Nonperturbative quantum gravity

Abstract

Asymptotic safety describes a scenario in which general relativity can be quantized as a conventional field theory, despite being nonrenormalizable when expanding it around a fixed background geometry. It is formulated in the framework of the Wilsonian renormalization group and relies crucially on the existence of an ultraviolet fixed point, for which evidence has been found using renormalization group equations in the continuum."Causal Dynamical Triangulations" (CDT) is a concrete research program to obtain a nonperturbative quantum field theory of gravity via a lattice regularization, and represented as a sum over spacetime histories. In the Wilsonian spirit one can use this formulation to try to locate fixed points of the lattice theory and thereby provide independent, nonperturbative evidence for the existence of a UV fixed point.We describe the formalism of CDT, its phase diagram, possible fixed points and the "quantum geometries" which emerge in the different phases. We also argue that the formalism may be able to describe a more general class of Hořava-Lifshitz gravitational models.

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].…”

Wilson loops in nonperturbative quantum gravity

“…The building blocks of the piecewise linear geometries used in dynamical triangulations are identical, equilateral 5 four-simplices, which by assumption are everywhere flat on the inside, like the building blocks of Regge calculus [17]. When these building blocks are glued together along identical boundary three-simplices or "faces" to construct a four-dimensional piecewise flat manifold, curvature will generically appear in a singular fashion along the two-dimensional subsimplices of the triangulation, the triangles or "hinges.…”

“…More complete descriptions can be found elsewhere; see [5] and references therein. In standard CDT, each spacetime T has a proper-time slicing with integer label t, and is assembled from four-simplices in a layered fashion, 12 where one layer of thickness Δt ¼ 1 is a piecewise flat piece of spacetime of topology S 3 × I, all of whose vertices are contained in either of its spatial boundary submanifolds at times t or t þ 1.…”

“…The action we use is the Regge form of the Einstein-Hilbert action [17], applicable to piecewise linear geometries. On causal dynamical triangulations with identical equilateral building blocks it assumes a particularly simple form (see [5] for details).…”

Quantum Gravity via Causal Dynamical Triangulations

“…For a more detailed description of the CDT construction we refer the interested reader to [22,23,24].…”