Euclidean dynamical triangulations running couplings and curvatu

Syracuse University SURFACE Dissertations - ALL SURFACE December 2019 Euclidean Dynamical Triangulations: Running Couplings and Curvature Correlation Functions Scott David Bassler Syracuse University Follow this and additional works at: https://surface.edu/etd Part of the Physical Sciences and Mathematics Commons Recommended Citation Bassler, Scott David, "Euclidean Dynamical Triangulations: Running Couplings and Curvature Correlation Functions" (2019).edu/etd/1104 This Dissertation is brought to you for free and open access by the SURFACE at SURFACE. It has been accepted for inclusion in Dissertations - ALL by an authorized administrator of SURFACE. For more information, please contact surface@syr. Abstract Quantum field theories have been incredibly successful at describing many fundamental aspects of reality with great precision, sometimes relying on the powerful computational tool of lattice methods.

Gravity has so far eluded a quantum field theory description, leading many to consider more exotic theories like String Theory. However, recent results in lattice quantum gravity have brought some renewed interest in the subject. After reviewing the progress made so far in Euclidean Dynamical Triangulations (EDT), a lattice theory of gravity, we examine how the couplings of the theory run with scale, and discover that their runnings are consistent with the asymptotic safety scenario for gravity with a 1-dimensional UV critical surface, making it maximally predictive. We also study two-point curvature correlation functions for gravity, and upon removing the disconnected contributions, we find universal behavior in these correlators, including a power-law drop-off with distance.

We also explore a means of extracting the coefficient of the R2 term in the low energy effective action, and find that this coefficient may be very large in magnitude, though a calculation in the low energy effective theory is still needed to test this possibility. This result implies that Starobinsky Gravity emerges naturally in the theory, which would have important implications for observational cosmology. Euclidean Dynamical Triangulations Running Couplings and Curvature Correlation Functions by Scott Bassler B. Applied Physics, Stockton University, 2013 Dissertation Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics.

Syracuse University December 2019 Copyright c Scott David Bassler 2019 All Rights Reserved Acknowledgments I thank my advisor Jack for helping guide me all these years. In working with him I’ve learned how to code, about lattice techniques, about QCD, about gravity, and all manner of physics I never dreamed would be relevant to the study of gravity. It has been a privilege to work for someone so intelligent and kind. I thank my professors at Syracuse for the numerous challenging but rewarding classes which I was lucky enough to be in.

The long hours on the fourth floor were not in vain! I thank my fellow graduate students for all of their help on handling assignments and life at Syracuse, both in my year and not, both in my department and not. I cannot hope to list them all, but special thanks go out to Greg, Michael, Craig, Shelby, Kyle, Alex, Raghav, Eric, Lindsay, Ohana, and of course Monica. Graduate school for physics is infamous for being a time that people struggle to find happiness. With all of you, it was easy to find.

I thank Walter for helping me foster my love of teaching at Syracuse, first as his teaching assistant, then as an apprentice at teaching conventions, and soon as a peer. I thank my undergraduate advisor Neil for convincing me that physics was a path worth following. I cannot help but wonder where I would be without his wisdom, which he gave when I needed it most. And of course, I thank my family: my mom, my dad, Danny, and Mikey.

For all my life, you have supported me in everything I’ve done, from choir to physics, from Woodbury to Syracuse. I love you all, and couldn’t ask for better parents or brothers. iv Contents 1 Introduction 1 1.1 Quantum Field Theory .2 A Quantum Field Theory of Gravity and Asymptotic Safety .3 Euclidean Dynamical Triangulations. 6 2 The Lattice Formulation 9 2.1 The Discretized Action .5 Evidence for Classical Behavior .1 Global Hausdorff Dimension .2 The de Sitter Solution .6 Towards the Continuum Limit .1 Setting the relative lattice spacing .2 The Spectral Dimension .3 Entropy Scaling of Black Holes.

35 3 Running of the Couplings 36 3.2 Restoring General Coordinate Invariance .3 Dimension of the UV critical surface .1 Unsubtracted Correlation Functions .4 Coefficient of the Power Law Decay. 67 5 Conclusions 74 6 Curriculum Vitae 80 vi List of Figures 1 A schematic view of the phase diagram for EDT as a function in the κ2 − β plane. 14 2 The phase diagram for QCD with Wilson fermions in the β − κ plane. β is the inverse strong coupling constant, 1/αs , and κ is the inverse bare fermion mass.

15 3 This histogram of N0 for the 4k β = 0 ensemble, with a clear double guassian structure. That the two peaks are the same height indicates that the ensemble is correctly tuned to the phase transition. 16 4 A plot of the peak of the volume correlator, cN4 (0), for both the 32k and 16k β = 0 ensembles. The value of κ2 that maximizes the slope is the critical value of κ2.

The red line shows the critical value for the 4k β = 0 ensemble, which should agree with the 16k and 32k ensembles if the correct Hausdorff dimension is chosen. That there is good agreement suggests that the chosen value of 4 is correct. 17 5 The history of the quantity N0 /N4 in Monte Carlo time at β = 0 for three different volumes. From top to bottom, the volumes are 4k, 8k, and 16k.

18 6 The volume distributions, cN4 (x), for three different volumes at β = 0, after rescaling assuming a Hausdorff dimension of 4. 20 7 The volume distributions, n4 (ρ), for three different volumes at β = 0, after rescaling assuming a Hausdorff dimension of 4. 21 8 The rescale n4 (ρ) distribution for several different β values, and therefore several different lattice spacings, as well as the de Sitter solution in black. As the continuum limit is taken, the lattice data gets closer to the de Sitter solution.

22 vii 9 A visualization of the geometries using a network visualization tool. The top left geometry is the coarsest at β = 1.5, the top right is the second coarsest at β = 0. The bottem left is the second finest at β = −0.6, and the bottom right is the finest at β = −0. For the coarser lattice, the baby universe branchings are easily identified as separate from the bulk, but this distinction diminishes for finer lattices.

24 10 The return probability as a function of diffusion time σ for three different volumes at β = 0. This appears to be insensitive to volume. 27 11 The return probability as a function of diffusion time σ for many different β values. 28 12 The return probability Ps as a function of rescaled diffusion time σr for many different β values, where the rescaling is such that the different curves lie on top of one another.

The amount by which σ must be rescaled gives the relative lattice spacing. 29 13 The spectral dimension as a function of distance scale probed in black, as well as the fit suggested by Ambjorn et al. 31 14 The spectral dimension at large distances DS (∞) for many different ensem- bles, and the extrapolation to infinite volume and zero lattice spacing in cyan. Especially fine ensembles are excluded.

33 15 The spectral dimension at large distances DS (∞) for many different ensem- bles, and the extrapolation to infinite volume and zero lattice spacing in cyan. The especially fine ensembles are included, and so a V12 term is included in the fit function. 34 16 The spectral dimension at short distances DS (0) for many different ensembles, and the extrapolation to infinite volume and zero lattice spacing in cyan. The especially fine ensembles are excluded.

37 viii 17 The spectral dimension at short distances DS (0) for many different ensembles, and the extrapolation to infinite volume and zero lattice spacing in cyan. The especially fine ensembles are included, necessitating the inclusion of a V12 term in the fit function. 38 19 GΛ as a function of κ2 for many different values of β. 42 20 GΛsub , Λ̂sub /10, and 10Ĝ plotted together as a function of κ2.

43 21 The correlator as defined in Eq.(30) for the 16k β = 0 ensemble. 48 22 The correlator defined in (31), which should have no measured correlation. 49 23 The modified correlator proposed by de Baker and Smit. 49 24 The connected correlator for the 32k β = 0 ensemble in the triangle approach.

51 25 The connected correlator for the 16k β = 0 ensemble in the triangle approach. 51 26 The connected correlator for the 8k β = 0 ensemble in the triangle approach. 52 27 The connected correlator for the 4k β = 0 ensemble in the triangle approach. 52 28 The connected correlator for the 4k β = 1.5 ensemble in the triangle approach.

53 29 The connected correlator for the 8k β = −0.8 ensemble in the triangle approach. 53 30 The connected correlator for the 32k β = 0 ensemble in the simplex approach. 54 31 The connected correlator for the 16k β = 0 ensemble in the simplex approach. 54 32 The connected correlator for the 8k β = 0 ensemble in the simplex approach.

55 33 The connected correlator for the 4k β = 0 ensemble in the simplex approach. 55 34 The connected correlator for the 4k β = 1.5 ensemble in the simplex approach. 56 35 The connected correlator for the 8k β = −0.8 ensemble in the simplex approach. 56 36 The correlator for the 4k β = 1.5 ensemble with the simplex discretization, zoomed in to highlight the “bump” that is absent for other ensembles.

58 37 The fit in the universal regime to the 32k β = 0 ensemble in the triangle discretization. The fit is done in the range [9,13]. 60 38 The fit in the universal regime to the 16k β = 0 ensemble in the triangle discretization. The fit is done in the range [8,12].

60 ix 39 The fit in the universal regime to the 8k β = 0 ensemble in the triangle discretization. The fit is done in the range [8,18]. 61 40 The fit in the universal regime to the 4k β = 0 ensemble in the triangle discretization. The fit is done in the range [7,11].

61 41 The fit in the universal regime to the 4k β = 1.5 ensemble in the triangle discretization. The fit is done in the range [7,18]. 62 42 The fit in the universal regime to the 8k β = −0.8 ensemble in the triangle discretization. The fit is done in the range [7,12].

62 43 The fit in the universal regime to the 32k β = 0 ensemble in the simplex discretization. The fit is done in the range [22,44]. 63 44 The fit in the universal regime to the 16k β = 0 ensemble in the simplex discretization. The fit is done in the range [20,38].

63 45 The fit in the universal regime to the 8k β = 0 ensemble in the simplex discretization. The fit is done in the range [19,32]. 64 46 The fit in the universal regime to the 4k β = 0 ensemble in the simplex discretization. The fit is done in the range [16,29].

64 47 The fit in the universal regime to the 8k β = −0.8 ensemble in the simplex discretization. The fit is done in the range [43,58]. 65 48 The power of the power law for both the simplex and triangle discretizations.