The effective action in 4-dim CDT. The transfer matrix approach

Abstract

Abstract:We measure the effective action in all three phases of 4-dimensional Causal Dynamical Triangulations (CDT) using the transfer matrix method. The transfer matrix is parametrized by the total 3-volume of the CDT universe at a given (discrete) time. We present a simple effective model based on the transfer matrix measured in the de Sitter phase. It allows us to reconstruct the results of full CDT in this phase. We argue that the transfer matrix method is valid not only inside the de Sitter phase ('C') but also in the other two phases. A parametrization of the measured transfer matrix/effective action in the 'A' and 'B' phases is proposed and the relation to phase transitions is explained. We discover a potentially new 'bifurcation' phase separating the de Sitter phase ('C') and the 'collapsed' phase ('B').

Signature change of the metric in CDT quantum gravity?

“…[17] a likely second order transition was identified for ∆ ∼ 0, thus raising the exciting possibility of defining a continuum limit in CDT. The newly reported bifurcation phase [20], however, exists in the parameter space between the physical phase C and the second order transition, and so being able to approach the second order transition from within the physical phase seems less likely. Therefore, it is important to establish the actual extent of the bifurcation phase in the CDT parameter space and check whether phases C and B meet directly in some region.…”

“…Inside the bifurcation phase the situation is quite different, and the measured transfer matrix takes the form [20] Starting within phase C and keeping κ 0 fixed while decreasing ∆, for some critical value ∆ c we observe a phase transition where c 0 changes from zero to some positive value.…”

“…It was argued in ref. [20] that the transition between phases A and C can be identified by the vanishing of the kinetic term of the effective transfer matrix (1.3). In phase C the kinetic term is Gaussian with a positive coefficient.…”

“…One can also perform the same study but keeping κ 0 fixed and varying ∆. In this case one encounters a new so-called bifurcation phase separating phase C from phase B [20], see figure 9. Within this new bifurcation phase the kinetic term of the effective transfer matrix (1.3) bifurcates from a single Gaussian characteristic of small spatial volumes to a sum of two shifted Gaussians for large volumes.…”

“…The transfer matrix approach was also used to parametrise the effective action in other phases of CDT, and to analyse the phase transitions. This led to the discovery of a new bifurcation phase, where the transfer matrix can again be used to construct a simplified model that explains the observed narrowing of the spatial volume profile [20].…”