Towards a non-perturbative renormalization of euclidean quantum gravity

Abstract

A real space renormalization group technique, based on the hierarchical baby-universe structure of a typical dynamically triangulated manifold, is used to study scaling properties of 2d and 4d lattice quantum gravity. In 4d, the β-function is defined and calculated numerically. An evidence for the existence of an ultraviolet stable fixed point of the theory is presented. May 1995 LPTHE Orsay 95/341 Permanent address: Institute of Physics, Jagellonian University, ul. Reymonta 4, PL-30 059, Kraków, Poland 2 Laboratoire associé au CNRS, URA-D0063 1 1. As is well known, the renormalization group (RG) is a tool providing deep insight into the structure of a quantum field theory. It is certainly worth applying this tool to quantum gravity. This work is devoted to the development and the application of the real space renormalization group technique in the context of euclidean quantum gravity. It is a direct continuation of the work [1].We choose the lattice gravity framework. More precisely we adopt the particularly promising dynamical triangulation approach [2]. The remarkable results obtained within a class of exactly solvable models in two dimensions strongly suggest that the dynamical triangulation recipe is the correct way of discretizing gravity (at least for fixed topology).In conventional statistical mechanics a real space renormalization group transformation has two facets:(a) geometry -cells of the body are "blocked" together.(b) matter fields -"block" fields are defined in terms of the original fields. On a regular lattice it is trivial to perform the step (a) in such a manner that the resulting lattice is identical, modulo rescaling, to the original one.Since the values of critical couplings depend on the lattice type this selfsimilarity feature of the transformation is important. On a random lattice an appropriate definition of (a) requires some thought. In this work we consider pure geometry, without matter fields, and consequently the geometrical aspect of the renormalization group.In ref.[1] a method of "blocking" triangulations that exploits the selfsimilarity feature of random manifolds has been proposed. Without repeating in detail the arguments presented in [1] let us briefly sketch the main idea 3 . The intuitive arguments given below will be replaced progressively by more precise ones later on. We do not wish to give an impression of complexity from the outset.2. In 2d one can show [4] that an infinite randomly triangulated manifold is a self-similar tree obtained by gluing together sub-structures called baby universes (BUs), defined as subuniverses separated from the remaining part of the universe by a narrow neck. We conjecture that a similar picture holds in 4d, at least in the neighborhood of the phase transition point. The results presented in this paper strongly support this conjecture.3 As we have learned from J. Ambjørn during the LATTICE '94 Conference, our ideas partly overlap with those discussed earlier for 2d in ref. [3]. 2We have proposed [1] to define the step (a) as the o...

A scenario for the c > 1 barrier in non-critical bosonic strings

“…In fact there is another equally valid reason to take them into account. In the dynamical lattice approach of 2-D gravity, there has been several (mostly numerical) attempts to perform real space renormalization [14,24,25,[46][47][48][49][50]. This amounts to replace a block of neighboring cells of the random lattice by a single, larger cell.…”

“…A problem with such procedures is that they inevitably lead to touching points, since a small "bottleneck" can be replaced by a single vertex, thus generating wormhole-like configurations. Some real-space renormalization procedures consist in removing such touching points, when they connect a small surface (a "baby universe") to the parent surface [46,47]. From the point of view that I presented here, it is perhaps better to keep these configurations, and to include branching interactions from the beginning in these models.…”

Two-loop prediction for scaling exponents in (2 + ϵ)-dimensional quantum gravity

“…These scaling exponents have been numerically measured in four dimensions by using a real space renormalization group technique [17]. What has been observed is the following scaling relation at an ultraviolet stable fixed point: On the other hand, the classical scaling relation corresponds to the exponent β 0 = 2.…”

“…This prediction can be compared with the scaling exponent which is measured in a recent numerical simulation [17] of four dimensional quantum gravity by putting ǫ = 2.…”

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