Scaling behaviour in random non-commutative geometries

Random non-commutative geometries are a novel approach to taking a non-perturbative path integral over geometries. They were introduced in arxiv.org/abs/1510.01377, where a first examination was performed. During this examination we found that some geometries show indications of a phase transition. In this article we explore this phase transition further for geometries of type $(1,1)$, $(2,0)$, and $(1,3)$. We determine the pseudo critical points of these geometries and explore how some of the observables scale with the system size. We also undertake first steps towards understanding the critical behaviour through correlations and in determining critical exponents of the system.Comment: 16 pages, 16 figures (v2: updated after review

Towards a Definition of Locality in a Manifoldlike Causal Set

It is a common misconception that spacetime discreteness necessarily implies a violation of local Lorentz invariance. In fact, in the causal set approach to quantum gravity, Lorentz invariance follows from the specific implementation of the discreteness hypothesis. However, this comes at the cost of locality. In particular, it is difficult to define a "local" region in a manifoldlike causal set, i.e., one that corresponds to an approximately flat spacetime region. Following up on suggestions from previous work, we bridge this lacuna by proposing a definition of locality based on the abundance of m-element order-intervals as a function of m in a causal set. We obtain analytic expressions for the expectation value of this function for an ensemble of causal set that faithfully embeds into an Alexandrov interval in d-dimensional Minkowski spacetime and use it to define local regions in a manifoldlike causal set. We use this to argue that evidence of local regions is a necessary condition for manifoldlikeness in a causal set. This in addition provides a new continuum dimension estimator. We perform extensive simulations which support our claims.Comment: 35 pages, 17 figure

Causal set d'Alembertians for various dimensions

We propose, for dimension d, a discrete Lorentz invariant operator on scalar fields that approximates the Minkowski spacetime scalar d'Alembertian. For each dimension, this gives rise to a scalar curvature estimator for causal sets, and thence to a proposal for a causal set action.Comment: 14 pages, 1 figure, published in Class. Quantum Grav (text and figure were updated to agree with the published version

Finite Size Scaling in 2d Causal Set Quantum Gravity

We study the $N$-dependent behaviour of $\mathrm{2d}$ causal set quantum gravity. This theory is known to exhibit a phase transition as the analytic continuation parameter $\beta$, akin to an inverse temperature, is varied. Using a scaling analysis we find that the asymptotic regime is reached at relatively small values of $N$. Focussing on the $\mathrm{2d}$ causal set action $S$, we find that $\beta \langle S\rangle$ scales like $N^\nu$ where the scaling exponent $\nu$ takes different values on either side of the phase transition. For $\beta > \beta_c$ we find that $\nu=2$ which is consistent with our analytic predictions for a non-continuum phase in the large $\beta$ regime. For $\beta<\beta_c$ we find that $\nu=0$, consistent with a continuum phase of constant negative curvature thus suggesting a dynamically generated cosmological constant. Moreover, we find strong evidence that the phase transition is first order. Our results strongly suggest that the asymptotic regime is reached in $\mathrm{2d}$ causal set quantum gravity for $N \gtrsim 65$.Comment: 32 pages, 27 figures (v2 typos and missing reference fixed

A closed form expression for the causal set d'Alembertian

Recently a definition for a Lorentz invariant operator approximating the d'Alembertian in d-dimensional causal set space-times has been proposed. This operator contains several dimension-dependent constants which have been determined for d=2,...,7. In this note we derive closed form expressions for these constants, which are valid in all dimensions. Using these we prove that the causal set action in any dimension can be defined through this discrete d'Alembertian, with a dimension independent prefactor.Comment: 20 pages + 20 pages appendix, to be published in CQ

Quantum Gravity on the computer: Impressions of a workshop

Computer simulations allow us to explore non-perturbative phenomena in physics. This has the potential to help us understand quantum gravity. Finding a theory of quantum gravity is a hard problem, but in the last decades many promising and intriguing approaches that utilize or might benefit from using numerical methods were developed. These approaches are based on very different ideas and assumptions, yet they face the common challenge to derive predictions and compare them to data. In March 2018 we held a workshop at the Nordic Institute for Theoretical Physics (NORDITA) in Stockholm gathering experts in many different approaches to quantum gravity for a workshop on "Quantum gravity on the computer". In this article we try to encapsulate some of the discussions held and talks given during this workshop and combine them with our own thoughts on why and how numerical approaches will play an important role in pushing quantum gravity forward. The last section of the article is a road map providing an outlook of the field and some intentions and goalposts that were debated in the closing session of the workshop. We hope that it will help to build a strong numerical community reaching beyond single approaches to combine our efforts in the search for quantum gravity.Comment: 22 pages, 1 figure, impressions from the workshop "Quantum gravity on the computer" at Nordita ( nordita.org/qg2018 ); v2: minor corrections, speakers contributions to workshop more distinguished, references adde

Extrinsic curvature in two-dimensional causal dynamical triangulation

Causal dynamical triangulation (CDT) is a nonperturbative quantization of general relativity. Hořava-Lifshitz gravity, on the other hand, modifies general relativity to allow for perturbative quantization. Past work has given rise to the speculation that Hořava-Lifshitz gravity might correspond to the continuum limit of CDT. In this paper we add another piece to this puzzle by applying the CDT quantization prescription directly to Hořava-Lifshitz gravity in two dimensions. We derive the continuum Hamiltonian, and we show that it matches exactly the Hamiltonian derived from canonically quantizing the Hořava-Lifshitz action. Unlike the standard CDT case, here the introduction of a foliated lattice does not impose further restriction on the configuration space and, as a result, lattice quantization does not leave any imprint on continuum physics as expected

Development of a model educational presentation on Zendium directed towards graduating dental and dental hygiene students

Thesis (M.S.)--Boston University, Henry M. Goldman School of Graduate Dentistry, 1984 (Dental Public Health).Includes bibliographical references (leaves 58-59).Future dental professionals receive very little information on new dental products.　The goal　of　this project was　to introduce Zendium, a new dentifrice manufactured by CooperCare, Oral-B, to graduating dental　studentS, post doctoral periodontic students and senior dental hygiene students.　This information was presented at　the dental and dental hygiene schools to fifteen (15) dental students, thirteen (13) periodontic students and eighty-one (81) dental hygiene students. Students were asked to complete two questionnaires to evaluate　the appropriateness of the presentation and their reaction to Zendium. Results indicated　that　the students felt　the presentation was appropriate in terms of the educational　level, length of presentation, use of visual　aids and speaker’s knowledge of　the product. Their response to Zendium was mixed, with the highest acceptance of the product by senior dental hygiene students. The students had several　concerns about　the product, including flavor, lack of ADA approval　and　the need for more　long-term clinical　studies　to be conducted in　the United States.　It is hoped　that　this information will aid CooperCare, Oral-B with their marketing strategies of Zendium to other dental professionals