The transfer matrix method in four-dimensional causal dynamical triangulations

TL;DR

This paper investigates the transfer matrix in four-dimensional causal dynamical triangulations (CDT), demonstrating its role in describing the universe's scale factor and connecting it to an effective minisuperspace action through computer simulations.

Contribution

It introduces an effective transfer matrix for CDT that depends solely on the scale factor, linking it to a minisuperspace action and enhancing understanding of the model's phase structure.

Findings

The CDT model predicts a de Sitter phase with a semi-classical geometry.

The effective transfer matrix accurately describes the scale factor dynamics.

Measurements relate the transfer matrix to an effective minisuperspace action.

Abstract

The Causal Dynamical Triangulation model of quantum gravity (CDT) is a proposition to evaluate the path integral over space-time geometries using a lattice regularization with a discrete proper time and geometries realized as simplicial manifolds. The model admits a Wick rotation to imaginary time for each space-time configuration. Using computer simulations we determined the phase structure of the model and discovered that it predicts a de Sitter phase with a four-dimensional spherical semi-classical background geometry. The model has a transfer matrix, relating spatial geometries at adjacent (discrete lattice) times. The transfer matrix uniquely determines the theory. We show that the measurements of the scale factor of the (CDT) universe are well described by an effective transfer matrix where the matrix elements are labelled only by the scale factor. Using computer simulations we determine the effective transfer matrix elements and show how they relate to an effective minisuperspace action at all scales.