aa r X i v : . [ h e p - t h ] N ov Nonlocal Dynamics of p -Adic Strings Branko DragovichInstitute of PhysicsPregrevica 118, 11080 Zemun, Belgrade, Serbia
Abstract
We consider the construction of Lagrangians that might be suitablefor describing the entire p -adic sector of an adelic open scalar string.These Lagrangians are constructed using the Lagrangian for p -adicstrings with an arbitrary prime number p . They contain space-timenonlocality because of the d’Alembertian in argument of the Riemannzeta function. We present a brief review and some new results. Nonlocal field theory with an infinite number of derivatives has recently at-tracted much attention. It is mainly based on ordinary and also on p -adicstring theory, which emerged in 1987 [1]. Various kinds of p -adic stringshave been considered, but the most interesting are strings whose worldsheetis p -adic while all other properties are described by real and complex num-bers. Four-point scattering amplitudes of open scalar ordinary and p -adicstrings are connected at the tree level by their product, which is a constant.Ordinary and p -adic strings are treated on an equal footing in this product(see, e.g. [2, 3] for a review). Some other p -adic structures have also beeninvestigated and p -adic mathematical physics was established (see [4] for arecent review).Unlike for ordinary strings, there is an effective nonlocal field theory forthe open scalar p -adic strings with a Lagrangian [5, 6] describing four-pointscattering amplitudes and all higher ones at the tree-level. It is worth notingthat this Lagrangian does not contain p -adic numbers explicitly, but only the1rime number p which can be regarded as either a real or a p -adic parameter.Because this Lagrangian is simple and exact at the tree level, it has beenessentially used in the last decade and many aspects of p -adic string dynamicshave been considered, compared with the dynamics of ordinary strings, andapplied to nonlocal cosmology (see, e.g. [7, 8, 9, 10, 11] and the referencestherein).This paper contains a review and some new results related to constructinga Lagrangian with the Riemann zeta function for the entire p -adic sectorof an open scalar string. Requiring of the Riemann zeta function in theLagrangian is motivated by the fact that it appears in the product over p of all four-point p -adic scalar string amplitudes. In constructing possibleLagrangians, we start from the Lagrangian for a single p -adic open scalarstring. An interesting approach to a field theory and cosmology based on theRiemann zeta function was proposed in [12]. p -Adic Open Scalar Strings The p -adic string theory started analogously to ordinary string theory withscattering amplitudes. Let v ∈ V = {∞ , , , ..., p, ... } . The crossing sym-metric Veneziano amplitude for scattering of two open scalar strings is definedby the Gel’fand-Graev-Tate beta function A v ( a, b ) = g v Z Q v | x | a − v | − x | b − v d v x , (1)where Q p is the p -adic number field and a = − α ( s ) = − s − , b = − α ( t )and c = − α ( u ) are complex-valued kinematic variables with the condition a + b + c = 1. We note that the variable x in the integrands is relatedto the string worldsheet: the worldsheets of ordinary and p -adic strings arerespectively treated by real and p -adic numbers (see, e.g. [2, 3] and [13] forthe basic properties of p -adic numbers and their functions). Hence p -adicstrings differ from ordinary strings only by the p -adic treatment only of theworldsheet. Integrating in (1), one obtains A ∞ ( a, b ) = g ∞ ζ (1 − a ) ζ ( a ) ζ (1 − b ) ζ ( b ) ζ (1 − c ) ζ ( c ) , (2) A p ( a, b ) = g p − p a − − p − a − p b − − p − b − p c − − p − c , (3)2here ζ is the Riemann zeta function. Expression (2) is for the ordinary caseand (3) is for the p -adic case.The Riemann zeta function ζ ( s ) = + ∞ X n =1 n s = Y p − p − s , s = σ + iτ , σ > , (4)has an analytic continuation to the entire complex- s plane excluding the point s = 1, where it has a simple pole with unit residue. Taking the product of p -adic string amplitudes (3) over p and using (4), we obtain (see, e.g. [20]) Y v A v ( a, b ) = Y v g v = const . (5)The product of p -adic amplitudes in (5) diverges [14], but it converges afteran appropriate regularization. Requiring that amplitude product (5) be finiteimplies that the product of coupling constants is finite, i.e. g ∞ Q p g p = const. There are three interesting possibilities: (i) g p = 1, (ii) g p = p p − , whichgives Q p g p = ζ (2), and (iii) g p = | mn | p , where m and n are any two nonzerointegers, and this gives g ∞ Q p g p = | mn | ∞ Q p | mn | p = 1.It follows from (5) that the ordinary Veneziano amplitude, which is a spe-cial function, can be expressed as the product of all inverse p -adic counter-parts, which are elementary functions. This is a consequence of the Gel’fand-Graev-Tate beta functions and is not a general property of string scatteringamplitudes. In the general case, the string amplitude product is a functionof kinematic variables.Another interpretation of expression (5) is related to an adelic string. Butan adelic string should have an adelic worldsheet. A scattering amplitudeof two open scalar strings with their adelic worldsheets has not yet beenobtained. Therefore, the concept of an adelic string with an adelic worldsheetis not well founded and remains questionable. But p -adic strings with a p -adic worldsheet are well defined, and the string amplitude product for openscalar strings has a useful meaning.The exact tree-level Lagrangian of the effective scalar field ϕ , which de-scribes the open p -adic string tachyon, is [5, 6] L p = m D g p p p − h − ϕ p − ✷ m ϕ + 1 p + 1 ϕ p +1 i , (6)3here p is a prime, ✷ = − ∂ t + ∇ is the D -dimensional d’Alembertian.The corresponding equation of motion for (6) has been investigated by manyauthors (see, e.g. [9] and the references therein).We now want to consider construction of Lagrangians that can be usedto describe entire p -adic sector of an open scalar string. In particular, anappropriate such Lagrangian should describe the scattering amplitude, whichcontains the Riemann zeta function. Consequently, this Lagrangian mustcontain the Riemann zeta function with the d’Alembertian in its argument.We should therefore seek possible constructions of a Lagrangian that containsthe Riemann zeta function and is closely related to p -adic Lagrangian (6).There are additive and multiplicative approaches; we mainly consider theadditive approach below. The prime number p in (6) can be replaced by any natural number n ≥ p replaced by n ∈ N . Thecorresponding sum of all Lagrangians L n is L = + ∞ X n =1 C n L n = m D + ∞ X n =1 C n g n n n − h − φ n − ✷ m φ + 1 n + 1 φ n +1 i , (7)whose concrete form depends on the choice of the coefficients C n and couplingconstants g n . We set C n g n n n − D n , n = 1 , , ... . The following simple cases lead to the Riemann zeta function: D n =1 , D n = ( − n − , D n = n + 1 , D n = µ ( n ) , D n = − µ ( n )( n + 1) , and D n =( − n − ( n + 1), where µ ( n ) is the M¨obius function.The case D n = 1 was considered in [15, 16] and the case D n = n + 1 wasinvestigated in [17].The variants with the M¨obius function µ ( n ) are described in [18] and [19].We recall that its explicit definition is µ ( n ) = , n = p m , ( − k , n = p p · · · p k , p i = p j , , n = 1 , ( k = 0) , (8)4nd it is related to the inverse Riemann zeta function by1 ζ ( s ) = + ∞ X n =1 µ ( n ) n s , s = σ + iτ, σ > . (9)The corresponding Lagrangian for D n = µ ( n ) is L = m D h − φ ζ (cid:16) ✷ m (cid:17) φ + Z φ M ( φ ) dφ i , (10)where M ( φ ) = P + ∞ n =1 µ ( n ) φ n = φ − φ − φ − φ + φ − φ + φ − φ − . . . .For D n = − µ ( n ) ( n + 1) the Lagrangian is L = m D n φ h ζ (cid:16) ✷ m − (cid:17) + 1 ζ (cid:16) ✷ m (cid:17) i φ − φ F ( φ ) o , (11)where F ( φ ) = P + ∞ n =1 µ ( n ) φ n − = 1 − φ − φ − φ + ... .The case with D n = ( − n − ( n + 1) was recently introduced in [20]. Werecall that + ∞ X n =1 ( − n − n s = (1 − − s ) ζ ( s ) , s = σ + iτ , σ > , (12)which has an analytic continuation to the entire complex- s plane withoutsingularities, i.e. the analytic expression [21] is(1 − − s ) ζ ( s ) = ∞ X n =0 n +1 n X k =0 ( − k (cid:18) nk (cid:19) ( k + 1) − s . (13)At point s = 1, one has lim s → (1 − − s ) ζ ( s ) = P + ∞ n =1 ( − n − n = log 2.Applying (12) to (7) and using analytic continuation we obtain L = − m D h φ n (cid:16) − − ✷ m (cid:17) ζ (cid:16) ✷ m − (cid:17) + (cid:16) − − ✷ m (cid:17) ζ (cid:16) ✷ m (cid:17)o φ − φ φ i . (14)5e now consider the case D n = ( − n − . The corresponding Lagrangianis L = m D h − φ (cid:16) − − ✷ m (cid:17) ζ (cid:16) ✷ m (cid:17) φ + φ −
12 log(1 + φ ) i . (15)The potential is V ( φ ) = − L ( ✷ = 0) = m D h φ − φ + 12 log(1 + φ ) i , (16)which has one local maximum V (0) = 0 and one local minimum at φ = 1. Itis singular at φ = −
1, i.e. V ( −
1) = −∞ , and V ( ±∞ ) = + ∞ . The equationof motion is (cid:16) − − ✷ m (cid:17) ζ (cid:16) ✷ m (cid:17) φ = φ φ , (17)which has the two trivial solutions: φ = 0 and φ = 1.The Riemann zeta function arising in the multiplicative approach is givenin the form of product (4). The initial Lagrangian is p -adic Lagrangian (6)with g p = p p − . The Lagrangian obtained in this approach [19] is similarto (11) above. These two Lagrangians describe the same field theory in theweek field approximation. In the preceding section, we presented some Lagrangians that can be used todescribe the p -adic sector of open scalar strings. They contain the Riemannzeta function and are also starting points for interesting examples of theso-call zeta field theory. The corresponding potentials, which are V ( φ ) = − L ( ✷ = 0), and equations of motions are considered in the cited references.All these zeta field theory models contain tachyons.The most interesting of the above Lagrangians are (14) and (15). Un-like the other Lagrangians, these have no singularity with respect to thed’Alembertian ✷ , and it is easier to apply the pseudodifferential treatment.This analyticity of the Lagrangian is expected to be useful in its applicationto nonlocal cosmology, in particular, using linearization procedure (see, e.g.,[22] and references therein). 6 cknowledgements The paper was supported in part by the Ministry of Science and TechnologicalDevelopment, Serbia (Contract No. 144032D). The author thanks organizersof the International Bogolyubov Conference “Problems of Theoretical andMathematical Physics” (August 21-27, 2009, Moscow-Dubna, Russia) for avery pleasant and useful scientific meeting.
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