Abstract
Abstract The Causal Dynamical Triangulation model of quantum gravity (CDT) 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 labeled 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.
Highlights
The time foliation present in CDT provides us with a transfer matrix
In Dynamical Triangulations (DT) the assumption is that this set of geometries is in a suitable sense dense in the set of continuous geometries when we take the link length a to zero, and in this way the link length a will act as an ultraviolet cutoff, just like in ordinary lattice field theory
CDT comes with a transfer matrix T |M|T
Summary
The use of piecewise linear geometries was introduced in the context of general relativity by Regge [23] as a natural tool to work with a discretized version of the Hilbert-Einstein action, but without the use of coordinates. The transfer matrix is defined on the vector space spanned by the set T3 of threedimensional triangulations. This space is infinite dimensional and has the natural scalar product. The measurements performed so far, using Monte Carlo simulations, have been concentrated on the measurement of the scale factor, or more conveniently the three-volume nti ≡ N3(ti) at the spatial slice at time ti, as well as the correlation between the three-volumes at time ti and time tj These “observables” can be expressed using the transfer matrix M. We will in the following try to determine the transfer matrix from the data, in the range where nt is not necessarily large and we will try to improve the expression (2.22)
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