Volume 209, number 2,3 PHYSICS LETTERS B 4 August 1988

LAGRANGIAN FORMULATION OF OPEN AND CLOSED p-ADIC STRINGS

R.B. ZHANG Department of Physics, University of Tasmania, Hobart, Tasmania 7001, Australia

Received 11 April 1988

p-adic field theories, analogous to the two-dimensional field theories for ordinary strings, are proposed for both open and closed p-adic strings. Using them we compute the tachyonic scattering amplitudes.

1. Introduction

The works of Freund and Olson [ 1 ] and Volovich [2] have raised considerable interest in non-archi- medean dynamical systems [ 3-7 ]. A lot of progress has been made, especially in the study of non-archi- medean string theories [4-7] . It appears quite rea- sonable that the p-adic strings corresponding to individual prime numbers may be regarded as build- ing blocks of the conventional string [4,5 ]. One ex- pects that the investigation on these building blocks wll lead to a deeper understanding of the underlying structure of string theory and provide new insights into other related areas, such as conformal field the- ory [ 6 ], etc.

In the p-adic string theory constructed by Freund and Olson [ 1 ] quantum amplitudes still take values in the complex number field, but the world-sheet now is the p-adic field (for the open string) or its qua- dratic extensions (for the closed string). It possesses the intriguing property that the tachyonic amplitudes exhibit only one real tachyonic mass pole, all the higher modes of the conventional string do not inter- fere with the tachyon. This property, being more field theoretical than stringy, inspired the construction of a non-local field theory to generate the p-adic string amplitudes [4,7 ]. However, here we will try a differ- ent approach of investigation.

We will propose a lagrangian formulation for building quantum field theories on p-adic fields. It is prescribed in the construction of field theories for both open and closed p-adic strings. These p-adic field

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theories, being parallel to the a-models for ordinary strings, render it possible to compute scattering am- plitudes systematically as vacuum expectation values of products of vertex operators. We will consider the open string first.

2. Open string

We start by explaining some definitions. Let ~i be a set of complex-valued functions defined on Qp:

~i: Q p ~ C .

It is possible that these functions are discretely val- ued, but they can be continuous in the sense that for x,yEQp, such t h a t l x - y [ =p-M<f i , we have IOi(x) _ ~ i ( y ) l <fi (here we did not give different nota- tions for the non-archimedean and archimedean norms, but this should not cause any confusion). The Fourier transform of a function is defined by

X(u) = ~ dx(b(x) exp(i2z~ux) . ( 1 ) Qp

Here dx is the real-valued translation invariant mea- sure defined in ref. [8], and the integration should be understood in terms of a Riemann sum. For ex- ample, the integration of a complex-valued function f (x) is

f d x f ( x ) = k L f dxf(x) Qp Ix[ =p-k

and

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Volume 209, number 2,3 PHYSICS LETTERS B 4 August 1988

f d x f ( x ) = lim p - ~ o - * I x l p k

X ~f(p~(ao+a,p+...+au_lpU--~)] ai

with aoe [ 1,2 ..... p - 1 ] and a,e [0, 1, 2 .... , p - 1 ], for i>~l.

In ( 1 ), u belongs to Qp, and plays the role of mo- mentum. It is not difficult to convince onself that the following relation holds:

0 ( x ) = ~ duX(u) exp ( - i 2zcux ) . (2)

Qo

As in previous works on p-adic strings [ 1,4,5,7 ], we will take the world-sheet of the open string to be Qp with a parametr izat ion x. Let the 0r (x) be the string coordinates in spacetime, which are real-val- ued functions. To construct a one-dimensional field theory for them, we Fourier t ransform them into the m o m e n t u m space by (1). Now we propose the fol- lowing m o m e n t u m space action:

S = f l - I I duX(-u) lu lX(u) , 0 < f l e ~ . (3) Q p

Note that in (3) , the "kinetic operator" is l ul, the momen tum, instead of m o m e n t u m squared as we would expect naively. As a mat ter of fact, there is no basic principle to prevent us from utilizing other ki- netic operators, but (3) is the right action to produce the tachyonic ampli tudes given in the literature.

The quantization of this theory can be carried out by using the path-integral method. We define the par- tition function by

Z = j [dX] e x p ( - S ) . (4)

The measure, [dX], though seems to be a highly in- volved object, in fact is just the same as the familiar functional integration measure in ordinary quantum field theory, because the X's are simply complex-val- ued functions. Functional differentiation can also be defined in the usual way. Now the propagator of the fields can be obtained:

( X'(u)X'(v) ) =fl~O~(u-v) /lul . (5)

The corresponding propagator in the configuration space then is given by

C 3 (x, y) = fl J du exp [ - i2 z~u ( x - y) ] / I u I -

Q p

There exists a divergence at I u l ~ 0. Dealing with it we introduce a regularization by defining

As(x,y)=fl J duexp[-i2zcu(x-y)] lul s ~, Qp

s ~ 0 . (6)

3s(x, y) can be easily evaluated by using p-adic F- funct ions:

As(x, y)=fl( Ix-y l - 'F(s) ,

F ( s ) - - ( l - p ~ - ' ) / ( 1 - p " ) . (7)

The 1 / s divergence at s ~ 0, sharing the same origin with the infrared divergence of the ordinary string coordinate propagator (in archimedean field theory all the two-dimensional bosonic massless theories have the same divergence), can simply be ignored. Upon choosing fl to be p lnp / (p- 1 ) one has

A(x-y) = - l n I x - y l . (8)

Next we introduce the tachyonic vertex operator,

V(k) = I d x e x p [ i k 0 ( x ) ] , (9) Qv

in analogy to that in ordinary string theory. Then am- plitudes can be constructed as expectation values of the products of vertex operators. To avoid diver- gences arising from coincident limits of propagators, we should "normal order" the vertex operator so that no self-contractions are allowed. Now we spell out explicitly the N-tachyon amplitude:

A(kl ..... kx)

= j [dY] V(kl )...V(kN) exp( - S ) , ( 10)

with 52k, = 0, k2 = 2. Elementary manipulat ions lead to

A(k,,. . . ,kN)= f [l dx, Iq lx,--xjI k'ki. (11) • t < . j

Q p

As noted by Freund and Olson [ 1 ], ( 11 ) is subject to an SL(2; Qp) gauge symmetry. Hence we can ar- bitrarily fix three of the N integration parameters, in- corporating a proper jacobian. A convenient choice is

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Volume 209, number 2,3 PHYSICS LETTERS B 4 August 1988

J = I (xl - x 2 ) (x, - x 3 ) (x2 - x 3 ) l ,

Ix, l=+oo, x 2 = l , x3=O. (12)

In particular the four-point amplitude reduces to

A= J dx I I -xlk2k' lxl k3k4, (13) Qo

which has been evaluated many times with the result

A= F[ ( 1 - p - ' ~ ¢ ° - l ) / ( 1 - p " ¢ ° ) , (14) ~=s,t,u

where s, t and u are the Mandelstam variables and o~ ( t ) = 1 + t/2, etc. Explicit formulae for more-point amplitudes are also given in ref. [4], we will not re- peat the calculations here.

3. Closed string

We shall now discuss the closed string. As pointed out in ref. [ 1 ], the world-sheet in this case is a qua- dratic extension of the field Qp. A standard theorem says that three distinct quadratic extensions of Qp ex- ist, namely Qp(~/a), Qp ( , ,~ ) and Qp(,,f@), with e the ( p - 1 ) - t h root of unity. (More details can be found in any p-adic analysis text book, for example in ref. [ 9 ]. ) For the sake of convenience, we will de- fine our field theory o n Qp(v/~); other cases can be realized by straightforward generalizations.

Let us define the following momentum space action:

S=y' f d u X ( - u ) [ u a l X ( u ) , (15) Qp(-j¢)

with u=u, + , f i u2 Q . The X, being complex-valued functions, are the Fourier transforms of the fields ¢:

X ( u ) = | d x 0 ( x ) exp[in(ug?+ax)] . Qp (-,/e)

The regularized propagator for the fields ~; can now be defined as

A~(x,y)=y f duexp{-i~r[u(97_37) / Qp(w e)

+ a ( x - y ) ] } lual s-' (16)

Then we have

d s ( x , Y ) = Y l 4 ( x - y ) ( . ~ - N ) l - ~ F ( s ) ~ ( s ) , (17)

where/ ' ( s ) = ( 1 + p ~ - ~ ) / ( 1 + p - ' ) . Ignoring the 1 / s d ivergence in ( 17 ) and sett ing 7 = 2 in p~ ( 1 - p - 2 ) results in

3(x, y) = - l n l ( x - y ) (97-Y) I • (18)

Using the tachyon vertex operator

V(k)= j dxexp[ ik¢(x, 9?)], k 2 - 2 , (19) Qp(x/e)

we can compute the N-point amplitude as we have done in (10), obtaining

A(kl . . . . . kN)

= J I~dx; I~ I ( x , - x j ) ( & - x j ) f *`k'. (20) i i<j Qp(v"e)

As in the open string case, (20) possesses an SL(2; Qp(x/a) ) gauge invariance. We fix the gauge by setting

IXlg?,l=+oo, X2=1, x3=O ,

and choosing the jacobian

J = I (x, --X2) (-~1 --9?2) (X, --X3) (9?1 --'~3)

× (x2 - x 3 ) (9?2 -9?3)[ - (21)

It is then a simple matter to get the familiar four-point amplitude

Z = f d x l ( 1 - x ) ( 1 - 9 ? ) Ik2k41xg?l k3k' Qp(x/e)

= I~ ( l - - q - " ( ° - l ) / ( 1 - q " ( ¢ ) ) , q=p2. ~ s , t , u

(22)

We conclude the paper with some remarks. It is possible to construct vertex operators for higher mass excitations of the bosonic strings in analogy to those in ordinary string theory, and to incorporate super- symmetry into the present formulation. We will re- port the results in a separate publication [ I 0 ]. The formulation we have proposed here is quite general, and may be applied to the construction of other kinds of p-adic field theories. For example, we can easily introduce a mass term and an interaction into ( 15 ) to obtain

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Volume 209, number 2,3 PHYSICS LETTERS B 4 August 1988

S:y-' f du(X(-u)(lu~l+mg)X(u) Qp (v, er")

+ ~.~ du, du2 X(RI)X(u2)X(u)

xX( -u~ -u2 -u ) ) .

T h e n the ques t ion is whe the r these theor ies have any-

th ing to do wi th the real world , apar t f r o m the i r ap-

p l ica t ions to strings. Also some under ly ing s y m m e t r y

pr incip les r e m a i n to be u n c o v e r e d which should ori-

enta te the cons t ruc t ion o f such theories .

References

[ 1 ] P.G.O. Freund and M. Olson, Phys. Len. B 199 ( 1987 ) 186.

[2] l.V. Volovich, Class. Quantum. Grav. 4 (1987) L83. [ 3 ] Y. Meurice, The classical harmonic oscillator on Galois and

p-adic fields, preprint ANL-HEP-PR- 1987-114; B. Grossmann, Phys. Lett. B 197 (1987) 101; I.Ya. Aref'va, B.G. Dragovic and I.V. Volovich, Phys. Lett. B 200 (1988) 512.

[4] P.H. Frampton and Y. Okada, The p-adic string N-point function, North Carolina preprint IFP-302-UNC (1987); Effective scalar field theory ofp-adic string, North Carolina preprint IFP-303-UNC ( 1988 ).

[ 5 ] P.G.O. Freund and E. Witten, Phys. Lett. B 199 (1987) 191. [6] J.L. Gervais, p-adic analyticity and Virasoro algebras for

conformal field theories in more than two dimensions, Ecole Normale preprint LPTENS 87/32 (1987).

[7] L. Brekke et al., Non-archimedean string dynamics, Chi- cago preprint EFI 87-101 (1987).

[8] I.M. Gel'fan& M.I. Graev and I.I. Piateskii-Shapiro, Rep- resentation theory and automorphic functions (Saunders, Philadephia, PA, 1969).

[9] N. Koblitz, p-adic numbers, p-adic analysis, and zeta func- tions (Springer, Berlin, 1984).

[ 10] R.B. Zhang, in preparation.

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