[Paper Review] Statistical discrete geometry

Seramika Ariwahjoedi, Valerio Astuti+3 more|arXiv (Cornell University)|Jul 28, 2016

Black Holes and Theoretical Physics50 references3 citations

TL;DR

This paper constructs statistical discrete geometry by applying statistical mechanics to Regge gravity, introducing a coarse-graining procedure that preserves atomism and background independence through UV and IR cut-offs. It argues that the thermodynamical limit—not the continuum limit—is the correct large-$n$ limit for recovering classical general relativity from discrete quantum gravity.

ABSTRACT

Following our earlier work, we construct statistical discrete geometry by applying statistical mechanics to discrete (Regge) gravity. We propose a coarse-graining method for discrete geometry under the assumptions of atomism and background independence. To maintain these assumptions, restrictions are given to the theory by introducing cut-offs, both in ultraviolet and infrared regime. Having a well-defined statistical picture of discrete Regge geometry, we take the infinite degrees of freedom (large n) limit. We argue that the correct limit consistent with the restrictions and the background independence concept is not the continuum limit of statistical mechanics, but the thermodynamical limit.

Motivation & Objective

• To develop a statistical mechanics framework for discrete Regge gravity that respects fundamental principles of quantum gravity.
• To address the classical limit of discrete quantum gravity while maintaining atomism and background independence.
• To resolve the tension between continuum limits and background independence in quantum gravity.
• To identify the correct large-$n$ limit—thermodynamical rather than continuum—for recovering general relativity from discrete geometry.

Proposed method

• Applies statistical mechanics to discrete (Regge) gravity by treating spacetime foliations as many-body systems with coarse-grained variables: lengths, areas, and volumes.
• Introduces UV and IR cut-offs to enforce finite degrees of freedom, ensuring consistency with atomism and preventing divergences.
• Imposes restrictions on the theory to maintain background independence, treating quanta of space as the space itself without a pre-existing stage.
• Uses a refinement procedure to take the infinite degrees of freedom limit, analyzing the behavior of macroscopic extensive variables.
• Compares the continuum limit and thermodynamical limit, showing the latter is consistent with background independence and finite informational entropy.
• Proposes that the equation of state derived from the thermodynamical limit may reproduce general relativity, particularly if entropy is proportional to area.

Experimental results

Research questions

• RQ1 Can a consistent statistical mechanics framework be constructed for discrete Regge gravity while preserving atomism and background independence?
• RQ2 What is the correct large-$n$ limit of discrete geometry that recovers classical general relativity?
• RQ3 Why is the thermodynamical limit preferred over the continuum limit in background-independent quantum gravity?
• RQ4 How do UV and IR cut-offs constrain the informational entropy and degrees of freedom in discrete geometry?
• RQ5 Can the equation of state derived from the thermodynamical limit of discrete geometry reproduce the laws of general relativity?

Key findings

• The thermodynamical limit, not the continuum limit, is the correct large-$n$ limit consistent with background independence and atomism in discrete gravity.
• UV and IR cut-offs are essential to truncate infinite degrees of freedom and prevent divergences, ensuring finite informational entropy.
• The extensive macroscopic variables—total length, area, and volume—diverge as $n \to \infty$ in the thermodynamical limit, consistent with background independence.
• The continuum limit violates both atomism (via UV divergence) and background independence (by fixing total size), making it inconsistent with the foundational principles.
• The thermodynamical limit may yield an effective equation of state that reproduces general relativity, particularly if entropy is proportional to area.
• The quantum version of this framework is under development, suggesting a path toward understanding the thermodynamics of quantum gravity.