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Quantum Field Theory on Multifractal Spacetime: Varying Dimension and Ultraviolet Completeness

Alessio Maiezza, Juan Carlos Vasquez

TL;DR

This work develops a quantum field theory on a multifractal spacetime where the effective spacetime dimension runs with energy. By implementing a scale-dependent smoothing of fields and a translationally non-invariant form factor, the authors achieve perturbative UV finiteness and eliminate UV renormalons, while enabling a consistent S-matrix via breaking vacuum translational invariance to evade Haag's theorem. The framework predicts dimensional reduction from $d^{eff}=4$ toward $d^{eff}=2$ at high energies, leading to asymptotic safety-like behavior with couplings approaching constant values beyond the scale $M$; low-energy QFT predictions remain intact. Phenomenologically, the model implies anisotropies in high-energy scattering and Lorentz-violating effects suppressed by the large scale $M$, offering potential experimental signatures, while also inviting further exploration in cosmology and beyond-Standard-Model physics.

Abstract

Inspired by various quantum gravity approaches, we explore quantum field theory where spacetime exhibits scaling properties and dimensional reduction with changing energy scales, effectively behaving as a multifractal manifold. Working within canonical quantization, we demonstrate how to properly quantize fields in such a multifractal spacetime. Our analysis reveals that a non-differentiable nature of spacetime is not merely compatible with quantum field theory but significantly enhances its mathematical foundation. Most notably, this approach ensures perturbative UV finiteness and improved behavior of the series expansion and enables rigorous construction of the S-matrix in the interaction picture by breaking vacuum translational invariance. The multifractal structure tames dominant, large-order divergence sources in the perturbative series and resolves the Landau pole problem through asymptotic safety, substantially improving the theory's behavior in the deep ultraviolet regime. Our formulation preserves all established predictions of standard quantum field theory at low energies while offering novel physical behaviors at high energy scales.

Quantum Field Theory on Multifractal Spacetime: Varying Dimension and Ultraviolet Completeness

TL;DR

This work develops a quantum field theory on a multifractal spacetime where the effective spacetime dimension runs with energy. By implementing a scale-dependent smoothing of fields and a translationally non-invariant form factor, the authors achieve perturbative UV finiteness and eliminate UV renormalons, while enabling a consistent S-matrix via breaking vacuum translational invariance to evade Haag's theorem. The framework predicts dimensional reduction from toward at high energies, leading to asymptotic safety-like behavior with couplings approaching constant values beyond the scale ; low-energy QFT predictions remain intact. Phenomenologically, the model implies anisotropies in high-energy scattering and Lorentz-violating effects suppressed by the large scale , offering potential experimental signatures, while also inviting further exploration in cosmology and beyond-Standard-Model physics.

Abstract

Inspired by various quantum gravity approaches, we explore quantum field theory where spacetime exhibits scaling properties and dimensional reduction with changing energy scales, effectively behaving as a multifractal manifold. Working within canonical quantization, we demonstrate how to properly quantize fields in such a multifractal spacetime. Our analysis reveals that a non-differentiable nature of spacetime is not merely compatible with quantum field theory but significantly enhances its mathematical foundation. Most notably, this approach ensures perturbative UV finiteness and improved behavior of the series expansion and enables rigorous construction of the S-matrix in the interaction picture by breaking vacuum translational invariance. The multifractal structure tames dominant, large-order divergence sources in the perturbative series and resolves the Landau pole problem through asymptotic safety, substantially improving the theory's behavior in the deep ultraviolet regime. Our formulation preserves all established predictions of standard quantum field theory at low energies while offering novel physical behaviors at high energy scales.

Paper Structure

This paper contains 17 sections, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fractality, smoothing and dimensional reduction
  3. Introducing a scale in Euclidean Green equation: Dimensional Reduction
  4. Canonical quantization with varying effective dimension
  5. Propagator
  6. Translations, rotations, and the vacuum
  7. Consistency of the theory
  8. Källén-Lehmann spectral representation.
  9. Gell-Mann and Low theorem and S-matrix construction.
  10. Possible Implications and Discussion
  11. Loop finiteness.
  12. Anisotropies in high-energy scatterings.
  13. Absence of the UV renormalons.
  14. Implications.
  15. Conclusions and outlook
  16. ...and 2 more sections

Figures (1)

  • Figure 1: Running of the gauge couplings of the standard model, which asymptotically reach constant values.