Volume 213, number 3 PHYSICS LETTERS B 27 October 1988

ON ADELIC FORMULAS FOR THE p-ADIC STRING

Paul H. FRAMPTON, Yasuhiro OKADA and Marcelo R. UBRIACO Institute of Field Physics. Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599, USA

Received 15 June 1988

For the p-adic bosonic string, a simple regularization procedure is applied to the four-particle adelic formula. Arguments that have been made against the validity of the five-particle adelic formula are criticised.

A completely fresh approach to string theory has been started within the last year [ 1-3 ]. By trading in the normal real variable x~ E which parametrizes the distance along an open string for a p-adic variable x~ Q~, certain very intriguing relationships have been obtained for the string scattering amplitudes. Most intriguing of all is surely the adelic formula [2 ] which states that the infinite product over all p-adic ampli- tudes, including the real amplitude itself, gives a sim- ple result. This adelic formula thus might be taken to suggest that the p-adic string is a "building block" of the usual string. Such a suggestion is of sufficient po- tential importance that is worth studying further the nature of the adelic formula. We shall discuss only the simplest bosonic string.

Let us begin with the four-particle string amplitude

A4 = B ( - ol(s), - o ~ ( t ) ) + B ( - c ~ ( t ) , - o L ( u ) )

+ B ( - o ~ ( u ) , -o~(s ) ) , (1)

with o~(s )=l+½s, c ~ ( s ) + e ~ ( t ) + ~ ( u ) = - I and B (a, b) is the Euler Beta function. The p-adic version A~ ~'~ may be written in the triple product form

A~4 p)= [I ( 1 - p - " ~ ° ' - ' ~ Q=~.'." l__p~Q ) j . (2)

If we overlook questions of convergence, it is sim- ple to derive from eqs. ( 1 ) and (2) that

A4 H A ] P ) = 1 , (3) P

which is the adelic formula.

For convergence of the infinite product

[-[ (1 - p - - - ) - ' = ~ ( z ) (4) P

one needs Re z> 1. Thus, in eq. (2) one would need simultaneously Re ol (Q) > 0 and Re 0¢ (Q) < - 1, ob- viously mutually incompatible. Thus, the product in eq. (3) converges nowhere and a regularization is necessary. Such a successful regularization should firstly render the infinite product convergent and should secondly be removable to arrive at eq. ( 1 ).

Such a regularization is the following: Replace the prime number p by p '- and finally let x--, 1 by analytic continuation. The requirement that numerator and denominator converge then become, respectively (we may take x real),

o l ( Q ) > l / x - 1 ( x > 0 ) , (5a)

c ~ ( Q ) < l / x - I ( x < 0 ) , (5b)

o t ( Q ) < - l / x ( x > 0 ) , (6a)

c~(Q) > - 1 / x ( x < 0 ) , (6b)

Eqs. (5) and (6) are incompatible for all x < 2 . For x > 2 they become compatible for some c~(Q)_ -½. For x > 3 they permit c~(x) = - ] consistent with the mass-shell condition c~(s) +c~(t) + c~(u) = - 1. As x becomes arbitrarily large, all 0 > c ~ ( Q ) > - i are permitted.

This is the simplest regularization we have found

260 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 213, number 3 PHYSICS LETTERS B 27 October 1988

and can be used to justify the adelic formula for four particles ~1.

On the adelic formula for five particles, there has been some adverse comments which we wish to cri- ticise. For example, Marinari and Parisi [ 5 ] present the following analysis. The five-point p-adic scatter- ing amplitude is

A~") = ( l - p - I ) 2

1 1 x Z

',o,k/l ( 1 - p ~ Q (1 _p.k~) 15 terms

- ( I - - P - ~ ) ( 2 - - P - ' ) Z I 0 lerms

+ ( 2 - p - ~ ) ( 3 - p -~) .

One may now define

~ " ' =A ~") I ] ( 1 _p -o )

10 terms

1 - -p~"

(7)

X ]-I (1--pC"~+c~'~) - l ( 1 - p - ~ ) - 5 (8) {ij, ik}

30 terms

One may now check that the infinite product over ~ P ) , so defined, is convergent in a finite neighbor- hood of the symmetric point oLij=- ½, for all i, j. Hence one may apparently deduce that

I~ A~ p) ~ ( ( I ) - 5 = 0 . (9) P

in a finite neighborhood and thus that the left-hand side of eq. (9) vanishes everywhere.

But this argument is dangerous and potentially misleading. Consider the much simpler case where the p-adic amplitude would be represented by

P - ' (10) B(P)(°t) = 1 1 - p " '

where a is a complex variable. Let us exhibit now a danger in studying the infinite product over B (p) ( a ) as a function of a. I f Re a < 0 we may expand

B(P) (a ) = ( 1 - p - I ) B ( P ) ( o ~ ) , ( 11 )

/~(r) (o~) = 1 + p - ~ +"+ .... (12)

~ In ref. 14], the authors derive an alternative four-particle adelic formula which, however, fails to maintain the s-t-u symmetry of the amplitude.

whereupon

]-I/~(p~ ( a ) (13) P

is convergent, and hence

H B (P) ( ° ~ ) ~ ( 1 ) - - 1 = 0 . (14) P

On the other hand, if Re c~ > 0 we expand

B ( P ) ( a ) = I + p - ' - ~ ( 1 + p - a + . . . ) , (15)

and in this case

1-I B ( p ) ( a ) (16) P

is, by itself convergent. In other words, there exists a boundary at Re a = 0

in the function defined in eq. ( l 0) such that an infi- nite product converges for Re a > 0 but not for Re ce < 0. This is an example of "branching" where an infinite product has different analytic continua- tion for different values of its complex variable. The same phenomenon may undoubtedly occur for A ~P) defined by eq. (7).

A related, though more careful, argument about the five-particle adelic formula is provided in Brekke et al. [ 6 ] who start from a particular kinematic region where the adelic formula converges. However, other kinematic regions may lead to different analytic con- tinuations due to the branching phenomenon dis- cussed above.

Any conclusion that the adelic formula

As l-[ As(P' = 1 , ( 1 7 ) P

first written explicitly in ref. [ 3 ], cannot be valid is therefore premature since the left-hand side of eq. (17) is intrinsically ambiguous and its uniqueness would require some principle to choose the appropri- ate prescription.

Elsewhere, adelic formulas for generalized four- particle amplitudes ofp-adic strings are discussed ] 7 ].

This work was supported in part by the US De- partment of Energy under Grant DE-FG05-85ER- 40219.

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Volume 213, number 3 PHYSICS LETTERS B 27 October 1988

References

[ 1 ] I.V. Volovich, Class. Quant. Grav. 4 ( 1987 ) L 83. [2] B. Grossman, Phys. Lett. B 197 (1987) 101;

P.G.O. Freund and M. Olson, Phys. Lett B 199 (1987) 186. [2] P.G.O. Freund and E. Witten, Phys. Lett. B 199 (1987) 191. [3] P.H. Frampton and Y. Okada, Phys. Rev. Lett. 60 (1988)

484.

[4] I.Ya. Aref'eva, B.G. Dragovic and I.V. Volovich, Steklov Mathematical Institute, Moscow, preprint IF-13/88 (March 1988).

[5] E. Marinari and G. Parisi, Phys. Lett. B 203 (1988) 52. [6] L. Brekke, P.G.O. Freund, M. Olson and E. Witten, Nucl.

Phys. B 302 (1988) 765. [7] P.H. Frampton, Y. Okada and M.R. Ubriaco, University of

North Carolina at Chapel Hill Report No. IFP-319-UNC (1988).

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