Scaling in four-dimensional quantum gravity

Ambjørn, J.; Jurkiewicz, J.

doi:10.1016/0550-3213(95)00303-a

Cited by 97 publications

(164 citation statements)

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“…The "BU surgery" algorithm proposed in [10,15] should be appropriate there. Performing more intensive MC studies one should also be able to estimate the corrections to the formula (14) coming from irrelevant directions at the fixed point, neglected here.(ii) Remember, that ν −1 has the significance of the intrinsic Haussdorf dimension d H of the manifold [15]. It would be interesting to look for the tdependence of d H and determine the class of theories satisfying the constraint d H = 4.…”

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confidence: 99%

“…8 the local algorithm becomes inefficient as one enters the "cold" phase of the theory, where the manifold develops a branched polymer structure. The "BU surgery" algorithm proposed in [10,15] should be appropriate there. Performing more intensive MC studies one should also be able to estimate the corrections to the formula (14) coming from irrelevant directions at the fixed point, neglected here.…”

mentioning

confidence: 99%

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confidence: 99%

“…(ii) Remember, that ν −1 has the significance of the intrinsic Haussdorf dimension d H of the manifold [15]. It would be interesting to look for the tdependence of d H and determine the class of theories satisfying the constraint d H = 4.…”

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Towards a non-perturbative renormalization of euclidean quantum gravity

1995

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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...

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Towards a non-perturbative renormalization of euclidean quantum gravity

1995

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“…1, the susceptibility has a peak around κ 2 = 1.2 ∼ 1.3, which grows higher as the system size is increased. This implies that the correlation length of the local curvature diverges at the critical point [6], where we may hope to take a continuum limit. Since κ 2 corresponds to the inverse of the gravitational constant, as is seen from (5), we call the large κ 2 phase as the weak coupling phase and the small κ 2 phase as the strong coupling phase .…”

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Singular vertices in the strong coupling phase of four-dimensional simplicial gravity

Hotta

Izubuchi

Nishimura

1996

Nuclear Physics B - Proceedings Supplements

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We study four-dimensional simplicial gravity through numerical simulation with special attention to the existence of singular vertices, in the strong coupling phase, that are shared by abnormally large numbers of four-simplices. The second order phase transition from the strong coupling phase into the weak coupling phase could be understood as the disappearance of the singular vertices. We also change the topology of the universe from the sphere to the torus. † based on the talk given at Lattice '95 in Melbourne and "The new trend in the quantum field theory" in Kyoto.

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Surplus Anomaly and Random Geometries

Białas¹,

Burda²,

Johnston³

NATO Science Series: B:

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Scaling in four-dimensional quantum gravity

Cited by 97 publications

References 70 publications

Towards a non-perturbative renormalization of euclidean quantum gravity

Towards a non-perturbative renormalization of euclidean quantum gravity

Singular vertices in the strong coupling phase of four-dimensional simplicial gravity

Surplus Anomaly and Random Geometries

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