[Paper Review] New aspects of two-dimensional quantum gravity

J. Ambjørn, R. Loll+3 more|arXiv (Cornell University)|Nov 21, 2009

Black Holes and Theoretical Physics47 references3 citations

TL;DR

This paper establishes a non-perturbative formulation of two-dimensional quantum gravity using causal dynamical triangulations (CDT), demonstrating its equivalence to a novel matrix model and a string field theory. It shows that the theory exhibits stochastic time dynamics due to topology change, with a non-Hermitian Hamiltonian made self-adjoint via stochastic quantization, leading to a discrete, positive spectrum and a continuum limit defined non-perturbatively.

ABSTRACT

Causal dynamical triangulations (CDT) can be used as a regularization of quantum gravity. In two dimensions the theory can be solved anlytically, even before the cut-off is removed and one can study in detail how to take the continuum limit. We show how the CDT theory is related to Euclidean 2d quantum gravity (Liouville quantum gravity), how it can be generalized and how this generalized CDT model has a string field theory representation as well as a matrix model representationof a new kind, and finally how it examplifies the possibility that time in quantum gravity might be the stochastic time related to the branching of space into baby universes.

Motivation & Objective

• To establish a non-perturbative formulation of two-dimensional quantum gravity using causal dynamical triangulations (CDT).
• To generalize CDT to allow local causality violations and topology change, enabling a richer quantum geometry.
• To show that the generalized CDT model admits a string field theory representation and a new type of matrix model in the continuum limit.
• To demonstrate that the stochastic time evolution in the model arises naturally from the Fokker-Planck equation and stochastic quantization.
• To define a non-perturbative Hamiltonian with a discrete, positive spectrum through a similarity transformation of the original non-Hermitian operator.

Proposed method

• Formalizing the CDT path integral over piecewise linear Lorentzian geometries with a causal structure, then rotating to Euclidean signature using the Regge action.
• Introducing a generalized CDT model that includes local causality violations and topology change via a modified action and measure.
• Deriving a Fokker-Planck equation for the stochastic time evolution of spatial loop lengths, leading to a non-Hermitian Hamiltonian.
• Applying a similarity transformation to the non-Hermitian Hamiltonian to obtain a self-adjoint operator in the standard $ L^2 $ space, ensuring a well-defined quantum theory.
• Establishing equivalence between the string field theory formulation and a new matrix model that directly describes the continuum limit.
• Using WKB analysis and spectral theory to study the eigenvalue spectrum of the transformed Hamiltonian, showing discrete and positive eigenvalues despite an unbounded potential.

Experimental results

Research questions

• RQ1 How does the inclusion of topology change and local causality violations affect the non-perturbative structure of 2D CDT?
• RQ2 Can a string field theory formulation be consistently derived for generalized CDT in two dimensions?
• RQ3 What is the role of stochastic time in the quantum dynamics of spatial loops in 2D quantum gravity?
• RQ4 How is the non-perturbative spectrum of the theory defined when the Hamiltonian is non-Hermitian?
• RQ5 What is the precise relation between the generalized CDT model and a new type of matrix model in the continuum limit?

Key findings

• The generalized CDT model in two dimensions is equivalent to a new matrix model that directly describes the continuum limit, enabling non-perturbative calculations.
• The string field theory formulation of the model is equivalent to the matrix model, providing a dual description of the same non-perturbative quantum gravity theory.
• The stochastic time evolution of spatial loops is governed by a Fokker-Planck equation, and the associated Hamiltonian is non-Hermitian but made self-adjoint via a similarity transformation.
• The transformed Hamiltonian $ ilde{H}(x) $ has a positive, discrete spectrum due to its factorization as $ R^ op R $, ensuring a well-defined quantum theory.
• The wave functions for energies below the potential barrier exhibit exponential decay, while those above show oscillatory behavior, with the latter still square-integrable under the correct measure.
• The average size of the spatial universe becomes infinite when topology change is allowed, signaling a dramatic change in quantum geometry compared to the strictly causal model.