Spectrum of the Laplace-Beltrami Operator and the Phase Structure of Causal Dynamical Triangulation

Abstract

We propose a new method to characterize the different phases observed in the non-perturbative numerical approach to quantum gravity known as Causal Dynamical Triangulation. The method is based on the analysis of the eigenvalues and the eigenvectors of the Laplace-Beltrami operator computed on the triangulations: it generalizes previous works based on the analysis of diffusive processes and proves capable of providing more detailed information on the geometric properties of the triangulations. In particular, we apply the method to the analysis of spatial slices, showing that the different phases can be characterized by a new order parameter related to the presence or absence of a gap in the spectrum of the Laplace-Beltrami operator, and deriving an effective dimensionality of the slices at the different scales. We also propose quantities derived from the spectrum that could be used to monitor the running to the continuum limit around a suitable critical point in the phase diagram, if any is found.

arXiv comment21 pages, 26 figures, 2 tables

References in corpus (16)

• Spectral Dimension of the Universe
• Emergence of a 4D World from Causal Quantum Gravity
• Quantum Einstein Gravity
• Nonperturbative Quantum Gravity
• A second-order phase transition in CDT
• Second- and First-Order Phase Transitions in CDT
• Lattice Quantum Gravity and Asymptotic Safety
• Characteristics of the new phase in CDT
• Evidence for Asymptotic Safety from Dimensional Reduction in Causal Dynamical Triangulations
• De Sitter Universe from Causal Dynamical Triangulations without Preferred Foliation
• New higher-order transition in causal dynamical triangulations
• Anderson transition and multifractals in the spectrum of the Dirac operator of Quantum Chromodynamics at high temperature
• The effective action in 4-dim CDT. The transfer matrix approach
• Signature Change of the Metric in CDT Quantum Gravity?
• Four-dimensional CDT with toroidal topology
• Euclidean Dynamical Triangulation revisited: is the phase transition really 1st order? (extended version)

Cited by in corpus (8)

• Quantum Gravity from Causal Dynamical Triangulations: A Review
• Towards reconstructing the quantum effective action of gravity
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• Approximate Killing symmetries in non-perturbative quantum gravity
• Compact Gauge Fields on Causal Dynamical Triangulations: a 2D case study
• Derivation of the deformed Heisenberg algebra from discrete spacetime
• Spectral Analysis of Causal Dynamical Triangulations via Finite Element Method
• Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models