Euclidean Quantum Field Formulation of p-Adic open string amplitudes

We study in a rigorous mathematical way p-adic quantum field theories whose N-point amplitudes are the expectation of products of vertex operators. We show that this type of amplitudes admit a series expansion where each term is an Igusa's local zeta function. The lowest term in this series is a regularized version of the p-adic open Koba-Nielsen string amplitude.


Introduction
The string amplitudes were introduced by Veneziano in the 60s, [1], further generalizations were obtained by Virasoro [2], Koba and Nielsen [3], among others. In the 80s, Freund, Witten and Volovich, among others, studied string amplitudes at the tree level over different number fields, and suggested the existence of connections between these amplitudes, see e.g. [4]- [5]. In this framework the connections with number theory, specifically with local zeta functions appears naturally, see e.g. [6]- [9], and the survey [10], see also [11]- [13].
The p-adic string theories have been studied over time with some periodic fluctuations in their interest (for some reviews, see [14], [15], [16], [17]). Recently a considerable amount of work has been performed on this topic in the context of the AdS/CFT correspondence [18,19,20,21]. String theory with a p-adic world-sheet was proposed and studied for the first time in [22]. Later this theory was formally known as p-adic string theory. The p-adic strings are related to ordinary strings at least in two different ways. First, connections through the adelic relations [23], and second, through the limit p tends to 1 [24]- [26].
The tree-level string amplitudes were explicitly computed in the case of p-adic string world-sheet in [27] and [28]. Since the 80s there has been interest in constructing field theories whose correlators are the p-adic tree-level string amplitudes (or p-adic Koba-Nielsen amplitudes). Spokoiny [25] and Zhang [29], see also [30], constructed formally quantum field theories whose amplitudes are expectation values of products of vertex operators. In [31] Zabrodin established that the treelevel string amplitudes may be obtained starting with a discrete field theory on a Bruhat-Tits tree. These ideas have been used by Ghoshal and Kawano in the study of p-adic strings in constant B-fields [32]. This article aims to provide a rigorous mathematical construction of a class of quantum field theories whose amplitudes are expectations of products of vertex operators. By using this approach, we carry out a mathematically rigorous derivation of the N -point Koba-Nielsen amplitudes, thus our approach is completely different from the one followed in [25,29,31].
The naive Euclidean version of the p-adic N -point amplitudes is given by where Qp dx j e kj ·ϕ(xj) is the tachyonic vertex operator of the j−th tachyon, with momentum k j = (k 0,j , . . . , k D−1,j ), and field ϕ(x j ), the dot denotes the standard Euclidean scalar product, and the action is given by It is important to note that in (1.1) the tachyonic fields must be functions not distributions. These amplitudes are exactly the ones considered in [25], [29], [32]. Since the integral Q N p d N x in the right-hand side of (1.1) is always divergent, it is necessary to introduce a cut-off, and to define the amplitude by a limit process.
The key observation is that the action (1.2) corresponds to a free quantum field. In the Archimedean and non-Archimedean cases, free quantum fields correspond to Gaussian probability measures on suitable infinite dimensional spaces. The reader may consult [33, Section 6.2] for the Archimedean case, and [34, Section 5.5], [35], [36] for the p-adic case. We construct Gaussian probability measure P D on suitable function space (L D R (Q p )) and propose that and B N R denotes an N -dimensional ball of radius p R . Following the standard approach in QFT, we expand the right-hand side of (1.3) around a suitable solution of the equations of motion. The main difficulty is that the solutions of these equation are distributions, and we are restricted to work with functions. We show that there is a change of variables in (1.3) such that (1.4) where P D is a probability measure. Here is an important difference with respect to the classical QFT, which is that the k cannot be considered as a coupling constant, and thus there is no a standard perturbative expansion for (1.4). By taking the classical normalization and using the expansion of the exponential function, we show that (1.4) admits a series expansion of the form where G 0 (k, x) is a constant and the G l (k, x)s are continuous functions in x, for l ≥ 1. The product 1 B N −3 R (x) G l (k, x) can be approximated by a test function in x depending of k, for l ≥ 1, without altering the analytic dependence of integral A (N ) where Φ is a test function is a particular case of a multivariate Igusa zeta function [37].
In [6]- [9] was established that the integral Z where Z (N ) (k) is a meromorphic function which is a regularized version of the p-adic Koba-Nielsen amplitude, [6]. Therefore where A (N ) (k) is the p-adic Koba-Nielsen string amplitude in the Euclidean signature, C0 Z0 is a positive constant. We know that there is a common domain of convergence in k for A (N ) (k) and all the integrals appearing in the series, but we do not know if the series converges. The study of the limit R → ∞ in the previous formula is an open problem.
In a forthcoming article, we plan to study the p-adic quantum field theories [36] attached to a non-Archimedean version of the open string action in a background gauge field [38]. This action has cubic and quartic terms in the dynamical fields, which generate interesting non-trivial one-loop quantum corrections which determine the beta functions and the effective action for the gauge fields. We would like to find the corresponding non-Archimedean version for this case. Finally, we expect that the results presented in this work have a natural counterpart in the case of standard Koba-Nielsen amplitudes.

Basic facts on p-adic analysis
In this Section, we collect some basic results on p-adic analysis that we use through the article. For a detailed exposition on p-adic analysis the reader may consult [39], [40], [16].
2.1. The field of p-adic numbers. Throughout this article p will denote a prime number. The field of p−adic numbers Q p is defined as the completion of the field of rational numbers Q with respect to the p−adic norm | · | p , which is defined as where a and b are integers coprime with p. The integer γ = ord p (x) := ord(x), with ord(0) := +∞, is called the p−adic order of x. We extend the p−adic norm to Q N p by taking We define ord(x) = min 1≤i≤N {ord(x i )}, then ||x|| p = p −ord(x) . The metric space Q N p , || · || p is a complete ultrametric space. As a topological space Q p is homeomorphic to a Cantor-like subset of the real line, see e.g. [39], [16].
Any p−adic number x = 0 has a unique expansion of the form where x j ∈ {0, 1, 2, . . . , p − 1} and x 0 = 0. By using this expansion, we define the fractional part {x} p of x ∈ Q p as the rational number In addition, any x ∈ Q N p {0} can be represented uniquely as For r ∈ Z, denote by B N r (a) = {x ∈ Q N p ; ||x − a|| p ≤ p r } the ball of radius p r with center at a = (a 1 , . . . , a N ) ∈ Q N p , and take B N r (0) := B N r . Note that B N r (a) = B r (a 1 )×· · ·×B r (a N ), where B r (a i ) := {x ∈ Q p ; |x i −a i | p ≤ p r } is the one-dimensional ball of radius p r with center at a i ∈ Q p . The ball B N 0 equals the product of N copies of B 0 = Z p , the ring of p−adic integers. We also denote by S N r (a) = {x ∈ Q N p ; ||x − a|| p = p r } the sphere of radius p r with center at a = (a 1 , . . . , a N ) ∈ Q N p , and take S N r (0) := S N r . We notice that Since (Q N p , +) is a locally compact topological group, there exists a Haar measure d N x, which is invariant under translations, i.e. d N (x + a) = d N x. If we normalize this measure by the condition Z N p dx = 1, then d N x is unique. Notation 1. We will use Ω (p −r ||x − a|| p ) to denote the characteristic function of the ball B N r (a). For more general sets, we will use the notation 1 A for the characteristic function of a set A.
if it is locally constant with compact support. Any test function can be represented as a linear combination, with complex coefficients, of characteristic functions of balls.
, the largest number l = l(ϕ) satisfying (2.1) is called the exponent of local constancy (or the parameter of constancy) of ϕ.
We denote by D l m (Q N p ) the finite-dimensional space of test functions from D(Q N p ) having supports in the ball B N m and with parameters of constancy ≥ l. We now define a topology on D as follows. We say that a sequence {ϕ j } j∈N of functions in D converges to zero, if the two following conditions hold: (1) there are two fixed integers k 0 and m 0 such that each ϕ j ∈ D k0 m0 ; (2) ϕ j → 0 uniformly. D endowed with the above topology becomes a topological vector space.
The corresponding R-vector spaces are denoted as L ρ where d N x is the normalized Haar measure on Q N p , + , for 1 ≤ ρ < ∞, see e.g. [39,Section 4.3]. We denote by L ρ R (U ) the real counterpart of L ρ (U ).
2.5. The Fourier transform. Set χ p (y) = exp(2πi{y} p ) for y ∈ Q p . The map χ p (·) is an additive character on Q p , i.e. a continuous map from (Q p , +) into S (the unit circle considered as multiplicative group) satisfying The additive characters of Q p form an Abelian group which is isomorphic to (Q p , +). The isomorphism is given by κ → χ p (κx), see e.g.
Given κ = (κ 1 , . . . , κ N ) and y = ( see e.g. [39,Section 4.8]. We will also use the notation F x→κ ϕ and ϕ for the Fourier transform of ϕ. The Fourier transform extends to L 2 . If f ∈ L 2 , its Fourier transform is defined as where the limit is taken in L 2 . We recall that the Fourier transform is unitary on We endow D ′ with the weak topology, i.e. a sequence {T j } j∈N in D ′ converges to T if lim j→∞ T j (ϕ) = T (ϕ) for any ϕ ∈ D. The map is a bilinear form which is continuous in T and ϕ separately. We call this map the pairing between D ′ and D. From now on we will use (T, ϕ) instead of T (ϕ).

A naive Euclidean version of the p-adic open string amplitudes
We set k : The naive Euclidean version of the p-adic N -point amplitudes is given by The amplitudes (3.1) are just expectation values of products of vertex operators. These amplitudes were proposed by Spokoiny [25] and Zhang [29], see also [30], [31]. In these articles the authors obtain the p-adic open Koba-Nielsen amplitudes from amplitudes (3.1) by a formal calculation. The central goal of this work is to provide a mathematical framework to understand these calculations.
Since there is l ∈ Z such that ϕ j (x j ) = 0 for |x j | p > p l , for some l ∈ Z, To fix this problem, it is necessary to introduce a cut-off and set where R is a positive integer and B N R = x ∈ Q N p ; x p ≤ p R . 3.1. The action and the Vladimirov operator.
The complex vector space L (Q p ) endowed with the topology inherited from D (Q p ) is called the p-adic Lizorkin space of test functions of the second kind, see [39,Chapter 7]. We set L R (Q p ) := D R (Q p ) ∩ L (Q p ). We define the inverse of D as Consider the equation This equation has a unique solution ψ ∈ L (Q p ). Set Then

Gaussian processes and free quantum fields
We define the bilinear form B where ·, · denotes the scalar product in L 2 (Q p ) .
Proof. We first notice that for ϕ ∈ L R (Q p ), we have Then B (ϕ, ϕ) = 0 implies that ϕ is zero almost everywhere and since ϕ is continuous ϕ = 0. Let (ϕ n , θ n ) ∈ L R (Q p ) × L R (Q p ) be two sequences such that ϕ n → 0 and θ n → 0 in L R (Q p ). We recall that the topology of L R (Q p ) agrees with the topology of D R (Q p ). Now, |ξ| p dξ =: I 1 (θ n , ϕ n ) + I 2 (θ n , ϕ n ) .
Now, the continuity of B follows from the fact that ϕ n uniform. −−−−−→ 0 and θ n uniform. −−−−−→ 0 imply that ϕ n 1 → 0, θ n 1 → 0, and θ n 1 → 0 as n tends to infinity. The convergence of the last sequence follows from We recall that D (Q p ) is a nuclear space cf. [41,Section 4], and since any subspace of a nuclear space is also nuclear, L R (Q p ) is a nuclear space that is dense and continuously embedded in L 2 R (Q p ), cf. [39, theorem 7.4.4]. Then we have the following Gel'fand triple: . We denote by B := B (L ′ R (Q p )) the σ−algebra generated by the cylinder subsets of L ′ R (Q p ) .
Consider the mapping This functional is a continuous, positive definite mapping, cf. Lemma 1, and C (0) = 1. Then C defines a characteristic functional in L R (Q p ) . By Bochner-Minlos theorem, there exists a unique probability measure P called the canonical Gaussian measure on (L ′ R (Q p ) , B) given by its characteristic functional as where (·, ·) is the pairing between L ′ R (Q p ) and L R (Q p ). The measure P corresponds to a free quantum field on L ′ R (Q p ). This identification is well-known in the Archimedean and non-Archimedean settings, see e.g. [ , D-times. The probability measure where Z 0 = L D R (Qp) dP D (ϕ) represents a free quantum field in L D R (Q p ).
Intuitively, the N -point amplitudes are the expectation values of the products of the vertex operators with respect to the measure (5.1): It is important to mention that e N j=1 kj ·ϕ(xj ) requires that each entry of ϕ (x j ) be a function, for this reason, the factor 1 L D R (Qp) is completely necessary in (5.1). Due to the divergence of the second integral in the right-hand side of (5.2), we define the N -point amplitudes as follows.
Our central goal is to show (in a rigorous mathematical way) that the ansatz proposed in the above definition allow us to obtain the p-adic open Koba-Nielsen amplitudes as the constant term of a series expansion of lim R→∞ A (N ) R (k) in functions depending on k. The precise statements of our main results are given in Theorems 1, 2.
By using that We now introduce the notation taking advantage that l is fixed. Here v j ∈ R and ϕ ∈ L R (Q p ).
We set where δ (· − x j ) denotes the Dirac distribution centered at x j .
Proof. We first recall that By using that N j=1 |v j | |ϕ (x j )| δ (x − x j ) ∈ L ′ R (Q p ), for any ϕ ∈ L R (Q p ), and by fixing θ ∈ L R (Q p ) such that θ (x j ) > 1 for j = 1, . . . , N , we have Finally, the continuity in x follows from the dominated convergence theorem by using that Proof. By Lemma 2, for R, N, k given, dP (ϕ l ) < ∞ is a well-defined and continuous function. Now, the announced formula is a consequence of Fubini's theorem.

5.2.
Some technical results. We set for a positive integer n, and recall that δ n (x) as n → ∞. We now introduce an approximation for A (N ) where I is a positive integer.
Proof. The proof is similar to the one given for Lemma 2. The result follows from where W ∈ L ′ R (Q p ) is distribution depending on x j , v j , for j = 1, . . . , N , but not on I, and where θ ∈ L R (Q p ) is a fixed positive function. Now, by using (5.6) and the dominated convergence theorem, we have We denote by l ϕ index of local constancy of ϕ. We pick I ϕ = max {I, l ϕ }, then p Iϕ Z p is a subgroup of p I Z p and

A change of variables.
Let ϕ L,m , ϕ be functions in L R (Q p ), for L ≥ 1 and m ∈ Q × p . We now use the measurable mapping as a change of variables in (5.7). There exist a measure P L,m such that Lemma 4. With the above notation, the following holds true: The equation Dϕ L,m = J L,m has a unique solution ϕ L,m ∈ L R (Q p ) given by Proof. (i) Denote by ∆ m (ξ) = Ω |mξ| p , m ∈ Q × p , the characteristic function of the ball B log p |m| −1 p . Then where ∆ L (ξ) = Ω p −L |ξ| p , which implies that J L,m is a test function satisfying By using that |m| −1 p Ω |m| −1 p |x| p ∈ L 1 (Q p ) and δ L (x) ∈ L ρ , 1 < ρ < ∞, and applying [39,Lemma 7.4.2] we have Ω |x| p (iii) Take θ ∈ L R (Q p ), by using the Cauchy-Schwarz inequality, By the second part J L,m − J L 2 → 0 as |m| p → ∞. (vi) If |x − x j | p > p −L for any j = 1, . . . , N , Proof. By using the formula for f 1 * J L (x), in the case |x − x j | p > p −L for any x j , see (5.9), and the continuity of the pairing and the continuity of the convolution, as |m| p → ∞. Now since ln |x| p is locally constant in Q × p , and the lim t→−∞ e t = 0, we have for I sufficiently large that for I sufficiently large.

Calculation of the second limit.
We now describe the measure P L,m . Take ϕ L,m ∈ L ′ R (Q p ), W ∈ L ′ R (Q p ), by using (4.1) and changing variables as W = W − ϕ L,m , we have We denote P the measure corresponding to C (g). We now recall that Therefore P L,m ⇒ P when |m| p → ∞, L → ∞. Now, if l(x) ∈ L 1 (R, P hn ) for any n and l(x) ∈ L 1 (R, ), by using the fact that the bounded continuous functions are dense in L 1 (R, P hn ) and L 1 R, P h , see e.g. [45,Proposition 1.3.22], in (5.11) we can assume that l(x) is an integrable function.
In conclusion, we have the following result.

A formula for A
By using this formula and the definition A We now introduce the 'convention' that the insertion points x 1 , x 2 ,. . . , x N −1 , x N , with N ≥ 4, belong to the p-adic projective line, and then by using the Möbius group, we may take the normalization In our framework, the convention x N = ∞ means that the N -point amplitudes do not depend on x N , then A where the momenta vectors satisfy N i=1 k i = 0 and We now consider the function By using that where F r (k, ϕ (x 2 ) , . . . , ϕ (x N −2 )) is a homogeneous polynomial of degree r in the variables k l,j , l = 0, . . . , D − 1, j = 2, . . . , N − 2, whose coefficients are polynomials in the ϕ (x 2 ) , . . . , ϕ (x N −2 ). By the dominated convergence theorem, Corollary 1, and where x = (x 2 , . . . , x N −2 ). Now by using that F r (k, ϕ (x 2 ) , . . . , ϕ (x N −2 )) are integrable continuous functions in x for k fixed, we conclude that is a continuous function in x. Therefore R (k) admits the following expansion in the momenta: To continue the study of the amplitudes A (N ) R (k), we introduce the following notation: These functions were introduced in [8], see also [6]. The functions Z (N ) (s) are holomorphic in a certain domain of C D0 and admit analytic continuations to C D0 (denoted also as Z (N ) (s)) as rational functions in the variables for Re(s ij ) > 0 for any ij, is a multivariate Igusa local zeta function. These functions admit analytic continuations as rational functions of the variables p −sij , [46]. If we take φ to be the characteristic function of B N −3 R , the ball centered at the origin with radius p R , the dominated convergence theorem and [8, Theorem 1], imply that i=2 dx i is the normalized Haar measure of Q N −3 p , k = (k 1 , . . . , k N ), k i = (k 0,i , . . . , k D−1,i ), i = 1, . . . , N , N ≥ 4, is the momentum vector of the i-th tachyon (with Minkowski product k i k j = −k 0,i k 0,j + k 1,i k 1,j + · · · + k D−1,i k D−1,j ) obeying In [8], [6], Remark 1. In [8], [6], the local zeta functions Z (N ) (s) were used to regularize Koba-Nielsen amplitudes A ). This implies that the mapping is a meromorphic function in s having the same poles of the mapping (6.4).

The limit lim R→∞ A (N )
R (k). We now apply the above-mentioned results to study the limit  φ (s) | sij =2 (p−1) p ln p ki·kj is a multivariate local zeta function. Furthermore, These amplitudes admit expansions of the type given in Proposition 1, where the functions G r (k, x) are replaced by continuous functions in x depending on k and λ. The behavior of these quantum field theories is completely different from the standard ones due to the fact that we are computing the correlation functions for a very particular class of observables, which are products of vertex operators.