From first to second-quantized string theory

Volume 202, number 2

PHYSICS LETTERS B

3 March 1988

FROM FIRST TO SECOND-QUANTIZED STRING THEORY T.R. M O R R I S Department of Physics, The Universtiy, Southampton S09 5NH, UK Received 13 November 1987

We outline a first quantization of closed bosonic strings which employs the Feynman sum over histories and describes the world- sheets through implicit functions. This method has a number of features: it has a local GL( 24, R) invariance. World-sheets do not explicitly appear. The method is applicable to multi-dimensional membranes, and it leads naturally to second quantization.

A major unresolved problem in closed string theory is to find a natural second-quantized description. First-quantized descriptions apply to a specific topology for the string world-sheet and (thus) a specific finite n u m b e r of external strings. However, the answer to fundamental questions such as " W h a t is the true ground state?" must lie in the non-perturbatire sector of string theory [ 1 ] which inherently involves condensates of an infinite n u m b e r o f strings. This is one o f m a n y reasons why it is necessary to have a second-quantized string theory [ 1-4]. One can define the term "second quantized" to apply to any theory in which it is not required to specify the number of strings (or particles, membranes, etc.), indeed in which the n u m b e r itself becomes a quantum variable. An example is the familiar q u a n t u m field theory description of particles. String field theory has also been developed [ 2 ]; however, althoug it is a second-quantized formalism, it is widely believed to be inadequate. One reason is that all string field theories require a specific parameterization o f the moduli space in order to break up Rieman surfaces into vertices and propagators [ 5 ]. In so doing they obscure fundamental symmetries o f string theory such as duality and modular invariance. More importantly, they do not make manifest the hopedfor geometric basis for a second-quantized string theory, and it is doubtful that any progress will be made in understanding non-perturbative effects without this, A number of ideas have been suggested for a seeond-quantized description other than the naive field 222

theory [ 3 ], but to date none have led to a satisfactory concrete formulation. In this letter we will put forward a formulation o f string theory which is second quantized (ie: does not require one to specify the number o f strings) but avoids picking a particular parameterization for the string world-sheet. Thus it incorporates duality and modular invariance at a fundamental level. Instead of describimg the world-sheet explicitly by a function of two world-sheet variables (tr) e.g. x ~(tr), where x u is a D-dimensional space-time coordinate, we will choose to describe it implicitly as the simultaneous solution o f a set o f smooth functions {J~}. fg(x) = 0 ,

k = l ..... D - 2 .

(1)

Any world-sheet for a closed string can be described in this manner - but we will find it necessary to impose a restriction: dfl df2 . . . d f D - z ~ 0

when f = 0 .

(2)

(Here d is the exterior derivative, the wedge product is understood a n d f i s the setf~ ..... fD-2 considered as a vector. We will later need to use the Dirac 5-function 5(f) as a density under f d ~ x and (2) is required for this to be well-defined). (2) imposes a number of restrictions on the surfaces that f can describe. O f these perhaps the most important is that only orientable world-sheets are allowed, To see this note that, because dfk vanishes along directions in the surface (when f = 0), the Hodge dual of (2) ~l: In components this is just (2) contracted with an ~-tensor.

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* ( d f l , dr2 ... d f D _ 2) -~ 0

PHYSICS LETTERS B

S= jdDxd(f) IdA df2 ...df,,_2t.

when f-- 0

defines a nowhere vanishing two-form on the worldsheet. This is only possible if the world-sheet is orientable. One can also show that (2) does not allow degenerate cases where the solution surface to (1) has dimension greater or less than two; the surface cannot have singularities (such as cusps or folds etc.) and the surface cannot intersect itself or other 2-surfaces. The description (1) of a surface is not unique: there are many functionsfwith the same solutions. In fact, one can show from (2) that all equivalent functions are related by D--2

f;(x)=

E Gkj(x)f~(x),

j=l

(3)

where f a n d f ' describe the same surface and Gk; is any ( D - 2) X ( D - 2) matrix such that det G ( x ) ~ 0

for all x .

3 March 1988

(4)

Thus this description has a local G L ( D - 2 , ~) invarlance. (Strictly this is a local GL+ ( D - 2 , ~ ) invariance, consisting of positive determinant matrices, together with a global change of orientation for thefk when det G is negative). In order to formulate a quantum theory for these functions we will define an action forfwhich is equal to the Nambu action [6] for the corresponding surface. I f we use this action in the Feynman path integral description (where now we integrate over all allowed f , ) we can expect to recover the first-quantized theory for the string. Actually by integrating over all fk we will automatically integrate over all topologies for the string world-sheet including any number of disjoint surfaces. Thus this formalism yields a second-quantized description (see our previous definition). Since the action depends only on the {J~} at the surface (described by their zeros) it must contain

(6)

The modulus is the square root of the complete contraction of the (D-2)-rank tensor with itself, thus LdA ...dfD_2 I = (O,,f~ ~2 f2...~,_2 fD-2

x a t,,,f, o'-'A...o""--'%_~)"~

(7)

(where by square brackets we mean a sum over all permutations multiplied by the sign of the permutation ( - but not weighted by 1 / ( D - 2 ) ! ) . Under a G L ( D - 2 , ~) transformation as in (3):

f~Gf,

60e)~c$0 e) [ d e t G l - ' ,

and effectively

Idfl...dfD_2l--,Idfl...dfD_21 Idet GJ . (In this last equation no derivatives act on G: if they did act on G then there would be an Z (for some j) which was not differentiated and this term would vanish against the d(f) in the integral (6)). The above equations show that eq. (6) indeed has a local G I ( D - 2 , ~) invariance. It is also equal to the Nambu action for the corresponding surface as we will now show. Let x'(g a) be the parameterised solution to (1). (a a (a= 1,2) are world-sheet coordinates). We append to {a a} a set {2j ( j = 1 ..... D - 2 ) } of linearly independent coordinates to form a local coordinate system (2 j, a a) for space-time. We choose the )d so that they vanish at the world-sheet. In terms of these coordinates fk--=fk( k g ) satisfies

A(0,o)=0.

(8)

According to definition (7) we may write D

Z

,a,V= 1 ~

IdA'"dfD-2 dxU dx" C2

~,ul ...,up ~ v t ...VD

D--2

600= I-[ d G ) . k=l

(5)

This d-function makes sense under a space-time integral, as a density, only if (2) is obeyed. The action must also have the local G I ( D - 2 , DR) invariance discussed above. There is only one non-trivial solution (in flat space):

XO,atfl ...0,,9_2J~)-2 al2D_tXStab~oX" x a , , A ...a.o_2fD_2O. . . . X~O.oX. (as follows from the standard formula for the product of two E-tensors). On the one hand evaluating the derivatives o f x u and x", and on the other, interpreting the result as a squared determinant one obtains 223

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Idf~...dfD_212 = ½ Y~ I O y f k O y x . " O y x " }

2 .

,up

(9)

The determinant is that of a D X D matrix with y-values running down columns and fk ( k = I, ..., D - 2 ) , x u and x ~ along the rows. Substituting (9) into (6) and changing coordinates to (J., tr) one obtains

S - J- dD-2,~, d2a 6 ( f )

3 March 1988

Clearly these equations must be equivalent to those derived from (I0); this one can show either as above or by using world-sheet derivatives of (1) and noting that, up to a conformal factor, (*df~... dfo-2)"*",

~:Lu X' v]. The above analysis proves the equivalence of (6) to the Nambu action at the classical level. We will sketch how the equivalence may work at the quantum level. We define the partition function

Z= J ~fexp(iS), _.-f d ° - 2 2 dZa •(f)

oy

X(I ~ OafkOaX,UOaXU ,] (Here 3j = O/O,a.Jand 0a ~ O/Offa). Due to the presence o f 6 ( f ) , 0oJ~ vanishes (see (8)), thus the above simplifies to

S = J dO-2), d2o" 6(/) (1 X $~

)l/2 IOA¢I2 la~x~O~x~l 2

.

Using (8) and the first determinant to convert the 6(./) to 6(;0, and writing the second determinant as l a, x ~ aax~l =

x,~

(where Ox/Oa' =~c, Ox/Oaz = x ' ) . One has finally

S = J d 2 o'[.~2x '2 - (.~.x')2] 1/2

=f

(10)

where ~ab=OaXUObX,, is the induced metric on the world-sheet. The above formula is the Nambu action [61. The action (6) can be functionally differentiated to derive equations of motion. One finds that the 6' ( f ) terms cancel and after some manipulation the equations can be put in the form

0=a(f) (*df, ...dfo_ 2) ×(*dA ... dfD-2) u~' O,O~,f,n (m=l,...,D--2). 224

(12)

where S is as given in (6) (and the integral o v e r / i s restricted to those that obey (2)). One can follow the above analysis (9),(10) to factor out the G L ( D - 2,~) gauge invariance. First one splits the functional integral into a sum of integrals over sets {J~} corresponding to different topologies for the world-sheet (consisting generally of disconnected pieces) plus an integral over sets {fk} which have no solution for (1). Since Sis zero for this latter integral, this integral just yields an infinite additive constant. For the former integrals the action S deafly splits up into a sum of Nambu actions over each disjoint surface. Thus the expression (12) becomes a sum over all products of connected topologies with a separate first-quantized functional integral for each connected piece. In other words (12) generates the vacuum diagram perturbation series expected for a second-quantized partition function. Let us now describe how this formulation may be used for particles, and (higher dimensional) membranes or p-branes [ 7 ]. Here p refers to the dimension of the word-volume [7]. Thus a 1-brahe is a particle, a 2-brane is a string, 3-brahe is a membrane and so on. The equations generalise straightforwardly. The p-brane is described by a set of smooth functions. J~(x)=0,

k = l, ..., D - p ,

subject to the condition

df~ dr2 ...dfo_p#O

when/=0.

(14)

The above condition restricts the description to orientable non-degenerate p-branes and is needed to define 6(f) which appears in the action

s = J dDx 6(f)I df~ dfE...dfD-p [ (11)

(13)

(15)

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3 March 1988

This has a local G L ( D - p , ~) invariance and is equivalent to the Nambu-style action

sumably possible to gauge-fix it to a form equivalent to

S= f dPa d ~ ,

s = + l f dDx[(Ouf~)2+m2~ 2]

(16)

where ~= a,,x"Obx,, is the induced metric, and S is the world-volume. And finally the equations of motion turn out to be 0 = d ( f ) (*dfl ...dfo_p)u'u''u

To carry out this program it would be necessary to replace (15 ) with an action containing auxiliarly fields which is classically equivalent. For example S = j d DX ~(f)

X (*df~ ...dfv_p) u' .up-,~ OuO,,f,,, ( m = l .....

D-p)

(17)

For the particle case ( p = 1 ), (16) reduces to the relativistic one-particle action S=fdz (2 2) 1/2 (and in (17) the Pl..4tp- 1 do not appear). The formalism describes only free particles with no interaction. This is because interacting particles would require worldlines that split into two (or more) lines and this cannot be described by smooth functions (13). On the other hand strings (and membranes and higher) have interactions that are smooth surfaces and it is notable that therefore (15) describes the full interacting theory with no free parameters (except of course for one overall multiplicative parameter that defines a fundamental mass scale. This is the mass in the case of the free particle, string tension for the string, etc.). We conclude by discussing some problems with this approach, and then summarise the features and possible extensions. It would appear that the major problem with this approach is the problem of finding a gauge fixing of (6). We outlined a way of factoring out the gauge invariance in (9), (10) but this is not satisfactory from the second-quantized viewpoint. It is not satisfactory because our "gauge-fixing" depended on choosing a particular diagram (world-sheet topology) in the perturbation series, and indeed reintroduced the string coordinate xU(a) and a particular parameterization for the world-sheet. Since our description (6), (12) is one in which world-sheets do not explicitly appear it would be much preferred if a gauge-fixing of (6) could be found which does not reintroduce them (at a fundamental level at least). It is also preferable if the procedure leaves a functional integral which we know how to calculate. For example, since the description of the free massive particle (equations (12) and (15) w i t h p = 1 ) is second quantized (describes any number of world-lines) it is pre-

× [ e - l ( d f t dfz

...dfD_t)Z+m2e]

can be shown to be equivalent classically to (15) by eliminating the field e(x). For a single world-line this action reduces to the one considered in ref. [ 8 ] where the authors showed that at the quantized level it gave the standard scalar field propagator. Another problem we have not solved to our satisfaction is that of introducing source terms to (15). However, as is well known, one can generate the Smatrix elements without using sources if one couples the action to background fields. For example the coupling to the metric is determined by diffeomorphism invariance to be any of a number of equivalent forms of which two are

SG = [(D--p)!D!p!] -I,2 f dDx v/G JOC) ×{ * [ (*df~ ...dfo_p) df~ ...dfo_p] } 1/2 =(p!)-1/2 f

dOx~(f)

X (~/AI''4AD ~ . . . . . . O

a~lp, ...ailpV p

X av,,+,fl ...Ov,,fo-pO,,,,+,A

...a,ofD-v),,2,

and in the case of the string the coupling of the antisymmetric tensor b,v is

SB = J dDx a ( f ) d f ...dfo-z ^ B

(18)

The examples above can be shown to reduce to the familiar world-sheet descriptions (following the analysis (9), (10)). It would be interesting to find out how to couple all the other fields (both massless and massive) to the {j~} and whether a compact way of writing the result can be found (through a "string field"). Another extension would be to formulate the supersymmetric case which describes closed superstrings. We hope to 225

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investigate these ideas further. It does not seem possible to formulate open strings this way since their world-sheets can not he described by s m o o t h functions (1) that satisfy ( 2 ) . A n o t h e r puzzle is to understand how to formulate non-orientable closed strings: in this case the two-form * ( d ~ . . . d f o _ z ) m u s t vanish along a curve in the world-sheet (cf. our arguments below ( 2 ) ) . I f we allow ( 2 ) to vanish at points or curves in the world-sheet then the action ( 6 ) can still be defined - because the points where the integrand is ill-defined are o f zero measure. But two sets o f functions {~} a n d {f~ } can describe the same surface and have different subsurfaces where ( 2 ) vanishes. Thus equivalent functions are no longer necessarily related by eq. (3), and there is an infinite degeneracy in the description which cannot be f o r m u l a t e d as a gauge invariance, it is unclear to us whether this represents a fatal obstruction to the formulation o f a nonorientable closed string theory with this m e t h o d . In s u m m a r y , we have described a m e t h o d o f quantizing orientable closed strings which avoids using some p a r a m e t e r i z a t i o n o f the string; in fact strings a n d world-sheets do not a p p e a r explicitly. By formulating the q u a n t u m theory in terms o f a q u a n t i z e d set o f functions (fk ( k = 1,..., D - 1 )) whose simultaneous zeros describe world-sheets, we a u t o m a t i c a l l y incorporate any n u m b e r o f world-sheets. At any one time the n u m b e r o f stings is itself a q u a n t u m variable. Thus the theory described is second quantized, a n d forms an a p p r o p r i a t e starting p o i n t for discussing e.g. n o n - p e r t u r b a t i v e effects. The f o r m a l i s m can be e x t e n d e d to w o r l d - v o l u m e s o f any d i m e n s i o n p, by using a set o f functions (fk ( k = 1 . . . . , D - p ) ) . F o r p = 1 the description is only that o f free particles with some given mass. But for strings a n d ( p - 1 ) - d i m e n sional m e m b r a n e s the f o r m a l i s m leads to the full interacting theories a n d with no free parameters. O u r second-quantized d e s c r i p t i o n o f strings m a n ifestly incorporates duality a n d m o d u l a r invariance (because we avoid particular parameterizations), thus we hope that further study will shed light on the geometric basis for string theory ( i f one exists). It m a y be possible to formulate the action in a b a c k g r o u n d

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string field (i.e. a field consisting o f all the string c o m p o n e n t fields) in such a way as to exhibit the fund a m e n t a l string field gauge invariance. As we have discussed, the theory has a local G L ( D - 2 , E) invariance - and the most pressing p r o b l e m is to find a way o f gauge fixing this so that we can p e r f o r m perturbative and non-perturbative calculations. We wish to t h a n k Douglas Ross a n d Bill Spence for encouraging discussions.

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