A first look at transition amplitudes in (2 + 1)-dimensional causal dynamical triangulations

Abstract

We study a lattice regularization of the gravitational path integral--causal\ndynamical triangulations--for (2+1)-dimensional Einstein gravity with positive\ncosmological constant in the presence of past and future spacelike boundaries\nof fixed intrinsic geometries. For spatial topology of a 2-sphere, we determine\nthe form of the Einstein-Hilbert action supplemented by the\nGibbons-Hawking-York boundary terms within the Regge calculus of causal\ntriangulations. Employing this action we numerically simulate a variety of\ntransition amplitudes from the past boundary to the future boundary. To the\nextent that we have so far investigated them, these transition amplitudes\nappear consistent with the gravitational effective action previously found to\ncharacterize the ground state of quantum spacetime geometry within the\nEuclidean de Sitter-like phase. Certain of these transition amplitudes\nconvincingly demonstrate that the so-called stalks present in this phase are\nnumerical artifacts of the lattice regularization, seemingly indicate that the\nquantization technique of causal dynamical triangulations differs in detail\nfrom that of the no-boundary proposal of Hartle and Hawking, and possibly\nrepresent the first numerical simulations of portions of temporally unbounded\nquantum spacetime geometry within the causal dynamical triangulations approach.\nWe also uncover tantalizing evidence suggesting that Lorentzian not Euclidean\nde Sitter spacetime dominates the ground state on sufficiently large scales.\n