The effective action in four-dimensional CDT

Abstract

We present recent results concerning the measurement and analysis of the effective action in four-dimensional Causal Dynamical Triangulations. The action describes quantum fluctuations of the spatial volume of the CDT universe (or alternatively the scale factor) after integrating out other degrees of freedom. We use the covariance of volume fluctuations to measure and parametrize the effective action inside the de Sitter phase, also called the C phase. We show that the action is consistent with a simple discretization of the minisuperspace action (with a reversed overall sign). We discuss possible subleading corrections and show how to construct a more complicated effective action comprising both integer and half-integer discrete proper time layers. We introduce a new method of the effective action measurement based on the transfer matrix. We show that the results of the new method are fully consistent with the covariance matrix method inside the de Sitter phase. We use the new method to measure the effective action in the small volume range and to explain the behaviour of the stalk part of the CDT triangulations. Finally we use the transfer matrix method to measure and parametrize the effective action inside the A and B phases, and to analyze the phase transitions. The results lead to an unexpected discovery of a new "bifurcation" phase separating the "old" C and B phases. We analyze geometric properties of triangulations inside the new phase and draw a new phase diagram.