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On the relation between Euclidean and Lorentzian 2D quantum gravity

J. Ambjorn, J. Correia, C. Kristjansen, R. Loll

TL;DR

This work analyzes how 2D Euclidean quantum gravity can be related to 2D Lorentzian quantum gravity by integrating out baby universes. Through a peeling decomposition of dynamical triangulations, the authors construct a mapping between Euclidean and Lorentzian histories and show propagators coincide under a renormalization of time and coupling constants. In the continuum limit, Lorentzian gravity emerges as a renormalized version of Euclidean gravity, with explicit relations between Euclidean and Lorentzian boundary cosmological constants and a dressing of the two loop correlator. The study links these 2D gravity theories to branched polymers and random walks, clarifying how baby universes shape universality and fractal structure in the two approaches.

Abstract

Starting from 2D Euclidean quantum gravity, we show that one recovers 2D Lorentzian quantum gravity by removing all baby universes. Using a peeling procedure to decompose the discrete, triangulated geometries along a one-dimensional path, we explicitly associate with each Euclidean space-time a (generalized) Lorentzian space-time. This motivates a map between the parameter spaces of the two theories, under which their propagators get identified. In two dimensions, Lorentzian quantum gravity can therefore be viewed as a ``renormalized'' version of Euclidean quantum gravity.

On the relation between Euclidean and Lorentzian 2D quantum gravity

TL;DR

This work analyzes how 2D Euclidean quantum gravity can be related to 2D Lorentzian quantum gravity by integrating out baby universes. Through a peeling decomposition of dynamical triangulations, the authors construct a mapping between Euclidean and Lorentzian histories and show propagators coincide under a renormalization of time and coupling constants. In the continuum limit, Lorentzian gravity emerges as a renormalized version of Euclidean gravity, with explicit relations between Euclidean and Lorentzian boundary cosmological constants and a dressing of the two loop correlator. The study links these 2D gravity theories to branched polymers and random walks, clarifying how baby universes shape universality and fractal structure in the two approaches.

Abstract

Starting from 2D Euclidean quantum gravity, we show that one recovers 2D Lorentzian quantum gravity by removing all baby universes. Using a peeling procedure to decompose the discrete, triangulated geometries along a one-dimensional path, we explicitly associate with each Euclidean space-time a (generalized) Lorentzian space-time. This motivates a map between the parameter spaces of the two theories, under which their propagators get identified. In two dimensions, Lorentzian quantum gravity can therefore be viewed as a ``renormalized'' version of Euclidean quantum gravity.

Paper Structure

This paper contains 6 sections, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The two-loop correlator and the peeling procedure
  3. The Euclidean case
  4. The Lorentzian case
  5. Renormalizing Euclidean gravity to Lorentzian gravity
  6. Discussion

Figures (2)

  • Figure 1: The minimal decomposition. In the left column the boundary before the decomposition and in the right column the boundary after the decomposition.
  • Figure 2: A representation of a Euclidean triangulation as a sequence of peeling moves along a "spiral-like" one-dimensional curve. Black rectangles are double links and grey circles baby universes.