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On a renormalization group scheme for causal dynamical triangulations

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Abstract

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 scheme—in 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 a previous paper (Cooperman, arXiv:gr-qc/1410.0026). 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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Notes

  1. Whenever I refer to a scale, I always refer to a length scale.

  2. This is what it means for a quantity to be dimensionful: one requires a standard unit of measure to ascertain its value.

  3. I choose to display factors of \(\hbar \), but I treat \(\hbar \) as having the value unity.

  4. See [12, 13] for the original formulation and [8] for a comprehensive review.

  5. Numerical measurements of \(\langle d_{\mathfrak {s}}(\sigma )\rangle \) interpreted within the model of Sect. 3.3 do not alone determine the scale \(\ell _{\mathrm {dS}}\) in units of the lattice spacing a.

  6. Technically, a standard Wick rotation of the action (2.7) yields an overall negative sign, giving a kinetic term of the wrong sign. This is the well-known unboundedness from below of the conformal mode of the metric tensor \(\mathbf {g}\). Evidently, the effective action describing the ensemble average quantum geometry of phase C on sufficiently large scales does not have this overall negative sign [15, 16].

  7. One might wonder why Benedetti and Henson chose to fit the spectral dimension of the stretched sphere, not that of Euclidean de Sitter space, to \(\langle d_{\mathfrak {s}}(\sigma )\rangle \). Their motivation was twofold: first, they pointed out that the random walker probes the quantum geometry more generally than does the observable \(N_{2}^{\mathrm {SL}}(\tau )\), and, second, they were interested in possibly finding evidence for a Hořava-Lifshitz-like continuum limit as suggested in [4]. The stretched sphere is not a solution of the Einstein equations except for \(g_{tt}=1\) but is a solution of the Hořava-Lifshitz equations. This second point suggests that their use of the finite size scaling Ansatz (3.12) is not necessarily compatible with their choice of the line element (3.42) since the finite size scaling Ansatz assumes isotropic scaling.

  8. One might wonder why I have not developed a coarse graining procedure for the implementation of a renormalization group transformation, especially since this is the standard technique applied to lattice-regularized quantum theories of fields. My motivation is two-fold. First, the coarse graining procedures developed by Johnston, Kownacki, and Krzywicki [34] and by Catterall, Renken, and Thorleifsson [39, 40, 43] for Euclidean dynamical triangulations and even the coarse graining procedure developed by Henson [31] 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.) Since subsequent iterations do not begin with a causal triangulation, the relevance of their outcomes is in doubt. Second, reasoning from the general expectations of Sect. 4.1, a coarse graining procedure for causal dynamical triangulations should function to reduce simultaneously the number \(N_{d+1}\) of \((d+1)\)-simplices and the number T of time slices while preserving the causal nature of the triangulation. Undoubtedly, one could design such a coarse graining procedure. Its relevance for ensembles of causal triangulations that one can currently simulate is likely quite limited: owing to the relatively small values of \(T_{\mathrm {ph}}\) obtained, very few iterations of the coarse graining procedure could be implemented. The method for implementation of renormalization group transformations that I have proposed should allow for much more incremental iterations.

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Acknowledgments

I wish to thank Jan Ambjørn, Steven Carlip, and Renate Loll for several useful conversations. I also wish to thank Renate Loll for comments on a draft of this paper. I acknowledge support from the Foundation for Fundamental Research on Matter itself supported by the Netherlands Organization for Scientific Research. This research was supported in part by the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported by the Government of Canada through Industry Cananda and by the Province of Ontario through the Ministry of Economic Development and Innovation. This research was also supported in part by the United States Department of Energy under grant DE-FG02-91ER40674 at the University of California, Davis.

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  1. Institute for Mathematics, Astrophysics, and Particle Physics, Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6526 AJ, Nijmegen, The Netherlands

    Joshua H. Cooperman

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Cooperman, J.H. On a renormalization group scheme for causal dynamical triangulations. Gen Relativ Gravit 48, 29 (2016). https://doi.org/10.1007/s10714-016-2027-4

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