Non-perturbative Lorentzian quantum gravity, causality and topology change

Abstract

We formulate a non-perturbative lattice model of two-dimensional Lorentzian quantum gravity by performing the path integral over geometries with a causal structure. The model can be solved exactly at the discretized level. Its continuum limit coincides with the theory obtained by quantizing 2d continuum gravity in proper-time gauge, but it disagrees with 2d gravity defined via matrix models or Liouville theory. By allowing topology change of the compact spatial slices (i.e. baby universe creation), one obtains agreement with the matrix models and Liouville theory.

The unitary conformal field theory behind 2D Asymptotic Safety

“…In earlier work [100][101][102] indications were found that suggest that Quantum Einstein Gravity in the continuum formulation based upon the EAA might be related to the discrete approach employing Causal Dynamical Triangulation [79,103]. In particular, the respective predictions for the fractal dimensions of spacetime were compared in detail and turned out similar [101,102].…”

“…Since the set of nondegenerate metrics with fixed signature forms a nonempty open subset in the space of all covariant symmetric 2-tensor fields [59], there is no a priori reason to expect that it has vanishing measure, and so this question has no obvious answer. It is known, however, that "sufficiently different" choices will lead to inequivalent theories [79].…”

The transfer matrix in four-dimensional CDT

“…However, each piecewise linear geometry used in the CDT path integral allows a rotation to Euclidean proper time. The set of Euclidean geometries we obtain in this way is a subset of the DT Euclidean geometries and this restriction seemingly cures some of the higher dimensional DT diseases, while in two dimensions the relation between the restricted theory and the full DT theory has been worked out in detail: one obtains the CDT theory from the DT theory by integrating out all baby universes (which results in a non-analytic mapping between the coupling constants of the two theories), and (somewhat surprisingly) one can restore the DT theory from the CDT theory by the inverse mapping [45,46]. Using four-simplices (which is the case having our attention in this article) as building blocks one can, for suitable choices of bare coupling constants, observe a four-dimensional (Euclidean) universe [47,48].…”

Wilson loops in nonperturbative quantum gravity

“…Would a corresponding Lorentzian calculation yield a uniform distribution of SOð3; 1Þ-holonomies on the noncompact group manifold of SOð3; 1Þ with respect to its Haar measure? Although CDT has a well-defined Wick rotation [1,22], which allows us to go back and forth between Euclidean and Minkowskian length and angle assignments for individual building blocks, this question cannot be answered straightforwardly. Courtesy of the Wick rotation, we can assign to a given lattice loop both an SOð4Þ-holonomy-as we have done in this paper-and a Lorentzian SOð3; 1Þ-holonomy, simply by computing the transition matrices Rðs i ; s j Þ in terms of the Minkowskian length assignments before Wick rotating.…”