform the average over ' - in closed form. v~e have been able to construct some 2-invariant states iterativcly. This construction will be described elsewhere. The amplitude we have calculated is invariant under the local symmetries of the world-sheet action, namely Weyl and 2d coordinate invariance. For strings to describe gauge theories and gravity, though, d-dimensional gauge and coordinate symmetries must appear. It is still a bit mysterious how these will show up in our formulation. For example, the analysis of sect. 5 is no help. since point-states do not couple to gravitons. Finally, off-shell quantities in general relativity depend on the gauge chosen, while in the present approach all amplitudes seem to be unambiguously defined. Presumably our computation has implicitly chosen some special gauge, and this will become clear when the origin of the gauge symmetry itself is explained. We would Like to thank Sidney Coleman for important suggestions. We would also like to thank David Gross and Michael Green for many interesting discussions. This work was supported in part by the National Science Foundation under grant numbers PHY-82-15249 and PHY-83-04629. by the Harvard Society of Fellow.s. and bv the Robert A. Welch Foundation.

Note added in proof Recently we have received [13], in which points of related interest are discussed.

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A. (;. ('ohol et aL / String t/ze,n3

Appendix W e are to establish (4.6). Begin by defining new c o o r d i n a t e s for M ' ( o ) c o r r e s p o n d ing to an e x p a n s i o n in terms of cosines and sines Z -- "'(0" i

,~1,/2

(.v,,~ - ( V ) ( : , ~ J (.v" ,, + (~"

n

-.,-,,.

,, > o.

1 , l 1 /2 )( ,,~ j =-x,,.

,, < o.

R e p l a c i n g x,,,.~ by );,,.r..i",,i. f we can rewrite eq. (4.5) as (1 - e 4,,,,a ) 24e ,.~.-~:/4.-,.,'.~

f d ,.a v X'--/7;5e4'-"~ • ;z- 1

II

X ,,H__1 exp{

2sinh2r, na

[cosh 2 v, n

a

( .v,;f +

-.,,, O ,, ] ,' /

2 sinh"2~rn )~ [cosh 2 ~r,, X ( )',;-f +- .i"], ) - 2, ,, ,),,, ] j . T h e string h a m i h o n i a n is t t = c ( p 2 + 2( N + :'~' ) - 4 where N and :'~: are tile n u m b e r o p e r a t o r s for the right-and left-moving modes, p2 c o m e s from the center of mass t r a n s l a t i o n of the string, and - 4 is the usual normal ordering constmlt. Recall that the euclidean p r o p a g a t o r for a h a r m o n i c oscillator of mass m and frequency w is lll]:

] 1,'2

rhea)

(yflc

=:""' I.~'i) =

~'(1

-

e 2~,)

""pl T a k i n g w = 2n;

m

= !" "r =)wr we can use this result to write the p r o p a g a t o r as

=

'"',1'<

-

xf(o)le

.

~'=" I.v,(o ))

U s i n g a s u m m a t i o n representation for the infinite p r o d u c t due to Euler [121

1--l(l-e

~"~)=

~

(-l)"e

: ' ~ ' ~ " " .... '

157

A.(;. Cohen et al. ,/ String theota

we car! o b t a i n a s i m p l e form for

.,-,(o

dX

Z m,

=fd" .,,(o)

(-1)

''' ' ''e 2'';~13'''' ''''' ....... le =~'

o)

tl 65_ Z

Et-1)"

o).

t t l , t~

References [1] 12] [3] 14] 151

le'l 171 [8]

I~1 IO1 11 ] 121 13i

54 Kaku and K. Kikkawa, Phys. Rev. I)1[} (1974) 1110. 1873 \V Siegel, Phvs I.ett. 142B(19841276: 149B(1984) 157, 162 I) l'riedan, String field theory. ('hicagc. preprint I-H 85-27 1 l:kmk~, and M. Pcskm. in Anomalies, topolog), and geometry, cd. A. White (World Scientific. Singapore, 1985) A M Pol',ako,.. Ph',s. l.ett. 103B (1981) 21)7 () Alxarez. Nucl Ph,.'s B216 11983) 125 ,1 Polchinski. E'~aluation of the one-h',op ~,tring path inlegral, Texas preprint UTl(i-13-S5, to be puhh@~ed in Comm. Math. Ph,,s. ( i Moore and P Nelson. Nucl. Phv~, B266 11986) 5,'4 F. ('orrigan and I).B l"airlie. Nucl Ph',s. Bgl 11975) 527; .x.1 (ircen. Nucl. Ph,,s, B12411977) 461: I Sus~,kind. unpublished 1. Bcr~,. Bull Amer. Math. Soc. 5 (1981) 131 R |:c',nman and A }libbs, Quantum me~ hani~.~ and path tnte graZs (Mc(ira;**-Hill. NY. 1965) 1) Mumford. 1)~ta h'~ture~ ~m theta l (llirkh~luser, Boston. 1983) A 1,~e,,tlin.[.ebcde', prcprinl N265