Non-Local Models with Entire Functions

• The paper establishes that non-local models with entire functions are defined via analytic functions of differentiation operators, rigorously constructed using the Borel transform and contour integration.
• It demonstrates that solutions to linear non-local equations can be decomposed into particular and homogeneous parts, with the zeros of the symbol playing a critical role in the spectral structure.
• The application to zeta-nonlocal field equations, especially in p-adic string theory, highlights the physical relevance and wider applicability of this analytic framework.

Non-local models with entire functions describe evolution equations in which the dependence on time or space derivatives is encoded by an analytic or entire function of the differentiation operator. These models are of particular interest in mathematical physics, including p-adic string theory, where the kinetic or interaction terms are specified by transcendental symbols such as the Riemann zeta function. The rigorous analysis of such non-local operators, specifically those defined as $f(\partial_t)$ for entire or holomorphic $f$, relies on modern techniques from the theory of entire functions, contour integration, and the Borel transform. This article presents a comprehensive exposition of these non-local models with entire functions, focusing especially on the methodology introduced by the Borel transform and its application to zeta-nonlocal field equations, as detailed in (Chávez et al., 2019).

1. Rigorous Construction of Non-local Operators via Entire Functions

The functional calculus for non-local operators is founded on a Borel transform framework. Consider an entire function $\phi(z) = \sum_{n=0}^\infty a_n z^n$ of exponential type $\tau_\phi$. Its Borel transform is defined by

$B(\phi)(s) = \sum_{n=0}^\infty a_n n! s^{-(n+1)},$

which converges for $|s| > \tau_\phi$. Alternatively, for $\mathrm{Re}\,s > \tau_\phi$,

$B(\phi)(s) = \int_0^\infty \phi(t) e^{-s t} \,dt.$

The Polya inversion formula (also known as the Fourier–Borel–Laplace representation) enables reconstruction of $\phi$ from $B(\phi)$:

$f$0

where $f$1 is a simple closed contour encircling all Borel singularities of $f$2.

Every $f$3 in the space $f$4, the set of entire functions of exponential type whose Borel singularities are contained within a simply connected domain $f$5, can also be written as $f$6:

$f$7

where $f$8 is a complex measure supported compactly in $f$9.

Given $\phi(z) = \sum_{n=0}^\infty a_n z^n$0 holomorphic in $\phi(z) = \sum_{n=0}^\infty a_n z^n$1 and $\phi(z) = \sum_{n=0}^\infty a_n z^n$2, the non-local operator $\phi(z) = \sum_{n=0}^\infty a_n z^n$3 is defined by

$\phi(z) = \sum_{n=0}^\infty a_n z^n$4

or equivalently, in contour form,

$\phi(z) = \sum_{n=0}^\infty a_n z^n$5

with $\phi(z) = \sum_{n=0}^\infty a_n z^n$6 chosen within $\phi(z) = \sum_{n=0}^\infty a_n z^n$7 enclosing all singularities of $\phi(z) = \sum_{n=0}^\infty a_n z^n$8. This definition is independent of the specific choice of $\phi(z) = \sum_{n=0}^\infty a_n z^n$9 due to Cauchy's theorem (Chávez et al., 2019).

2. Existence and Structure of Solutions for Linear Non-local Equations

For the equation $\tau_\phi$0, the existence and structure of solutions are determined within $\tau_\phi$1, with $\tau_\phi$2 and $\tau_\phi$3:

• The mapping $\tau_\phi$4 is linear but not continuous in the compact-uniform topology.
• The operator is surjective on $\tau_\phi$5: for any $\tau_\phi$6, one can construct $\tau_\phi$7, avoiding zeros of $\tau_\phi$8.
• The general solution decomposes into particular and homogeneous parts. Where $\tau_\phi$9 are the zeros of $B(\phi)(s) = \sum_{n=0}^\infty a_n n! s^{-(n+1)},$0 in $B(\phi)(s) = \sum_{n=0}^\infty a_n n! s^{-(n+1)},$1 of multiplicities $B(\phi)(s) = \sum_{n=0}^\infty a_n n! s^{-(n+1)},$2,

$B(\phi)(s) = \sum_{n=0}^\infty a_n n! s^{-(n+1)},$3

where each $B(\phi)(s) = \sum_{n=0}^\infty a_n n! s^{-(n+1)},$4 is a polynomial of degree less than $B(\phi)(s) = \sum_{n=0}^\infty a_n n! s^{-(n+1)},$5, and $B(\phi)(s) = \sum_{n=0}^\infty a_n n! s^{-(n+1)},$6 encloses Borel singularities of $B(\phi)(s) = \sum_{n=0}^\infty a_n n! s^{-(n+1)},$7 and zeros of $B(\phi)(s) = \sum_{n=0}^\infty a_n n! s^{-(n+1)},$8 within $B(\phi)(s) = \sum_{n=0}^\infty a_n n! s^{-(n+1)},$9 (Chávez et al., 2019).

Homogeneous solutions in $|s| > \tau_\phi$0 are spanned by expressions of the form

$|s| > \tau_\phi$1

with polynomial coefficients determined by the structure of the kernel.

3. Zeta-Nonlocal Field Equation and Explicit Solution Construction

A key example is the non-local field equation with the Riemann zeta function as symbol:

$|s| > \tau_\phi$2

with real parameter $|s| > \tau_\phi$3, symbol $|s| > \tau_\phi$4, and source $|s| > \tau_\phi$5 of exponential type. For $|s| > \tau_\phi$6, $|s| > \tau_\phi$7 is analytic except for simple poles at $|s| > \tau_\phi$8, so the nonlocal operator is defined in

$|s| > \tau_\phi$9

with the Borel singularities of $\mathrm{Re}\,s > \tau_\phi$0 and $\mathrm{Re}\,s > \tau_\phi$1 inside $\mathrm{Re}\,s > \tau_\phi$2.

A function $\mathrm{Re}\,s > \tau_\phi$3 of type $\mathrm{Re}\,s > \tau_\phi$4 solves the equation if and only if

$\mathrm{Re}\,s > \tau_\phi$5

with $\mathrm{Re}\,s > \tau_\phi$6 encircling all Borel singularities of $\mathrm{Re}\,s > \tau_\phi$7 and zeros $\mathrm{Re}\,s > \tau_\phi$8 of $\mathrm{Re}\,s > \tau_\phi$9 within $B(\phi)(s) = \int_0^\infty \phi(t) e^{-s t} \,dt.$0 (Chávez et al., 2019).

For $B(\phi)(s) = \int_0^\infty \phi(t) e^{-s t} \,dt.$1, $B(\phi)(s) = \int_0^\infty \phi(t) e^{-s t} \,dt.$2 admits a power series expansion, yielding the infinite-order ODE

$B(\phi)(s) = \int_0^\infty \phi(t) e^{-s t} \,dt.$3

with solution structure as above.

4. Extension to General Analytic Forcing Terms

The Borel machinery also extends to $B(\phi)(s) = \int_0^\infty \phi(t) e^{-s t} \,dt.$4 that are analytic but not of exponential type. For such $B(\phi)(s) = \int_0^\infty \phi(t) e^{-s t} \,dt.$5, membership in the class $B(\phi)(s) = \int_0^\infty \phi(t) e^{-s t} \,dt.$6 is required: i.e., $B(\phi)(s) = \int_0^\infty \phi(t) e^{-s t} \,dt.$7 must have a Laplace transform $B(\phi)(s) = \int_0^\infty \phi(t) e^{-s t} \,dt.$8 analytic in a right half-plane and admitting analytic continuation to an angular sector.

For finite radius $B(\phi)(s) = \int_0^\infty \phi(t) e^{-s t} \,dt.$9, restrict the contour to $\phi$0 and set

$\phi$1

which belongs to $\phi$2 of type $\phi$3, and $\phi$4 uniformly on compacts in a sector $\phi$5. The associated solutions

$\phi$6

converge to a limit $\phi$7 as $\phi$8 in $\phi$9. The extended operator $B(\phi)$0 is defined by

$B(\phi)$1

This definition is independent of the choice of the angle $B(\phi)$2 and extends the operational calculus to analytic functions on certain Runge domains, bypassing classical initial value theory (Chávez et al., 2019).

5. Origin in p-adic String Theory and Physical Motivation

These non-local models arise naturally in the context of p-adic open string theory, where the tachyon scalar field $B(\phi)$3 obeys

$B(\phi)$4

with $B(\phi)$5 a prime number and $B(\phi)$6 the d'Alembertian. Dragovich (2008) proposed assembling all primes to form a zeta-nonlocal Lagrangian in $B(\phi)$7 dimensions:

$B(\phi)$8

so that the linearized field equation reads

$B(\phi)$9

The Borel transform-based approach rigorously establishes the existence and explicit form of solutions for the linearized zeta-nonlocal equation in the function spaces $f$00 and $f$01 (Chávez et al., 2019).

A major implication is that any entire or holomorphic symbol $f$02 admits this general treatment, regardless of growth conditions that challenge classical pseudo-differential calculus. The decomposition of solutions into particular and homogeneous parts clarifies how zeros of the symbol correspond to discrete spectral modes.

6. Outlook and Open Problems

Current methods provide a complete rigorous theory for linear zeta-nonlocal equations in one variable. Several directions remain for future research:

• Extending to nonlinear equations, such as $f$03 or general $f$04, requires new techniques, potentially fixed-point or heat kernel methods in function spaces akin to $f$05.
• Generalizing to field-theoretic models involving higher-dimensional $f$06 and explicit spacetime nonlocality remains open.
• The Borel–Laplace functional calculus outlined here is well-suited to any operator with entire or holomorphic symbol in a Runge domain, suggesting broader applicability to nonlocal theories in mathematical physics.

These developments reinforce the role of entire function-based nonlocal models as a mathematically coherent and physically motivated class of nonlocal field theories, with robust solution theory and a pathway to further analytic and physical generalizations (Chávez et al., 2019).