Physical model of dimensional regularization

TL;DR

This paper constructs specific fractals of dimension 4-epsilon to model dimensional regularization in quantum field theory, exploring geometric and boundary condition effects on amplitude calculations and implications for quantum gravity.

Contribution

It explicitly constructs fractals suitable for dimensional regularization, analyzing boundary effects and extending the model to Minkowski space, with implications for quantum gravity.

Findings

Fractals of dimension 4-epsilon approximate quantum-field amplitudes.

Boundary conditions significantly influence power-law behavior.

Extension of the model to 4D Minkowski space and implications for quantum gravity.

Abstract

We explicitly construct fractals of dimension 4-epsilon on which dimensional regularization approximates scalar-field-only quantum-field-theory amplitudes. The construction does not require fractals to be Lorentz-invariant in any sense, and we argue that there probably is no Lorentz-invariant fractal of dimension greater than 2. We derive dimensional regularization's power-law screening first for fractals obtained by removing voids from 3-dimensional Euclidean space. The derivation applies techniques from elementary dielectric theory. Surprisingly, fractal geometry by itself does not guarantee the appropriate power-law behavior; boundary conditions at fractal voids also play an important role. We then extend the derivation to 4-dimensional Minkowski space. We comment on generalization to nonscalar fields, and speculate about implications for quantum gravity.