Scaling in quantum gravity

TL;DR

This paper investigates the critical behavior of quantum gravity through the 2-point function, linking fractal dimensions and entropy exponents, and verifies these relationships explicitly in 2D gravity.

Contribution

It establishes and verifies the quantum gravity analog of Fisher's scaling relation connecting 2-point function behavior, fractal dimension, and entropy exponent in 2D gravity.

Findings

Confirmed the exponential decay of the 2-point function determines the fractal dimension.

Derived the integral of the 2-point function relates to the entropy exponent.

Verified the quantum Fisher scaling relation in 2D gravity.

Abstract

The 2-point function is the natural object in quantum gravity for extracting critical behavior: The exponential fall off of the 2-point function with geodesic distance determines the fractal dimension $d_H$ of space-time. The integral of the 2-point function determines the entropy exponent $\gamma$, i.e. the fractal structure related to baby universes, while the short distance behavior of the 2-point function connects $\gamma$ and $d_H$ by a quantum gravity version of Fisher's scaling relation. We verify this behavior in the case of 2d gravity by explicit calculation.