[Paper Review] Renormalization in quantum theories of geometry

J. Ambjørn, J. Gizbert-Studnicki+3 more|arXiv (Cornell University)|Feb 5, 2020

Noncommutative and Quantum Gravity Theories1 references4 citations

TL;DR

This paper investigates the applicability of Wilsonian renormalization group methods in quantum gravity, focusing on Causal Dynamical Triangulations (CDT). It argues that the absence of a fixed background geometry complicates defining quantum length and scale, which undermines standard renormalization procedures. Despite extensive simulations, no evidence for a UV fixed point was found, suggesting a need for rethinking quantum geometric observables near the Planck scale.

ABSTRACT

A hallmark of non-perturbative theories of quantum gravity is the absence of a fixed background geometry, and therefore the absence in a Planckian regime of any notion of length or scale that is defined a priori. This has potentially far-reaching consequences for the application of renormalization group methods a la Wilson, which rely on these notions in a crucial way. We review the status quo of attempts in the Causal Dynamical Triangulations (CDT) approach to quantum gravity to find an ultraviolet fixed point associated with the second-order phase transitions observed in the lattice theory. Measurements of the only invariant correlator currently accessible, that of the total spatial three-volume, has not produced any evidence of such a fixed point. A possible explanation for this result is our incomplete and perhaps naive understanding of what constitutes an appropriate notion of (quantum) length near the Planck scale.

Motivation & Objective

• To assess whether Wilsonian renormalization group methods can be applied in non-perturbative quantum gravity theories lacking a fixed background geometry.
• To investigate why previous CDT simulations failed to detect a UV fixed point despite expectations from asymptotic safety scenarios.
• To explore whether the lack of evidence for a UV fixed point stems from an inadequate understanding of quantum distance and scale at the Planck scale.
• To evaluate whether conventional phase transition paradigms (e.g., Landau-type) apply to quantum geometric phase transitions in CDT.
• To consider alternative observables—such as quantum Ricci curvature—beyond the spatial volume profile for detecting fixed points.

Proposed method

• Adapting the Wilsonian renormalization group framework to quantum gravity by tracking how bare coupling constants evolve under scale transformations in lattice-regulated CDT models.
• Using Monte Carlo simulations to measure the total spatial three-volume correlator as the primary observable to probe the existence of a UV fixed point.
• Analyzing the behavior of the volume profile across phase transitions, particularly the $C_{dS}$--$C_{b}$ transition, to detect scale-invariant behavior.
• Assessing whether quantum geometric observables like distance and volume exhibit anomalous scaling, which could invalidate standard renormalization assumptions.
• Comparing CDT phase transitions to topological phase transitions in condensed matter physics, especially regarding non-local order parameters and long auto-correlation times.
• Evaluating the role of gauge-invariant and diffeomorphism-invariant observables in the absence of a fixed background metric.

Experimental results

Research questions

• RQ1 Can standard Wilsonian renormalization group methods be consistently applied in quantum gravity theories without a fixed background geometry?
• RQ2 Why has no UV fixed point been detected in CDT simulations despite theoretical expectations from the asymptotic safety scenario?
• RQ3 To what extent does the lack of a well-defined quantum notion of distance at the Planck scale invalidate conventional renormalization procedures?
• RQ4 Are the $C_{dS}$--$C_{b}$ phase transitions in CDT analogous to conventional Landau-type phase transitions, or do they exhibit topological characteristics?
• RQ5 Can alternative geometric observables, such as quantum Ricci curvature, provide better signals for a UV fixed point than the spatial volume profile?

Key findings

• No evidence for a UV fixed point was found in CDT simulations using the total spatial three-volume as the primary observable.
• The measured lattice version of the volume profile $ ilde{ ho}_{ ext{cl}}$ was found to be too small and failed to increase sufficiently near the $C_{dS}$--$C_{b}$ phase transition line.
• The absence of a fixed point may stem from an incomplete or naive understanding of how quantum length and volume behave at the Planck scale.
• Quantum distance and volume may exhibit nonclassical or anomalously scaling behavior, challenging the assumptions of standard renormalization group methods.
• The $C_{dS}$--$C_{b}$ transition in CDT shares features with topological phase transitions, including non-local order parameters and long auto-correlation times.
• The possibility remains open that the asymptotic safety scenario does not hold in this framework, or that the choice of global observables like volume is insufficient for detecting fixed points.