A Phase Transition in Quantum Einstein Gravity
Jan Smit University of Amsterdam
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We study the imaginary-time path integral over geometries with spherical topology by summing over approximate Einstein spaces. At positive curvature these consist of an arbitrary number of four-spheres, glued together such that they correspond to `branched-polymer' tree graphs which specify their abundance. At negative curvature, geometries of arbitrary size are constructed by gluing hyperbolic `cups', with an assumed abundance inspired by an earlier proposed large-volume behavior of the partition function in the so-called crumpled phase of the time-space symmetric Euclidean dynamical triangulation model (SDT). Using the semi-classical effective action of the Einstein theory at one-loop order, with a finite UV-cutoff, we construct model partition functions that depend on the four-volume and the gravitational coupling, and study the transition between phases of positive and negative curvature. Qualitative features of the average curvature found in numerical simulations of SDT also appear in this continuum formulation, such as a first-order phase transition with rather strong finite-size effects.