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Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation

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Causal Dynamical Triangulations (CDT) is an approach to Quantum Gravity based on the sum over histories line of research which gives a quantization of classical Einstein gravity using a discrete approximation to the gravitational path integral, and spacetimes are approximated by Minkowskian equilateral triangles. This approach was developed by Renate Loll, Jan Ambjørn, and Jerzy Jurkiewicz. We have used a more recent version of CDT, called CDT without preferred foliation, to write a numerical simulation of quantum spacetimes in (1+1) dimensions. This talk comprises the full-fledged numerical implementation from how we can initialize triangulated spacetime in the form of a data structure to Monte-Carlo moves. Finally, how you can contribute to the further development of our software.

You can find more information from our project's webpage at https://projects.physicslog.com/cdt/ .

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Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation

  1. 1. Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation Damodar Rajbhandari, B.Sc. Department of Physics St. Xavier’s College, Nepal PRI Science Discussion Series Prithivi Narayan Campus, Nepal September 22, 2018 In the Collaboration with Prof. Dr. Udayaraj Khanal1 and Dr. Jonah Maxwell Miller2 1 Tribhuvan University, Nepal and 2 Los Alamos National Laboratory, USA
  2. 2. Introduction Model overview Implementation details Results Future work At last Table of Contents: Introduction Model overview Implementation details Results Future work At last AIM of this talk: To introduce (1+1) dimensional Causal Dynamical Triangulations model, numerical implementation details, and how you can contribute to the further development of our open source code. Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 2of 26
  3. 3. Introduction Model overview Implementation details Results Future work At last History What you always need to know about Quantum Gravity? Sir Newton days = Absolute space, Absolute time, Gravity = Classical Mechanics, Newton laws of gravitation Prof. Einstein days = Flat space + Relative time, Gravity, Quantum theory = Special Relativity, Newton laws of gravitation, QM Prof. Einstein days = Flat spacetime + Gravity, QM + Special Relativity = GR, QFT Finally, ⇓ + = Quantum Gravity Photos credit: Prof. Einstein photo by “Getty Images” and Prof. Feynman photo by “The Nobel Foundation” Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 3of 26
  4. 4. Introduction Model overview Implementation details Results Future work At last History Paths towards Quantum Gravity Einstein pointed out that quantum effects will lead to modifications of his theory [Einstein, 1918]. Covariant line of research. For eg. High derivative theory, Supergravity, and so on. Canonical line of research. For eg. Wheeler’s quantum geometrodynamics, Loop quantum gravity, and so on. Sum over histories line of research. For. eg. Hawking’s Euclidean quantum gravity, Quantum Regge calculus, Sorkin’s causal sets theory, Causal Dynamical Triangulations (CDT), CDT without preferred foliation (also known as Locally CDT, LCDT) , and so on. Extension from above line of researches. For e.g. M-theory (includes String theory), Twistor theory, Non-commutative geometry and so on. Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 4of 26
  5. 5. Introduction Model overview Implementation details Results Future work At last Background General Relativity: An approach to understand Gravity Spacetime tells matter how to move and matter tells spacetime how to curve. -John Archibald Wheeler Figure: Spatial curvature due to mountain Image credit: Jonah M. Miller Einstein Field Equations: Rµν − 1 2 Rgµν + Λgµν = 8πGTµν Vacuum Field Equations: Rµν − 1 2 Rgµν + Λgµν = 0 Einstein-Hilbert Action of vacuum universe: SEH = 1 16πG (R − 2Λ)d(2) V SEH = 1 16πG √ −g(R − 2Λ)d2 x Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 5of 26
  6. 6. Introduction Model overview Implementation details Results Future work At last Background Feynman Path Integral: An approach to Quantum Mechanics Figure: Quantum paths, inspired by [Israel and Lindner, 2012] The path integral for an one dimensional motion of an quantum particle evolution with time is [Feynman et al., 2010]: K[x0, x1, t] = x1 x0 D[x]eiS[x]/ where, S[x] = t1 t0 Ldt. S[x] is the action of the system, and L is the Lagrangian. D[x] does not mean the differential form of the co-ordinates x. Rather it is the differential form of all possible paths between endpoints x0 and x1. Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 6of 26
  7. 7. Introduction Model overview Implementation details Results Future work At last Introduction Definition of CDT: An approach to Quantum Gravity Based on the “Sum Over Histories Line of Research” Attempt to quantize Einstein’s “General Relativity” Using discrete approximation to the “Gravitational Path Integral Spacetimes are approximated by “Minkowskian Equilateral Triangles” Invented by Renate Loll, Jan Ambjørn & Jerzy Jurkiewicz Initially, it was named as Lorentzian Dynamical Triangulations Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 7of 26
  8. 8. Introduction Model overview Implementation details Results Future work At last Introduction [Vacuum] Standard CDT From [Jurkiewicz, 2015] From [Israel, 2012] From [Israel, 2012] Key priors Background independence Non-perturbative approach Always have causal structure Few free parameters (Λ, α, G) Foliated structure of spacetime Everything is close to GR! Credits: Bottom right figures by (left)[Jordan, 2013] & (right)Sabine Hossenfelder Figure →: Visualization of the phase diagram of CDT in (3+1)D, where κ0 is bare gravitational coupling and ∆ is asymmetry parameter Main results Emergence of spacetime Dimension reduction Non-trivial phase structure Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 8of 26
  9. 9. Introduction Model overview Implementation details Results Future work At last Introduction [Vacuum] CDT without preferred foliation: Jordan & Loll What is special in this approach? Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 9of 26
  10. 10. Introduction Model overview Implementation details Results Future work At last Introduction [Vacuum] CDT without preferred foliation: Jordan & Loll What is special in this approach? CDT main results DOES NOT rely on distinguished time slicing! [Jordan and Loll, 2013]! Everything is very close to GR! Figure: CDT Figure: LCDT Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 9of 26
  11. 11. Introduction Model overview Implementation details Results Future work At last Introduction So, we can have another type of Minkowski triangle. Huh? Not valid Valid Valid Not valid Red are time-like edges and blue are space-like edges. ∆tts for CDT and ∆sst added in LCDT. Figure: ∆tts type orientations Figure: ∆sst type orientations - Figures adapted from [Jordan and Loll, 2013] Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 10of 26
  12. 12. Introduction Model overview Implementation details Results Future work At last Theoretical Concepts Discretizing the smooth manifold using Regge Calculus Figure: Regge Calculus, adapted from [Misner et al., 1973] Gauss-Bonnet formula tells us that in two dimensions [Oprea, 2007]: Total curvature = R √ −gd2 x = 2πχ(M) where χ(M) is Euler Characteristics. So, E-H action becomes, ∴ SR = 1 16πG R √ −gd2 x − 1 8πG √ −gd2 x = 1 16πG (2πχ (M)) − Λ 8πG i Vi where Vi is the volume of ith triangle. Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 11of 26
  13. 13. Introduction Model overview Implementation details Results Future work At last Theoretical Concepts Topological Torus Figure: Triangulated torus, adapted from Wikipedia1 Our universe possesses toroidal topology with hole (genus) one. Euler characteristics χ(M) will be χ(M) = 2 − 2 × genus = 0 Our Regge action becomes ∴ SR = − Λ 8πG i Vi 1Image credit: https://commons.wikimedia.org/wiki/File:Torus-triang.svg Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 12of 26
  14. 14. Introduction Model overview Implementation details Results Future work At last Theoretical Concepts Regge action in-terms of volume for both triangle types i Vi = Vol( tts) + Vol( sst) where, Vol( tts) = √ 4α + 1 4 a2 Vol( sst) = α(4 + α) 4 a2 and a is lattice spacing between two vertices of a Minkowski triangle. Now, the action becomes [Ruijl, 2013] SR = − Λa2 32πG Ntts × √ 4α + 1 + Nsst × α(4 + α) Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 13of 26
  15. 15. Introduction Model overview Implementation details Results Future work At last Theoretical Concepts Wick-rotating the Regge action Figures inspired by [Israel and Lindner, 2012] Change Minkowski signature of (−+) to Euclidean signature (+ +). For example: The Pseudo-Pythagorean theorem for Minkowski spacetime is ds2 = −dt2 + dx2 . If we wick rotate the time axis i.e. t → −it and put it in the equation we can have, ds2 = +dt2 + dx2 which is Pythagorean theorem for Euclidean space and time . Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 14of 26
  16. 16. Introduction Model overview Implementation details Results Future work At last Theoretical Concepts Reconstructing the asymmetry parameter in Regge action applying Wick rotation It can be done by re-defining l2 t = +αa2 i.e. α → −α. i.e. √ 4α + 1 → √ −4α + 1 α(4 + α) → −α(4 − α) To have imaginary and non-vanishing if we bound on α as 1 4 < α < 4. Rewriting the mapping as −1(4α − 1) → i √ 4α − 1 −α(4 − α) → i α(4 − α) Then we can relate iSR [T ] → −SE [T ]. Thus, SE [T ] is the Euclidean version of Regge action and given by SE [T ] = Λa2 32πG √ 4α − 1 × Ntts + α(4 − α) × Ntss such that, 1 4 < α < 4. Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 15of 26
  17. 17. Introduction Model overview Implementation details Results Future work At last Theoretical Concepts Redefining Feynman path integral in-terms of gravitational propagator Images credit: Jonah M. Miller Define gravitational propagator with natural units (i.e. c = = 1) as [Ambjørn et al., 2012], G[g0 µν , g1 µν , t] = G D[gµν ]eiS[gµν ] where gµν is the metric tensor , G represents the class of all possible geometries of the Lorentzian spacetime. D[gµν ] is the differential form of all possible spacetime geometries between g0 µν and g1 µν . S[gµν ] is the classical action i.e. Einstein Hilbert action Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 16of 26
  18. 18. Introduction Model overview Implementation details Results Future work At last Theoretical Concepts Mapping Feynman path integral into a Partition function We discretized our spacetime geometry. So, the discretized gravitational path intergral is [Ambjørn et al., 2001] G[g0 µν , g1 µν , t] = T 1 C(T ) eiSR [T ] where T is the class of inequivalent Lorentzian triangulations which is corresponds to inequivalent discretized spacetime geometries. We have chosen the measure factor 1/C(T ) which will keep track of the repeated number of triangulations. After putting iSR [T ] = −SE [T ] in discretized gravitational path integral, it will transform into the partition function as, Z = T 1 C(T ) e−SE [T ] For simplicity, we choose α = 1 and lattice spacing a = 1. The Euclidean action becomes, SE [T ] = √ 3Λ 32πG (Ntts + Nsst ). Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 17of 26
  19. 19. Introduction Model overview Implementation details Results Future work At last Blueprint CDT is a Monte-Carlo method that uses lattices to represent the structure of spacetime! Figure: Beginning with a homogeneous 1+1 dimensional universe (left), large vacuum fluctuations in circumference arise (right), adapted from [Israel and Lindner, 2012] Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 18of 26
  20. 20. Introduction Model overview Implementation details Results Future work At last Constructing Quantum universe in a laptop Build spacetime data structure of triangulated topological 2-torus Figure: (1+1)D Minkowski triangulated spacetime equivalent to topological 2-torus. Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 19of 26
  21. 21. Introduction Model overview Implementation details Results Future work At last Constructing Quantum universe in a laptop Causality test: Ensure exactly one light cone in a vertex Figure: Violation of causality, adapted from [Jordan and Loll, 2013] Figure: Validation of causality, adapted from [Jordan and Loll, 2013] Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 20of 26
  22. 22. Introduction Model overview Implementation details Results Future work At last Constructing Quantum universe in a laptop Detailed Balance: Monte Carlo Method Main idea is that, when reaching equilibrium, it should satisfy the detailed balance condition: Peq(Ti )P(Ti → Tj ) = Peq(Tj )P(Tj → Ti ) where, Peq(T ) = 1 C(T ) e−SE [T ] Z We split transition probability P(Ti → Tj ) into two separate parts [Ruijl, 2013]: P(Ti → Tj ) = g(Ti → Tj )A(Ti → Tj ) where g(Ti → Tj ) is trial probability that pick the suitable region at random in the spacetime and A(Ti → Tj ) is the probability of accepting the move. Now, using all the above equations: A(Ti →Tj ) A(Tj →Ti ) = e−(SE [Tj ]−SE [Ti ]) × g(Tj →Ti ) g(Ti →Tj ) In Ising model, trial probabilities will cancel out! But, this is not in our case. So, we need to calculate it for every moves. Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 21of 26
  23. 23. Introduction Model overview Implementation details Results Future work At last Constructing Quantum universe in a laptop Monte Carlo moves: Generates strong gravity Alexander move: Figure: Before Alexander move Figure: After Alexander move g(Ti → Tj ) = 1 N ∗ Ne(a) + 1 N ∗ Ne(b) and g(Tj → Ti ) = 1 N + 1 where N = no. of vertices and Ne(x) = number of neighbors of x. Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 22of 26
  24. 24. Introduction Model overview Implementation details Results Future work At last Constructing Quantum universe in a laptop Monte Carlo moves: Generates strong gravity Collapse move: Figure: Before collapse move Figure: After collapse move g(Ti → Tj ) = 1 N ∗ Ne(a) + 1 N ∗ Ne(b) and g(Tj → Ti ) = 2 (N − 1)(Ne(a) + Ne(b) − 4)(Ne(a) + Ne(b) − 5) Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 23of 26
  25. 25. Introduction Model overview Implementation details Results Future work At last Run the code Usage: How to run our code? ””” Path : / F u l l c d t o n e p l u s o n e / debugging . py Time−stamp : <2018−07−20 15:54:46 ( Damodar)> Author : Damodar Rajbhandari ( d p h y s i c s l o g @ g m a i l . com) D e s c r i p t i o n : This i s a f i l e f o r debugging purpose only . Load i t i n the python 3. x i n t e r p r e t e r and play around with the program . HAPPY HACKING! ””” # Importing S c i e n t i f i c l i b r a r i e s : import numpy as np import s c i p y as sp import random # Importing c l a s s data s t r u c t u r e and f u n c t i o n s : import t r i a n g u l a t i o n s as t # In o r d e r to v a l i d CIRCULAR DEPENDENCY, s u b t r i a n g l e p r o p e r t y should be # imported f i r s t and then , t r i a n g l e p r o p e r t y . import s u b t r i a n g l e p r o p e r t y as s t import t r i a n g l e p r o p e r t y as tp import i n i t i a l i z a t i o n import s t a t e m a n i p u l a t i o n as sm import move factory as m import u t i l i t i e s as ut Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 24of 26
  26. 26. Introduction Model overview Implementation details Results Future work At last Hope to finish these in near future Future Work: To generate quantum spacetime ensembles We have completed Data structure Causality test Alexander move Collapse move We are working on Remaining Monte-Carlo moves Volume control terms to fixed the spacetime topology Metropolis-Hastings algorithm Anyone who is interested in this project can contribute to our open source scientific community at projects.physicslog.com/cdt Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 25of 26
  27. 27. Introduction Model overview Implementation details Results Future work At last Questions session Acknowledgments Deep gratitude to my academic supervisors Prof. Dr. Udayaraj Khanal and Dr. Jonah Maxwell Miller. Without their many hours of tireless advising and checking, my understanding of the subject would still be unbearably murky. Thanks to Prof. Dr. Jerzy Jurkiewicz, Prof. Dr. Stevan Carlip, Dr. Joshua Cooperman, Dr. Ben Ruijl and Mr. Adam Getchell for the helpful discussions. Thanks to Mr. Drabindra Pandit (Head of Department of Physics), Dr. Vinaya Kumar Jha, Dr. Binod Adhikari, Dr. Prem Raj Dhungel, Mr. Basu Dev Ghimire, entire Physics department and my dear friends especially Bindesh Tripathi for the encouragements. Special thanks to my family especially to my uncle, aunt and Dipika for always believing that I can do my best. Heartily thanks to Swastika Shrestha for always building up my courage and encouraging me to do my best. Thank You For Listening! I Invite You To Ask Any Questions You May Have... Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation 26of 26
  28. 28. Introduction Model overview Implementation details Results Future work At last References References I Ambjørn, J., Gørlich, A., Jurkiewicz, J., and Loll, R. (2012). Nonperturbative quantum gravity. Physics Reports, 519(4):127–210. Ambjørn, J., Jurkiewicz, J., and Loll, R. (2001). Dynamically triangulating lorentzian quantum gravity. Nuclear Physics B, 610:347–382. Einstein, A. (1918). ¨Uber Gravitationswellen. Sitzungsberichte der K¨oniglich Preußischen Akademie der Wissenschaften, pages 154–167. English translation at http: //einsteinpapers.press.princeton.edu/vol7-trans/25. Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation
  29. 29. Introduction Model overview Implementation details Results Future work At last References References II Feynman, R. P., Hibbs, A. R., and Styer, D. F. (2010). Quantum Mechanics and Path Integrals. Dover Publications, Mineola, New York, emended ed. edition. Israel, N. S. and Lindner, J. F. (2012). Quantum gravity on a laptop: 1+1 dimensional causal dynamical triangulation simulation. Results in Physics, 2:164 – 169. Jordan, S. (2013). Globally and Locally Causal Dynamical Triangulations. PhD thesis, Radboud University. Jordan, S. and Loll, R. (2013). Causal dynamical triangulations without preferred foliation. Physics Letters B, 724(1):155 – 159. Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation
  30. 30. Introduction Model overview Implementation details Results Future work At last References References III Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973). Gravitation. W. H. Freeman San Francisco. Oprea, J. (2007). Differential Geometry and its Applications. Pearson Education, Inc., second edition edition. Ruijl, B. (2013). A numerical simulation in 1+1 dimensions of Locally Causal Dynamical Triangulations. Master’s thesis, Radboud University, Nijmegen. available at http://www.ru.nl/publish/pages/760962/thesis_ final_version.pdf. Damodar Rajbhandari (dphysicslog@gmail.com) PRI Science Discussion Series, Prithivi Narayan Campus, Nepal | September 22, 2018 Quantum Gravity on a computer: An introduction to (1+1) dimensional Causal Dynamical Triangulations without preferred foliation

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