Abstract
Flatness -- the absence of spacetime curvature -- is a well-understood property of macroscopic, classical spacetimes in general relativity. The same cannot be said about the concepts of curvature and flatness in nonperturbative quantum gravity, where the microscopic structure of spacetime is not describable in terms of small fluctuations around a fixed background geometry. An interesting case are two-dimensional models of quantum gravity, which lack a classical limit and therefore are maximally "quantum". We investigate the recently introduced quantum Ricci curvature in CDT quantum gravity on a two-dimensional torus, whose quantum geometry could be expected to behave like a flat space on suitably coarse-grained scales. On the basis of Monte Carlo simulations we have performed, with system sizes of up to 600.000 building blocks, this does not seem to be the case. Instead, we find a scale-independent "quantum flatness", without an obvious classical analogue. As part of our study, we develop a criterion that allows us to distinguish between local and global, topological properties of the toroidal quantum system.
Highlights
In classical general relativity, spacetime is characterized in terms of its curvature properties
It was unclear whether a meaningful notion of curvature can be defined in quantum gravity beyond perturbation theory, in the sense of a finite, renormalized quantum curvature operator that remains well-defined in aPlanckian regime. This has changed with the advent of the quantum Ricci curvature, introduced in [1,2]. It was subsequently implemented in full, four-dimensional quantum gravity, formulated in terms of causal dynamical triangulations (CDT), leading to the remarkable result that the large-scale curvature of the nonperturbatively generated quantum universe is compatible with that of a de Sitter space [3]
We will investigate the quantum Ricci curvature of two-dimensional CDT quantum gravity on a torus, describing (1 þ 1)-dimensional universes whose spatial slices are compact one-spheres or circles of variable length L, and where for the convenience of the computer simulations, we cyclically identify the time direction
Summary
Spacetime is characterized in terms of its curvature properties. We will investigate the quantum Ricci curvature of two-dimensional CDT quantum gravity on a torus, describing (1 þ 1)-dimensional universes whose spatial slices are compact one-spheres or circles of variable length L, and where for the convenience of the computer simulations, we cyclically identify the time direction This model was solved analytically in [8] and is well studied; it has a spectral dimension dS of at most 2 and a Hausdorff dimension dH of almost surely 2 [9], the latter in agreement with earlier theoretical [8,10] and numerical [11] results.
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