A renormalization group for dynamical triangulations in arbitrary dimensions

Renken, R.L.

doi:10.1016/s0550-3213(96)00611-6

Cited by 11 publications

(11 citation statements)

References 19 publications

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“…My motivation is two-fold. First, the coarse graining procedures developed by Johnston, Kownacki, and Krzywicki [30] and by Catterall, Renken, and Thorleifsson [35,36,39] for Euclidean dynamical triangulations and even the coarse graining procedure developed by Henson [27] for causal dynamical triangulations are in fact not well-suited to causal dynamical triangulations: the triangulation obtained after just a single iteration of the procedure is no longer necessarily a causal triangulation. (Indeed, the output of Henson's coarse graining procedure might not even be a simplicial manifold.)…”

Section: Construction Of the Renormalization Group Flowsmentioning

confidence: 99%

On a renormalization group scheme for causal dynamical triangulations

Cooperman

2016

Gen Relativ Gravit

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The causal dynamical triangulations approach aims to construct a quantum theory of gravity as the continuum limit of a lattice-regularized model of dynamical geometry. A renormalization group schemein concert with finite size scaling analysis-is essential to this aim. Formulating and implementing such a scheme in the present context raises novel and notable conceptual and technical problems. I explored these problems, and, building on standard techniques, suggested potential solutions in the first paper of this two-part series. As an application of these solutions, I now propose a renormalization group scheme for causal dynamical triangulations. This scheme differs significantly from that studied recently by Ambjørn, Görlich, Jurkiewicz, Kreienbuehl, and Loll.

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Section: Construction Of the Renormalization Group Flowsmentioning

confidence: 99%

On a renormalization group scheme for causal dynamical triangulations

Cooperman

2016

Gen Relativ Gravit

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“…This amounts to adding to the action a term of the form −µ v ln o(v)/5, where o(v) is the number of 4-simplices containing a vertex v. For µ = −5, −1, 1, 5, this seems to lead merely to a shift of the critical line. This setting has been revived recently by Renken [178,177] in the context of a renormalization group analysis.…”

Section: Influence Of the Measurementioning

confidence: 99%

“…This step can only be performed once, which severely limits the power of the procedure. Renken [178,177] has applied a different blocking move, involving node deletion, and studied the RG flow using the volume and the vertex order as observables.…”

Section: Renormalization Groupmentioning

confidence: 99%

Discrete Approaches to Quantum Gravity in Four Dimensions

Loll

1998

Living Rev. Relativ.

179

175

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The construction of a consistent theory of quantum gravity is a problem in theoretical physics that has so far defied all attempts at resolution. One ansatz to try to obtain a non-trivial quantum theory proceeds via a discretization of space-time and the Einstein action. I review here three major areas of research: gauge-theoretic approaches, both in a path-integral and a Hamiltonian formulation; quantum Regge calculus; and the method of dynamical triangulations, confining attention to work that is strictly four-dimensional, strictly discrete, and strictly quantum in nature.

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“…larger volumes) are needed to determine which phase it is in or whether it approaches a fixed point. At a strongly first order transition, such as in three dimensions with µ = 0, it is not necessary to go to large volumes to see what the phase is, so there are no intermediate flows [8]. In four dimensions, the corresponding transition appears second order at small volumes, and there are intermediate flows similar to the one in the figure.…”

Section: Results For the Transition Linementioning

confidence: 93%

“…When combined with Monte Carlo simulation it allows a non-perturbative determination of the flows and phase structure. A formulation of the renormalization group applicable to dynamical triangulations has been developed and established in previous papers [6,4,7,8]. It is useful to recall some details of this scheme.…”

Section: Computational Techniquesmentioning

confidence: 99%

Phase structure of dynamical triangulation models in three dimensions

1998

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The dynamical triangulation model of three-dimensional quantum gravity is shown to have a line of transitions in an expanded phase diagram which includes a coupling µ to the order of the vertices. Monte Carlo renormalization group and finite size scaling techniques are used to locate and characterize this line. Our results indicate that for µ < µ1 ∼ −1.0 the model is always in a crumpled phase independent of the value of the curvature coupling. For µ < 0 the results are in agreement with an approximate mean field treatment. We find evidence that this line corresponds to first order transitions extending to positive µ. However the behavior appears to change for µ > µ2 ∼ 2.0 − 4.0. The simplest scenario that is consistent with the data is the existence of a critical end point.PACS numbers: 04.60.+n, 05.70.Jk, 11.10.Gh I. DYNAMICAL TRIANGULATIONS WITH A MEASURE TERMTriangulations provide a discretization of curved Euclidean spacetimes. In the Regge approach, a specific simplicial lattice is chosen a priori and the properties of the spacetime are determined by the length of the lattice links. A complementary approach, dynamical triangulations, fixes the length of the links to an invariant cut-off and allows the lattice connectivity to determine the geometry of the spacetime. This paper uses the latter approach. A lattice representation of the Einstein-Hilbert action iswhere N 0 is the number of vertices and N D the number of simplices in the triangulation (the latter is then also the volume), D is the dimension of the system and α and β are corresponding chemical potentials. The coupling β serves as the cosmological constant while α is related to Newton's gravitational constant. The connectivity of a triangulation plays the same role as the metric in continuous manifolds in the sense that points joined by a link are considered close together while those not connected are considered further apart. Continuum theories of quantum gravity generally involve an integration over all possible physically inequivalent metrics. In dynamical triangulation models, a sum over triangulations (i.e a sum over all allowed connectivities) fulfills the same need. Consequently, the partition function for the dynamical triangulation model of quantum gravity iswhere the sum is over all possible triangulations with fixed topology, S is the action given above and the term ρ(T ) allows for the possibility of a non-trivial measure in the space of triangulations. Matter can be coupled to the gravity by adding appropriate terms to eqn. (1) and including a sum over the matter degrees of freedom in eqn.(2). These models of quantum gravity have proven themselves in two dimensions where analytical results are available both in the continuum and on the lattice. They agree in all cases where comparisons are possible. This agreement occurs with trivial measure ρ(T ) = 1. In higher dimensions, it may be necessary to utilize a non-trivial measure term in order to obtain an acceptable lattice theory. In this paper, the three-dimensional model is modified s...

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A renormalization group for dynamical triangulations in arbitrary dimensions

Abstract: A block spin renormalization group approach is proposed for the dynamical triangulation formulation of quantum gravity in arbitrary dimensions. Renormalization group flow diagrams are presented for the three-dimensional and four-dimensional theories near their respective transitions.

Cited by 11 publications

References 19 publications

On a renormalization group scheme for causal dynamical triangulations

On a renormalization group scheme for causal dynamical triangulations

Discrete Approaches to Quantum Gravity in Four Dimensions

Phase structure of dynamical triangulation models in three dimensions

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