Volume 214, number 1 PHYSICS LETTERS B l0 November 1988

SCATTERING A M P L I T U D E S I N p - A D I C S T R I N G T H E O R Y ~"

Zvonimir H L O U S E K and Donald SPECTOR 1 Newman Laboratory for Nuclear Studies, Cornell University, Ithaca, NY 14853-5001, USA

Received 21 July 1988

In this letter we describe some features of the p-adic string theory. We show that different equivalent forms of real string scat- tering amplitudes lead to inequivalent p-adic string theories. Thus there is no canonical p-adic generalization of string theory, but we are able to give a prescription for what p-adic generalizations are possible. The reason for this multiplicity ofp-adic theories is that the linear combinations of multiplicative character which are used to build the integral representations of string scattering amplitudes behave quite differently in the p-adic and real number fields.

Any scattering amplitude in a string theory can be written as an integral, over an entire domain field, o f a linear combinat ion o f the products o f multiplicative characters on the domain field. Customarily, one considers the case in which the domain field is the real or complex number field, and the multiplicative character is the usual absolute value o f the difference o f the positions o f the sources. Recently, an alternative has been suggested, which is to use the p-adic number field as the domain field [ 1-5 ]. The norm on these number fields is ultramet- ric, obeying I x + y I p < max ( I x I p, I Y I p), where the norm is defined with respect to a chosen fixed prime number p. Importantly, the rational numbers are dense in the p-adic number field. Because o f many unusual properties, the set o f multiplicative characters over the p-adic number field is larger and richer than the corresponding set over real numbers. For a detailed exposition o f the properties ofp-adic numbers, we recommend that the reader consult refs. [6,7 ].

There are several reasons for considering the situation in which the world-sheet is taken to be p-adic. First, as we shall see later, the p-adic version o f the string theory is inherently nonlocal, and so one might hope it will shed light on a possible nonlocal phase o f the field theory. Secondly, in analogy with mathematics, where the p- adic number fields (i.e., the collection ofp-adic fields for all prime numbers) and the real number field can be organized in a structure known as adele, there is a hope that the p-adic theories together with the real theory can be organized in a similar structure. As a matter o f fact, that such a possibility exists has been shown by Freund and Witten [4 ]. Their original observation was that the product o f all p-adic four-point tachyon- tachyon scat- tering amplitudes and the four-point t achyon- tachyon scattering amplitude equals one. Unfortunately, such a simple relation is not true in general [ 8 ]. In analogy with mathematics, it appears that the original adelic relation o f Freund and Witten is a rather special case, the analogue of principal ideles. In fact, we will see here that p- adic string theories need not even obey the adelic relation at the level of the four-point function. Further research on the nature o f the adelic structure o f string theory is clearly needed.

In this letter, we present a systematic procedure for writing down the various possible p-adic string scattering amplitudes, and so give a classification ofp-adic string theories. We will show how to set up a p-adic theory for any desired choice o f multiplicative characters and give the explicit expression for some sample amplitudes. For a more detailed discussion of these theories, for elucidation o f the associated effective field theories, and for the

~r Work supported by the National Science Foundation under grant PHY82-09011. Address after August 23, 1988: Institute for Theoretical Physics, Princetonplein 5, P.O. Box 80 006, NL-3508 TA Utrecht, The Netherlands.

0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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Volume 214, number 1 PHYSICS LETTERS B 10 November 1988

details of the calculation ofp-adic integrals, we refer the reader to ref. [ 7 ]. We reserve a few comments on the adelic structure that can be associated with the p-adic theories for our concluding remarks.

We will insist that the p-adic string amplitudes are complex-valued functions so that we have the usual notion of spacetime. An alternative approach, where the spacetime is also valued in the field Qp ofp-adic numbers has been explored in refs. [ 1,2 ].

We start by giving three standard representations of real string scattering amplitudes, all of which are equiva- lent for real string theory. The most familiar form is the Koba-Nielsen form for the scattering of N strings, given by

fi04 i----FI4 B~N= dz, [Zi la ' i -Ll l - -z i [ a2'-I I-I I z i - zJ l "0-1 , (1) "= "= 4 ~ i < j < ~ N

where the integration variables are restricted to obey 1 > z4 > z5 >... > zN. There is another form, equivalent in the real string case to ( 1 ), which is obtained from the Koba-Nielsen form by changing the variables, z4 = x2, z5 = ZaX3, etc. The new variables xi all range in the unit interval on the real line. This is the Bardakgi-Ruegg form of the scattering amplitude. It reads

I IN--2 Bg R= I ] dx~x~ t~-I I-I I I -xsx j+ , . . . x j - , I a'/-' , (2)

i=2 2<~i<j<~N-- I 0

where a ( i ) = (kl +k2+.. .+k~) z - 1. In both the formulas (1) and (2) one must still sum over all noncyclic permutations, i.e., all compatible channels. There is a third formula which gives the full scattering amplitude (i.e. with this summation already carried out). It is best written in the Koba-Nielsen form:

A N = i ~ 4 d Z ` Izil . . . . ~ll__Z;[a2,-, I-I Iz;-zJl a°-' (3) • = 4~<i<j<~N

Strikingly, although the above formulas are equivalent in the real case, we will see that they lead to inequivalent p-adic theories!

For our purposes, the important feature to recognize is that the above three expressions for scattering ampli- tudes can be written in the form

~ d g ( D ) × H (multiplicative over D ) , characters D where the domain field D has been taken to be the real number field above. The measure d/ t (D) is the usual Mrbius invariant measure and represents the integration over the position of sources, and the last factor in- cludes absolute values of polynomials of positions on the world-sheet. Thus, to obtain the p-adic generalization of the amplitude, our prescription is simple: we change the domain field to the field of p-adic numbers, the measure to the appropriate Mrbius invariant measure on the p-adic field, and the combination of multiplicative characters to the multiplicative characters on the p-adic number field. However, to do this effectively, we note that we would like to extend the region of integration for each variable to be all of R (so we can easily replace this by ~p) while preserving the feature that the integrand only involves multiplicative characters. To do this in the real case, we just need include combinations of sign functions which vanish outside the desired interval, and have value 1 in the desired interval [ 9 ]. The p-adic sign function is a p-adic character. However, there exist two distinct (for our purposes ) quadratic extensions of the p-adics. Define e to be a nontrivial ( p - 1 ) th root of unity in ©p; then based on whether we extend by the square root of z= e or z = p or Cp, we obtain different p-adic sign functions [ 10]. It may seem that there are actually three possibilities, but this is not so, as the situations z=p and z= ep lead to the same amplitudes and theories because Cp and p can reverse their roles [ 10 ].

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Volume 214, number 1 PHYSICS LETTERS B 10 November 1988

We make one observation before writing down p-adic scattering amplitudes. Note that i f F ( x ) = IF(x) I and real for xe [0 ,1 ] , then we can use the sign function to write f ~ dxF(x ) - fRdx IF(x)1½[l_+sign(_+l) × sign (x) sign (1 - x ) ]. However, this rewriting of the integral is not unique. We can just as well use the com- bination ½ [ 1 _ sign( _+ 1 ) sign(x) sign( 1 - x ) ]G(x) for any "nice" function G(x) with the property G(x) = 1 for xe [0,1 ]. As far as the real string case is concerned, this freedom in the choice of the function G(x) is not important, but it will become relevant for the p-adic generalizations. Also, the simple procedure given above works nicely for the four-point function. However, additional ingredients will be needed to obtain similar expressions for higher-point functions. Finally in the real string case, we can use either + sign ( + 1 ) or - sign ( - 1 ) because they are identical, but in the p-adic case this will no longer be true. By requiring that the four-point function in any p-adic theory have the right pole structure, it is necessary to use the combination - s ign( - l ). In the p-adic case we will have - s i g n ~ ( - 1 ), which equals " - 1 " for the choice r=e , and which obeys -sign~( - l ) = - s ign , ( p - l ) for the choice r=p, ~p [ 10 ]. For a discussion of Mrbius invariance and various combinations of sign functions, one can refer to ref. [ 7 ].

Thus, we can write the real scattering amplitudes in forms which will be easy to generalize. The amplitudes AN, B~ TM , and B ~R given in ( 1 ), (2), and (3) above can be written as

A u = f f i dzi[zi l~"-l l l -zi] a2i-' 1~ [z,-zJ[ ~'~-1 , (4) i=4 4<~i<j<~N R

BKN N= i~=4dzi I z i l a " - l l l - z i l a2`-I I-I I z , - z J l " ' - t "= 4<~i<j<~N

N ×I-I ½ [ 1 - s i g n ( - 1) sign(zi) s ign(1-z i ) ] 1-I ½ [ l - s i g n ( - 1) s ign(zi -z j ) ] , (5)

i = 4 4<~i<j<~N

f N--2 [ J--l Iaij--1N B~ R= I-[ dz~ Izil ~(°-~ I~ 1--k~__ Zk 1--[ ½ [ 1 - s i g n ( - 1 ) s i g n ( z i ) s i g n ( 1 - z , ) ] . (6)

i=2 2 <~i<j<~N-- 1 i = 2

Since we must ultimately sum (5) and (6) over all noncyclic permutations to get the full, dual scattering ampli- tude, we can use either the appropriate integral given above or a noncyclic permutation thereof. In some cases in which we write down an N-point amplitude, we will take advantage of this freedom. Remember also that in eqs. (5) and (6), it is possible in addition to include an arbitrary number of functions of the form G(x) with the property G (x) - 1 for xe [ 0,1 ]. For simplicity, we have not listed these factors here, but we will exploit this freedom in our examples.

It is worth mentioning here that using the form (5) or the form (6) it is possible to include Chan-Paton charges in the open string case. Also, if one considers the scattering of arbitrary excitations of the string, the corresponding amplitudes will be of the same form, with some polarization factors standing in front. There is probably another way to include the scattering of arbitrary states of the string, which uses the amplitude in form AN. This other possibility has been worked out completely for the scattering of four arbitrary states of the string [ 11 ]. It exploits a set of remarkable identities which exist for certain linear combinations of three beta functions.

Now the prescription for writing down the p-adic generalization of the string theory is: replace R by Q,, replace I I by the p-adic norm I I p, and replace all the sign functions by the corresponding p-adic sign functions, making sure to examine both choices of the p-adic sign function.

The simplest case is to consider (4). It is straightforward but tedious to evaluate the amplitude explicitly. The N = 4 and the N= 5 cases can serve as examples:

A(40) = f dx Ixl~'a-~ I 1 - - X I ~ 24-1 , (7)

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Volume 214, number 1 PHYSICS LETTERS B 10 November 1988

A~P)= f d x d y [xl~"-I [ 1 - x l ~ ~'-' ly[~,'~-' [ 1 -ylT, ~ - ' [ x -y l~ 4~-~ . Qp

After carrying out the integrations, we have

A~p) - l - p 1 l - p 1 1 - p 1 p - 2 l_p--------~ + - - - - + - - - - + - - , p 1--pb p 1--p~ p P

( 1 - p ) / p ( 1 - p ) / p p - 2 ( 1 - p ) / p p - 2 p - 3 A~ p) = E + Z + - - - - ,

i(ij)(kl)] 1 __pao l__pakt P (ij)'" 1 --pa~j p p

(8)

(9)

(10)

where the double sum (pairs [ (ij), (k / ) ] ) is over the 15 possible (compatible) simultaneous, double dual channels, and the single sum ( ( i j ) ) is over the 10 possible single channels where three strings are emitted si- multaneously. The last term is the five-point contact term. This contact term is the most striking feature of the above amplitudes. It is a direct consequence of the p-adic time ordering of the emission times on the world- sheet. (The set ofp-adic numbers with given norm is not a set of measure zero!) In the real string case, one has only the single-channel contributions. From the two examples given above it is quite obvious what the general N-point p-adic string amplitude AN looks like. First one draws all the diagrams with N - 3 propagators and three- point vertices. Then one draws all the possible diagrams with N - 4 propagators and four-point vertices, etc., until just the contact term is left. Thus the effective field theory associated with this p-adic theory has a propa- gator proportional to 1 / ( 1 --pa), and the p-adic theory is inherently nonlocal.

Next, we consider the class of p-adic theories based on the unsummed Koba-Nielsen formula (5). These amplitudes include the characters with the sign function:

f, 4 B~N(P)= dzi Izi p a " - l l l - z i l ~ 2 ' - I I ] Iz~-zJl~ "-~ "= 4<~i<j<~N

Qp

N ×I-I ½ [ 1 - s i g n , ( - 1) signdze) sign~(1-z~)] 1-I ½ [ 1 - s i g n a l - I ) s ign~(z , - z j ) ] , (11)

i=4 4<~i<j<~N

where for simplicity we have omitted p-adic generalizations of the function G(z4, ..., ZN). TO examine the theory we compute the four-point and five-point functions,

n KN(p) = f dx Ixl~,-' I 1 - x l ~ - ' ½ [ 1 - s ign , ( - 1 ) sign~ (x) sign~( 1 - x ) ] . G ( x ) , ( 12) Qp

B KN(p) = f d x d y Ixl~;4-~ I 1 - x l ~ 2'-~ [yl~"-I I I - y l ~ 25-1 I x - y l ~ "~-~

× ½ [ 1 - signal - 1 ) s ign~(x-y) ] ½ [ 1 - sign~( - 1 ) s igndx) signal 1 - x ) ]

× ½ [ 1 - signal - 1 ) sign~(y) signal 1 - y ) ].G(x, y ) , (13)

where the functions G(x) and G(x, y) are as yet unspecified. Everywhere, we have to distinguish between the cases z= e and z =p or p~, because they correspond to independent quadratic extensions of the p-adic field Qp. For the simplest choice, G (x) - 1, the four-point function when explicitly evaluated reads (we display the cases z=E and the case z=p or z=pE side by side)

B~N(p)_ lp - -1 ( 1 1 ) , l l - - p ( 1 l_~pb) uRN(p) . . . . 1 + + -- - - - "u 4rmeP'P-- 2 p ~ " 2 p s inh(a lnp) + s inh (b lnp )

The difference between the two situations is the absence or presence of a contact term. As before, the theory is nonlocal. The five-point function for the z= e case and G(x, y) = 1,

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Volume 2 14, number 1 PHYSICS LETTERS B 10 November 1988

1 1 +paz3

To obtain the full amplitude one must still sum over all noncyclic permutations. From the expression for the amplitude, it is clear that the theory can be described in terms of an effective field theory of two mutually interacting scalar fields. Writing down this effective field theory is equivalent to giving the diagrammatic rules for writing down the amplitudes. The case of the five-point function in the remaining situation, ~=p, can be handled similarly.

Lastly, we describe the remaining set of theories which start from the Bardakci-Ruegg formula (6), for which a nice theory exists. We have

N-2

B;RR’P’ - - ,Q2 dz, ]ZiJiCi)-’

N-2

x,r2 f [l-sign,(-1) sign,(zi) sign,(l-zi)].G(zi). (14)

In the defining formula ( 14) we have also included factors G(.z,), which are not the most general additional factors one might include, but which will suffice for our purposes. (Remember that as real functions, the addi- tional factors must equal unity in the region of integration ZiE [ 0, 1 ] . ) If we define Q( zi) = i[ 1 - sign,( - 1) xsign,(zi) sign,(l-zi)].G(z,),thenwecanwrite

The restriction to the additional factors of the form G( z2) sG(z,) . . . . .G(z~_~) now has a clear interpretation, namely that the function Q( Zi) can be associated with the measure dzi. Different values of G are distinguished by such features as how they treat the set 1 zi I p= 1. In the real case, this set has measure zero; in the padic case, the set I Zi I p= 1 has nonzero measure, and so depending on how the function G( z,) treats that set, we will obtain a different p-adic theory. Nonetheless, much of the structure of the amplitudes can be obtained independently of our choice of G (zi ) .

To analyze the structure of the N-point amplitude, we must study the behavior of the integrand for the various cases that I zi I p < 1, I z, I p = 1, and I zj I p > 1. We note first that for I zi I p> 1, the function Q( zi) = 0, regardless of our choice of G( zi) . The contribution to the amplitude from the region of integration in which all the variables obey I Zi I pc 1 is simply the product of N- 3 independent one-variable integrals. Furthermore, there is a partial separation of the integrals whenever at least some of the I zi I p < 1.

For a generic value of O(z), we define the integrals

H(a)= j. dz ]z];-‘Q(z), J4(u23)= j- dz I l-~]~~-‘Q(z) , I~I,lC I I.4p=l

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Volume 214, number 1

J5 (a23, a34, a24) = f 1:2 b,= Iz3 Iv= I

PHYSICS LETTERS B

d z l l . I~-'~-I l -- 'a2 I1' l1 --Z3 1~34- l1 _ z 2 z 3 [ ~ 2 4 - 1 ~ , ~ ( z 2 ) O ( z 3 ) ,

10 November 1988

f N--2 j--I ] aU-I

JN(atZ.N--,1)= 1--I (dzig2(z,)) I ] 1--]--[ ' I z 2 l t . . . . . . Is . '~ 2 I t ' = 1 i = 2 2<.i<j<~N--I

where by a [2,N-[] we mean the set of all pairs ajj in which 2 ~< i < j - 1 ~< N - 1. We introduce a formal product denoted by "o" by which we can combine the above integrals. This product, as we will shortly see, implicitly describes the structure of the p-adic Bardakgi-Ruegg string theory. Using " X " to denote the ordinary multipli- cation of complex numbers, we define "o" by

H(a) oH(b) = H ( a ) x H ( b ) , H(a)°JN(btk.q) = H ( a ) XJN(btk.q ) ,

J,v(atkj]) oH(b) =JN(atk.n) XH(b) , Jn(a~k.n) °JN(atc,,j) =Jg+N--3(atk.,,.,] ) • ( 1 5)

Thus the o product is a kind of convolution. The o product is associative, but it is not commutative; the order of the/-/'s and JN'S is important; for example, HOJMOJNV~JMoHOJN.

In terms of this product, the p-adic Bardak¢i-Ruegg N-point amplitude can be simply represented. After sim- plifying the relevant integrals, one finds

B~, R = [H(a (2) ) -I- J4 (a23) ]o [H(a(3 ) ) + J4 (a34) ] . . . . . [ n ( a ( N - 2 ) ) +J4(aN--2,N--~ ) ] • (16)

Thus, for example, B~R=H(a(2))H(a(3))+H(a(2))J4(a34)+Ja(a23)H(a(3))+Jh(a23, a34 , a24). Of course, one still must sum over noncyclic permutations in order to recover the full dual amplitude.

It is apparent from the formula (16) that to compute the N-point amplitude, in addition to knowing the integrals for the amplitude with fewer than N tachyons, we need to know one additional integral, JN(at2.N J ). This integral corresponds to the new N-point contact interaction that (of course) first appears in the N-point amplitude. The formulas in ( 15 ) already indicate the generic structure of the effective field theory for this string theory. One can either interpret the expression H + J4 as a kind of propagator, with the o product encoding the (nonlocal) interactions among the associated fields; or one can view H and J4 as two different kinds of propa- gators representing two different fields, with the field associated with the propagator H having only cubic inter- actions (but with both kinds of fields) while the field associated with J4 has (arbitrarily) high-order self- interactions. While this is the generic structure, in particular cases the structure may simplify.

To go further, we must specify the function g2, and the value of z in the expression sign~. Various choices are examined in our paper in ref. [ 7 ], and here we describe only the one for which the theory is the simplest in form. We choose the function G (z) = ½ [ 1 - sign~ ( - z) ], and choose z = ~. The resulting function

g2(z) = ½ [ I -s ign, ( - 1 ) sign, (z) sign,( 1 - z ) ]. ½ [ 1 - s i g n , ( - z ) ]

vanishes for I zlp~> 1. The region of integration for each variable reduces to the set [Zip< 1. Since for the case at hand the function G (z) = 0 for I zlp = 1, we see explicitly from the defining equation for

JN that with our current choice for G(z), JN = 0 identically for all N. On the other hand, H is unchanged from the previous case. Thus we find

, , - 1 ( 1 1 ) , H~(a ) = ~ - - ~ p--g-+--1 + ~-Z-1- 1 J~NG(a[2'N--1])=O"

Hence all the zi integrals are completely decoupled. The four- and five-point functions are

BR(p) B4c =Ha(a(2) ) , u ~ ( P ) = H ~ ( a ( 2 ) ) X H ~ ( a ( 3 ) ) a.., 5G

More generally, we have the physically interesting result that the N-point function is simply expressible in terms of the four-point function,

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Volume 214, number 1 PHYSICS LETTERS B 10 November 1988

1 ) 1~, BR (p) __ "UNG -- 1=211 Ba4R~P)[a(I)] = l=2U a'(l) __p- -a( l ) '

where the channel var iable is a ( l ) = ~.~--,m=l ( ~ 1 k l ] ~ 2_ 1 . To calculate the N-point ampl i tude in this theory, we only need to know how to calculate the four-point function. It is obvious that the corresponding field theory will be very s imple (besides being nonlocal ) , and will only have cubic self-interactions. We can interpret the combina- t ion ( 1 - p ~ ) - t _ ( 1 + p a ) - l as being a propagator .

In this letter, we have classified the p-adic string theories. The most impor tan t lesson o f this exercise is that the connect ion between p-adic and real string theories is in no sense canonical , but ra ther there are many differ- ent possible p-adic theories. However , we have also given a prescr ipt ion for de te rmining the var ious possibil i t ies, and we have indica ted the nonlocal effective scalar field theories in terms o f which they can be described. Ob- viously, these theories may be interest ing in themselves; al ternatively, they may be examined in the context of adeles, which give addi t ional in format ion and structure to the collection o f all p-adic fields and the real number field. This possible connect ion is, at present, poor ly unders tood. In order to unders tand the connect ion o f the theories, one must devote more s tudy to adeles. Perhaps the original adelic relat ion is a special case o f some relat ion true for general p-adic string theories. The full impl ica t ions await more careful examinat ion, as physi- cists become famil iar with the necessary concepts. Perhaps we have an oppor tuni ty to learn about a nonlocal phase o f string theory.

We thank Hikaru Kawai for useful discussions. We also thank the referee for advis ing us o f the proper chro- nological order for the references.

References

[ 1 ] l.V. Volovich, Class. Quantum Grav. 4 (1987) 83; CERN preprint CERN-TH.4781/87 (1987). [2] B. Grossman, Phys. Lett. B 197 (1987) 101. [3] P.G.O. Freund and M. Olsen, Phys. Lett. B 199 (1987) 186. [ 4 ] P.G.O. Freund and E. Witten, Phys. Lett. B 199 (1987) 191. [ 5 ] P. Frampton and Y. Okada, Phys. Rev. Lett. 60 ( 1988 ) 484; Phys. Rev. D 37 ( 1988 ) 3077. [6] N. Koblitz, p-adic numbers, p-adic analysis, and zeta functions (Springer, Berlin, 1984). [7] Z. Hlousek and D. Spector, Cornell preprint CLNS 88/832 (1988). [8] E. Marinari and G. Parisi, Phys. Lett. B 203 (1988) 52. [9] L. Brekke et al., Nucl. Phys. B 302 (1988) 765.

[ 10 ]I.M. Gel'fand, M.I. Graev and I.I. Pyatetskii-Shapiro, Representation theory and automorphic functions (Saunders, London, 1986 ). [ 11 ] Z. Hlousek, Cornell preprint, Phys. Rev. D, to be published.

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