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Diffusion method in field theories with fakeons

Gianluca Calcagni

TL;DR

This work demonstrates that the diffusion method, when applied to classicized dynamics of fakeons, requires exactly two initial conditions per physical field to specify the nonlocal evolution. By formulating both complex- and real-mass fakeon models within a Balakrishnan--Komatsu diffusion representation, the authors show that the nonlocal equations of motion are uniquely determined by two initial data, consistent with previous direct calculations. They develop the diffusion framework for interacting and linear toy models, revealing how the diffusion flow selects a single physical mode among many higher-derivative roots, particularly in perturbative regimes. The results solidify the connection between fakeon prescriptions and a well-posed Cauchy problem, with potential applications to nonlocal gravity theories and infrared nonlocal gravity, where similar $oxed{oxdot^{-n}}$-type nonlocalities appear. The analysis also clarifies the practical implementation of diffusion in both active and passive slicing and highlights how the method complements existing approaches to nonlocal dynamics in quantum gravity and related frameworks.

Abstract

We adapt the diffusion method employed in fundamentally nonlocal field theories to determine the number of initial conditions for the classicized dynamics of unitary field theories with fakeons, characterized by inverse powers of the d'Alembertian operator $\Box$. We show that this number is two and we recover all the results obtained with a direct calculation elsewhere, including explicit solutions of linear toy models. Possible applications to nonlocal gravity are discussed.

Diffusion method in field theories with fakeons

TL;DR

This work demonstrates that the diffusion method, when applied to classicized dynamics of fakeons, requires exactly two initial conditions per physical field to specify the nonlocal evolution. By formulating both complex- and real-mass fakeon models within a Balakrishnan--Komatsu diffusion representation, the authors show that the nonlocal equations of motion are uniquely determined by two initial data, consistent with previous direct calculations. They develop the diffusion framework for interacting and linear toy models, revealing how the diffusion flow selects a single physical mode among many higher-derivative roots, particularly in perturbative regimes. The results solidify the connection between fakeon prescriptions and a well-posed Cauchy problem, with potential applications to nonlocal gravity theories and infrared nonlocal gravity, where similar -type nonlocalities appear. The analysis also clarifies the practical implementation of diffusion in both active and passive slicing and highlights how the method complements existing approaches to nonlocal dynamics in quantum gravity and related frameworks.

Abstract

We adapt the diffusion method employed in fundamentally nonlocal field theories to determine the number of initial conditions for the classicized dynamics of unitary field theories with fakeons, characterized by inverse powers of the d'Alembertian operator . We show that this number is two and we recover all the results obtained with a direct calculation elsewhere, including explicit solutions of linear toy models. Possible applications to nonlocal gravity are discussed.
Paper Structure (15 sections, 1 theorem, 71 equations, 1 figure)

This paper contains 15 sections, 1 theorem, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fakeons with interactions
  3. Fakeons with complex masses
  4. Fakeons with real masses
  5. Diffusion method
  6. Diffusion method for linear toy models
  7. Active versus passive slicing
  8. Localized system
  9. Higher-derivative dynamics
  10. Nonlocal dynamics
  11. Examples
  12. Tachyonic fakeon two-derivative models
  13. Standard fakeon two-derivative model
  14. Higher-order fakeon model
  15. Discussion

Key Result

Theorem 1

The number of initial conditions is $\mathcal{N}=2$ for each physical field in any theory with fakeons whose classicized Lagrangian is second-order in time derivatives when nonlocality is switched off.

Figures (1)

  • Figure 1: On the slice $r_*=0$, only the root $\Omega_{01}^2=0$ (filled dot) is a solution of the characteristic equation (\ref{['masterlainf2']}) of the linear nonlocal system, while the roots of $\mathcal{A}(\Omega_{0i}^2)=0$ are not (empty dots). On the slice $r_*=+\infty$, all the roots $\Omega_{*i}^2$ of the higher-derivative parent system are solutions of (\ref{['masterlainf2']}) (filled dots) but only $\Omega_{*1}^2$ is smoothly connected to $\Omega_{01}^2=0$ by the function $\Omega_1^2(r_*)$ (solid curve). Since the direction of diffusion is only towards increasing values of the extra direction, it is not possible to "go upstream" from any slice with finite $r_*>0$ back to the $r_*=0$ slice. This selects $\Omega_{*1}^2$ as the only physical root of the nonlocal system.

Theorems & Definitions (1)

  • Theorem