Scalar Quantum Field Theory on Fractals

TL;DR

This paper constructs measures for scalar quantum fields on fractals, revealing continuum limits with correlation functions that vary as irrational powers of distance, indicating quantum fields on fractional-dimensional spaces.

Contribution

It introduces a method to create scale-invariant scalar field measures on fractals, including a non-Gaussian fixed point, and analyzes their correlation functions with irrational exponents.

Findings

Correlation functions follow power laws with irrational exponents.

Most measures are Gaussian, with some non-Gaussian fixed points.

Exponents depend on subdivision schemes, suggesting fractional-dimensional quantum fields.

Abstract

We construct a family of measures for random fields based on the iterated subdivision of simple geometric shapes (triangles, squares, tetrahedrons) into a finite number of similar shapes. The intent is to construct continuum limits of scale invariant scalar field theories, by imitating Wiener's construction of the measure on the space of functions of one variable. These are Gaussian measures, except for one example of a non-Gaussian fixed point for the Ising model on a fractal. In the continuum limits what we construct have correlation functions that vary as a power of distance. In most cases this is a positive power (as for the Wiener measure) but we also find a few examples with negative exponent. In all cases the exponent is an irrational number, which depends on the particular subdivision scheme used. This suggests that the continuum limits corresponds to quantum field theories (random fields) on spaces of fractional dimension.