an open source code for causal dynamical

AN OPEN SOURCE CODE FOR CAUSAL DYNAMICAL TRIANGULATIONS WITHOUT PREFERRED FOLIATION IN (1+1)- DIMENSIONS WITH ELEMENTARY EXPOSITIONS

A Project Work Submitted to Office of Controller of Education, Tribhuvan University, Balkhu in the Partial Fulfillment of the Requirements for the Degree of Bachelor of Science in Physics

by Damodar Rajbhandari Symbol No.: xxxxxxxxx T.U. Regd. No.: x-x-xxx-xxx-xxx July 2018

© Damodar Rajbhandari, July 2018 Copyright in this work rests with an author. Please ensure that any reproduction or re-use in accordance with the relevant Tribhuvan University copyright legislation.

To my late Father, who taught me the real meaning of the purpose of life at a very young age.

RECOMMENDATION

It is certified that Mr. Damodar Rajbhandari has carried out the project work entitled “AN OPEN SOURCE CODE FOR CAUSAL DYNAMICAL TRIANGULATIONS WITHOUT PREFERRED FOLIATION IN (1+1)- DIMENSIONS WITH ELEMENTARY EXPOSITIONS” under our supervision. We recommend this project in the partial fulfillment for the requirement of Bachelor’s Degree of Science in Physics.

Prof. Dr. Udayaraj Khanal (Supervisor) Professor Department of Physics Tribhuvan University, Nepal

Date: ....................................

Dr. Jonah Maxwell Miller (Co-supervisor) Postdoctoral Researcher Los Alamos National Laboratory Los Alamos, USA

CERTIFICATE OF APPROVAL

We certify that we have read this project work entitled “AN OPEN SOURCE CODE FOR CAUSAL DYNAMICAL TRIANGULATIONS WITHOUT PREFERRED FOLIATION IN (1+1)- DIMENSIONS WITH ELEMENTARY EXPOSITIONS” carried out by Mr. Damodar Rajbhandari and in our opinion it is very good in scope and quality as a project work in the partial fulfillment for the requirement of Bachelor’s Degree of Science in Physics.

Prof. Dr. Udayaraj Khanal (Supervisor) Professor Department of Physics Tribhuvan University, Nepal

Dr. Jonah Maxwell Miller (Co-supervisor) Postdoctoral Researcher Los Alamos National Laboratory Los Alamos, USA

Internal Examiner

External Examiner

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The final copy of this report has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the mentioned discipline.

Acknowledgments First and foremost, I would like to express my deep gratitude to my academic supervisors Prof. Dr. Udayaraj Khanal and Dr. Jonah Maxwell Miller whose expertise, patience, encouragement and invaluable guidance through out my research. Without their many hours of tireless advising and checking, my understanding of the subject would still be unbearably murky. Also, thanks to Prof. Dr. Jerzy Jurkiewicz (Jagiellonian University, Poland), Prof. Dr. Steven Carlip (University of California, Davis, USA), Dr. Ben Ruijl (ETH Zurich, Switzerland), Dr. Joshua Cooperman (Radboud University, Netherlands) and Mr. Adam Getchell (University of California, Davis, USA), who provided invaluable advices. A special thanks must be extended to Mr. Drabindra Pandit (Head of Department of Physics), Dr. Vinaya Kumar Jha, Dr. Binod Adhikari, Dr. Prem Raj Dhungel and Mr. Basu Dev Ghimire whose everlasting effort on “Encouraging students to involve in research activities” gave me a lot of inspiration in my heart. Next, I would also like to thank the entire Physics department at St. Xavier’s College, including all of the faculty, staffs and my dear friends (especially Bindesh Tripathi). On a personal note, I would like to thank my parents especially to my Uncle and Aunt, for continued love, encouragement, best wishes, support, and putting me through the best education possible. I appreciate their sacrifice and I wouldn’t have been able to get to this stage without them. I personally thanks to my sister Dipika Rajbhandari for always being a part of my life. It will be tasteless if I haven’t mentioned Swastika Shrestha. Without her unending love, emotional support and motivation up to this day, I might never have had the courage to start this research work. For me, she is the isotropic and homogeneous in my universe. Lastly, I believe having the pure level of knowledge and ideas matters much to me. During the period I learned, all the research articles should need to be free in order to grow from that point and to avoid repetition of the work. I very much appreciate giving credits in every knowledge I mentioned in my report because it’s a way of respecting the author’s heartfelt dedication and hard work, and also by looking at my references, readers may get a straightforward path if they want to follow my work. There are many others that deserve to be acknowledged for their help and support during my research and I apologize if I have forgotten to mention some of them. Thank you, everyone.

Abstract 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 report comprises the detailed derivation of the model and the numerical implementation from how we can initialize triangulated spacetime in the form of a data structure to MonteCarlo moves. Finally, we prove that our implemented code does work up to where we ended up.

Table of contents List of Figures

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List of Abbreviations

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1 INTRODUCTION 1.1 Quantum Gravity . . . . . . . . 1.2 Paths towards Quantum Gravity 1.3 Need for Quantum Gravity . . . 1.4 Quantum Gravity is hard . . . . 1.5 Concluding remarks . . . . . . .

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2 CAUSAL DYNAMICAL TRIANGULATIONS 2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Model overview . . . . . . . . . . . . . . . . . . . . . . 2.3 Recovering the Einstein-Hilbert action . . . . . . . . . . 2.4 Discretizing the Einstein-Hilbert action . . . . . . . . . . 2.5 Wick-rotating the Regge action . . . . . . . . . . . . . . 2.6 Mapping Feynman path integral into a partition function 2.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . 3 IMPLEMENTATION DETAILS 3.1 Blueprint . . . . . . . . . . . 3.2 Data structure . . . . . . . . 3.2.1 ID creation . . . . . 3.2.2 Destroying object . . 3.2.3 Vertex . . . . . . . . 3.2.4 Timelike edge . . . . 3.2.5 Spacelike edge . . . 3.2.6 TTS triangle . . . . . 3.2.7 SST triangle . . . . . 3.3 Causality validation . . . . . 3.4 Monte-Carlo method . . . . 3.4.1 Detailed balance . . 3.5 Monte-Carlo moves . . . . . 3.5.1 Alexander move . . . 3.5.2 Collapse move . . . 3.6 Concluding remarks . . . . .

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TABLE OF CONTENTS

4 RESULTS 4.1 Usage . . . . . . . . . . . . . . 4.2 Initial data structure . . . . . . . 4.3 Causality code . . . . . . . . . . 4.4 Monte-Carlo moves works . . . 4.4.1 Testing Alexander move 4.4.2 Testing Collapse move . 4.5 Desire volume . . . . . . . . . . 4.6 Concluding remarks . . . . . . .

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5 FUTURE WORK 5.1 Other moves . . . . . . . . . . . . . . . . . . . . . . 5.2 Metropolis-Hastings algorithm . . . . . . . . . . . . 5.3 Volume control . . . . . . . . . . . . . . . . . . . . 5.4 Contribution to an open source scientific community 5.5 Motivation to start this project work . . . . . . . . . 5.6 Concluding remarks . . . . . . . . . . . . . . . . . .

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A SUPPLEMENT DERIVATIONS A.1 4-Volume in-terms of Jacobian . . . . . . . . . . . . A.2 Relationship between the Jacobian and the metric . . A.3 Co-ordinate based 4-volume element . . . . . . . . . A.4 Relate determinant of the metric tensor to its cofactor A.5 Relation of space-like interval and time-like interval . A.6 Area of Minkowskian equilateral triangle . . . . . . . A.6.1 Area of tts triangle type . . . . . . . . . . . . A.6.2 Area of sst triangle type . . . . . . . . . . . .

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B SUPPLEMENT CODES 57 B.1 Another approach to initialize data structure . . . . . . . . . . . . . . . . . . . 57 C SUPPLEMENT CODE OUTPUTS C.1 Vertices . . . . . . . . . . . . C.2 Timelike edges . . . . . . . . C.3 Spacelike edges . . . . . . . . C.4 TTS triangles . . . . . . . . . References

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List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 3.1

(1+1) dimensions spacetime of Causal Dynamical Triangulations . . . . . . . (1+1) dimensions spacetime of Causal Dynamical Triangulations without preferred foliation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of Regge Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . The four triangle types which can be constructed from using link lengths, timelike edges (red) and, space-like edges (blue). . . . . . . . . . . . . . . . . . . Violation of causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validation of causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time orienting Lorentzian triangles . . . . . . . . . . . . . . . . . . . . . . . The time orientation of a given triangle determines the time orientation of its direct neighbors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∆tts types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∆sst types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lattice illustration of an Ising 2D model where up and down arrows represents the spin up and spin down respectively . . . . . . . . . . . . . . . . . . . . . Rotating real time to imaginary time. . . . . . . . . . . . . . . . . . . . . . . Rotating the positive complex plane to its lower half. . . . . . . . . . . . . . Solid line represents the path followed by a classical particle and dashed curve represents the paths followed by a quantum particle . . . . . . . . . . . . . . Quantum paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum spacetime geometries . . . . . . . . . . . . . . . . . . . . . . . . .

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Beginning with a homogeneous (1+1) dimensional universe (left), large vacuum fluctuations in circumference arise (right) . . . . . . . . . . . . . . . . . Topologically cylindrical spacetime cut from initial to final time slice and projected to a flat surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1+1)D Minkowski triangulated spacetime equivalent to topological 2-torus . Two dimensional lightcone where the bold point is an event . . . . . . . . . . Understanding future and past timelike edges . . . . . . . . . . . . . . . . . Before Alexander move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . After Alexander move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Before collapse move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . After collapse move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The spectral dimension ds of the universe as a function of diffusion time σ. . . . 50

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A.1 Light path observed by observer O0 relative to the source of light . . . . . . . . 54 iii

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LIST OF FIGURES A.2 Light path observed by observer O relative to earth . . . . . . . . . . . . . . . 54 A.3 Minkowski triangle of type tts . . . . . . . . . . . . . . . . . . . . . . . . . . 55 A.4 Minkowski triangle of type sst . . . . . . . . . . . . . . . . . . . . . . . . . . 56 B.1 Triangulation of (1+1)D spacetime before any move or anti-move was made . . 58

List of Abbreviations ATLAS A Toroidal LHC ApparatuS CDT

Causal Dynamical Triangulations

CMS

Compact Muon Solenoid

CTCs

Closed Timelike Curves

D

Dimensional

GR

General Relativity

LCDT

Locally Causal Dynamical Triangulations

LHC

Large Hadron Collider

LIGO

Laser Interferometer Gravitational-Wave Observatory

MC

Monte-Carlo

QFT

Quantum Field Theory

QG

Quantum Gravity

QM

Quantum Mechanics

TU

Tribhuvan University

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CHAPTER

1

INTRODUCTION “Einstein’s 1905 paper came out and suddenly changed people’s thinking about space-time. We’re again in the middle of something like that. When the dust settles, time-whatever it may be could turn out to be even stranger and more illusory than even Einstein could imagine.” - Carlo Rovelli

The primary goals of this report are two-fold: (1) To introduce and provide the detailed derivation of Causal Dynamical Triangulations without preferred foliation in (1+1) dimensions. And, how it can be a possible candidate for Quantum Gravity. (2) To provide you the detail numerical implementation of the model along with the explanation of the implemented codes. The first four chapters are devoted to the above goals and aim to introduce the ideas in a broad spectrum. In chapter 1, we introduce quantum gravity and its insights. In chapter 2, we introduce Causal Dynamical Triangulations and its historical background along with the mathematical formalism required to write the numerical simulation of this model. In chapter 3, we develop the possible strategies in order to start writing the code. In chapter 4, we show the implemented codes and proving that it does works. Finally, in chapter 5, we discuss further development of the code and the application part of it.

1.1

Quantum Gravity

In the early twentieth century, the two frontiers of today’s Physics were born. The General Theory of Relativity by Einstein1 which describes gravity as the large scale structure of the universe, and the quantum theory by some notable physicists which describes the physical 1 Hilbert isn’t in the popular story like Einstein is, but he and Einstein discovered general relativity almost at the same time.

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CHAPTER 1. INTRODUCTION

phenomenon in a very small scale. By that time, Einstein realized the description of gravity within the mathematical framework of differential geometry can be regarded as a field, called Gravitational field [1]. There have been many experimental verifications of classical tests of general relativity over the years, including gravitational lensing of light rays by the sun during an eclipse, the measurement of “frame dragging” via satellites like Gravity Probe B, lunar ranging experiments, the detection of the decay of the Hulse-Taylor pulsar system, and the Pound-Rebka experiment. But, the LIGO detection being only the most recent and one of the most dramatic. Now, gravitational field has been experimentally verified by LIGO and Virgo teams on 11 February 2016, and was first observed in 14 September 2015 in two LIGO detectors situated at Handford, Washington, and Livingston, Louisiana then, named the signal as GW150914 (means “Gravitational Wave” and the date of observation 2015-09-14) with the merger of the binary black hole as it’s source [2]. On the other hand, quantum theorists were building the tools sufficient to explain quantum theory in-terms of a field theoretic approach, called as Quantum Field theory. This was developed to combine special relativity and quantum mechanics. This was necessary for understanding, for example, how light and matter interact. It is also important for studying large collections of particles. The physics of very small, isolated particles, can be modeled very well with single particle quantum mechanics. We want to note that quantum field theory wasn’t really fully developed for many decades after quantum mechanics was discovered. Finally, this results the famous Standard Model in Particle Physics, and theorized Higgs Mechanism in 1960s which helps to discovered Higgs boson with mass of about 125.09 ± 0.24 GeV [3] in July 2012 at Large Hardon Collider (LHC) and confirmed in March 2013 by ATLAS and CMS experimental groups at CERN [4]. We know the Higgs Boson is responsible for giving its mass to fundamental particles. When studying very small, dense objects, we need both quantum mechanics and general relativity. We call the unknown theory required to study this combination “Quantum Gravity”. This means we want to see gravitational properties by probing the universe directly to the Planck scale where one can find that the Quantum effects are much relevant. The first attempts to quantize general relativity date back to the early 1930s [5] and physicists are still hoping that the gravitational field needs to be quantized. Because Einstein picture of gravity couples universally to all forms of energies [6]. One may expect that using the standard tools offered by QFT, one can able to quantized any fields from Klein-Gordon field (helps to formulate relativistic quantum mechanics) to Higgs field (offers Higgs boson particle). Unfortunately, this did not work out by proceeding with a naive quantization of GR, thus appears to be perturbatively non-renormalizable [7, 8, 9]. The main difficulty one may find during quantization of GR is that it is background free (which means like special relativity or, even QFT we suppose the background geometry to be fixed Minkowski metric or, fixed curved spacetime but in the case of GR the background geometry is dynamically changing by how much matter and energy contained in the region), always have causal structure (which means a meaningful direction of time) and spacetime diffeomorphism group (which maps one spacetime point into another).

1.2

Paths towards Quantum Gravity

Specifically, there are three directions to formulate a candidate theory of Quantum Gravity i.e. covariant, canonical and sum over histories [5]. Even though some of the candidate theories may not be included in any one of these. For example, String theory, Twistor theory, Non-

1.2. PATHS TOWARDS QUANTUM GRAVITY

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commutative geometry, and so on. We stated the three directions is because the majority of the candidates belongs to (or, emerges from) these categories. The covariant line of research is the attempt to build the theory using a flat Minkowski or some other metric space as a background geometry and then, begin to fluctuate using quantization algorithm over that metric. This means, the spacetime metric gµν (X) as a sum of ηµν + hµν where ηµν is the Minkowski metric with signature (−, +, +, +) and then, we quantize hµν using standard relativistic QFT, for example perturbation theory, effective field theory and renormalization group approaches [5]. This approach examples includes high derivative theory and supergravity. The search converged successfully to string theory in the late eighties. String theory (more general version called M-theory) uses the approaches given by covariant research but it is not field theoretic because spacetime points are replaced by extended structures such as strings and branes. And the main idea to construct this theory is a quantum theory of all the interactions and suggest graviton as a fundamental particle for gravitational field, and is due to an excitation of closed strings thus, try to be in the position of “theory of everything” in-order to complete Standard Model. But, most of the quantum gravity candidates specifically deals with the behavior of quantum spacetimes and then, extends from phenomenology to quantum cosmology. The canonical line of research is the attempt to build the theory using the Hamiltonian formalism carries an appropriate Hilbert space which have a representation of the operators corresponding to the full metric without background metric to be fixed then, identifies canonical variables and conjugate momenta [6]. This means, one will start with the standard Einstein’s Field equation along with natural units (i.e. Newton’s gravitational constant, velocity of light and Planck constant) to be unity, Gµν = 8πTµν ,

(1.1)

such that the left-hand side is the geometry of spacetime which is purely classical in the sense that it’s just an ordinary function of the spacetime points indicates no fixed background metric while the right hand side is a stress-energy tensor which has quantum operators. Thus, one can think of this form Gµν |Ψi = 8π Tˆµν |Ψi , (1.2) which we can interpret as an eigenvalue equation. But this equation 1.2 no longer makes sense because the ten operators components of stress-energy tensor (Tµν ) do not commute with each other [10] thus suggests simultaneous eigenstates will not exists. Thus, we suspect that it has no any solutions. One way one can replace is with the expectation of this tensor which was suggested by Møller (in 1962) and Rosenfeld (in 1963) as [10], Gµν = 8π hΨ|Tˆµν |Ψi ,

(1.3)

which lead to introduce “semiclassical gravity”. But the main difficulty is that Tµν is a function of classical field gµν and thus relate to Gµν . And one can find an uncertainty relation of the form ∆gµµ ∆Gµµ V (4) ≥ 1 [10]. This relation arise from commutation relation between the operators in stress-energy tensor thus, results into non-unique set of solutions. To resolve the issue, observing the ordinary Hamiltonian procedure that there is the commutation relations between the dynamical variable and it’s first time derivative, we need to find those variable in-order to have an unique sets of solution. Thus, this rose to the approaches like Wheeler’s quantum geometrodynamics and loop quantum gravity.

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CHAPTER 1. INTRODUCTION

The sum over histories line of research uses “Feynman quantization of general relativity” approach which was first suggested by John Archibald Wheeler and introduced by his doctoral student Charles William Misner in 1957 as [11], Z exp(i/~)(Einstein action)d(field histories). (1.4) Instead of using perturbative approaches that turns into non-renormalizable for gravity, a new window has been opened on non-perturbative approach using Feynman path integral for gravitation. In Feynman path integral, we have a resultant path to be a superposition of all the possible paths. We can define in-terms of resultant spacetime as a superposition of all possible spacetimes. Thus, new approaches born. For example, Hawking’s Euclidean quantum gravity, quantum regge calculus, Sorkin’s Causal sets theory and Causal Dynamical Triangulations. There are a number of candidates for the Quantum gravity than the above mentioned, with which had some successes and going under development to produce a consistent theory for quantum gravity. And thus, unravel the mystery of where do we come from! In this work, we investigate quantum gravity using Causal Dynamical Triangulations without preferred foliation model. Before we dive into the method, we need to have a careful understanding of the necessity of Quantum gravity.

1.3

Need for Quantum Gravity

In the seminal works of Penrose and Hawking in the singularity theorems [12], every black hole exists singularities where spacetime curvature becomes infinitely strong and unpredictable, and the starting point of the very early universe described by well-known big bang model is a singularity, as well. The singularity theorems don’t simply imply that black holes possess singularities. They imply that no matter what the initial conditions of the universe were, singularities will form. This is profound because it implies general relativity drives itself into a regime where predictability fails. This may imply there’s a more complete theory to which we don’t have access to that takes over such a small scales, i.e., quantum gravity.

1.4

Quantum Gravity is hard

Due to the enormous efforts are taken for building a complete theory of Quantum gravity from about a century, some believe there’s no any way to quantize GR. And, considering the possibility that gravity is purely classical interaction thus classical field are incompatible with quantum mechanics. This is due to the lack of experimental evidence. To verify gravity has quantum entities in short length scales, proposed tests have been invented under the phenomenology of specific QG model and cosmological observations. But, none of these yet provide conclusive shreds of evidence. Still there are other way one can suggest by just looking the Cavendish experiment but taking the two masses to be very small and emphasizing the gravitational interaction between them via graviton (a force mediator for gravitational field) such that we can study superposition between them [13]. However this approach does not clarify how the quantum coherence will be observed due to gravity. Thus, results different modified interferometry techniques (like molecular interferometry [14], Stern-Gerlach interferometers

1.5. CONCLUDING REMARKS

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[15], Mach-Zehnder interferometers [16], and so on) to understand this question. But these are still under development. The main reason it is hard because general relativity and quantum mechanics have different, seemingly incompatible principles at their cores. General relativity prizes the equivalence principle and locality. Quantum mechanics, on the other hand, prizes conservation of probability, or unitarity. It is not all clear that these ideas are compatible. The black hole firewall result is one demonstration of this. Thus, results the difficulty with renormalizing gravity using standard QFT tools.

1.5

Concluding remarks

By knowing that quantum gravity exists, sure us one day any one of the candidates will be the successful theory for QG. And we have still a long way to go in-order to understand the properties of Quantum Gravity.

CHAPTER

2

CAUSAL DYNAMICAL TRIANGULATIONS “Quantum gravity is notoriously a subject where problems vastly outnumber results.” - Sidney Coleman

One can attempt to construct a theory of quantum gravity by defining a non-perturbative quantum field theory as a sum over histories. This technique is a modification of Feynman path integral formulation which was first put forwarded by Stephen Hawking and Gary William Gibbons for their quantum gravity approach “Euclidean quantum gravity”[17]. In-short, instead of superposing the usual Lorentzian spacetimes, they used Euclidean spaces which have four space dimensions. The hope of Euclidean quantum gravity was that spacetime could be assumed to be Euclidean and that time would emerge dynamically. Unfortunately, this did not work out. In the 1990s, this work was combined with quantum Regge calculus so that it could be formulated non-perturbatively. This was called Dynamical Triangulations [18]. This technique failed due to unboundedness in the gravitational action [19]. The way dynamical triangulations failed is quite interesting. It produced spacetimes which were all crumpled up and maximizing their curvature. Also the path integral knows nothing about causality (i.e. light cone structure of the spacetime) and there’s no specification that any test matter will bounded by light cones. In an effort to resolve these issues, Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz reformulated this construction, but with Lorentzian signature imposed from the start. They only map into Euclidean signature via Wick rotation after constructing the theory. Thus, Causal Dynamical Triangulations was born (initially, it was named as Lorentzian Dynamical Triangulations). This approach provides concrete evidence that one must include causal structure, in the sense of having a well-defined light cones everywhere [20]. This model imposed preferred foliation of time so that we can track of time direction (see figure 2.1). And time labeling of each and every vertices can be found by looking at time slicing. But the question is, does CDT rely on distinguished time slicing? If it does, that’s theoretically a problem because general relativity has no such constraint. To answer this question, Renate Loll and Samo Jordan introduced the more general version of CDT which is called CDT without preferred foliation (equivalent to say, Locally causal dynamical triangulations) and found the answer to be hope6

2.1. DEFINITION

7

fully no [21] (see figure 2.2). And, everything is very close to General Relativity because we have more freedom in our triangulated spacetime. The main motivation for choosing this approach is that it is relies on very few fundamental physical principles and attempts to quantize gravitational degree of freedom without introduction of additional variables, extra dimensions or new symmetries widely used by String theory (for example). Moreover, it comes with definite numerical approximation scheme. To study the numerical simulations of (1+1) dimensional CDT with non-foliated structure, we requires new Monte Carlo moves [22], which are significantly more difficult to implement than the generalized Pachner moves used in standard CDT [9].

Figure 2.1: (1+1) dimensions spacetime of Causal Dynamical Triangulations, adapted from [21].

2.1

Figure 2.2: (1+1) dimensions spacetime of Causal Dynamical Triangulations without preferred foliation, adapted from [21].

Definition

Causal dynamical triangulations (CDT) is an approach to the quantization of classical theory of Einstein’s gravity based on a discrete approximation to the gravitational path integral, in which the spacetimes contributing to the sum over histories are approximated by locally flat simplicial manifold. The numerical simulation was successfully implemented by CDT’s authors [23, 24, 25] and Rajesh Kommu independently verified the CDT algorithm with initial important results, notably the emergence of classical spacetime and short scale dimensional reduction [26]. However, the suggestion of using Monte Carlo calculations was made by Martin Roˇcek and Ruth M. Williams in 1982 [27].

2.2

Model overview

This chapter provides a full-fledged derivation of one candidate theory of Quantum Gravity: Causal Dynamical Triangulations without preferred foliation in (1+1) dimensions. First, we recover the exact Einstein-Hilbert action. Then, we discretize the Einstein-Hilbert action using Regge calculus and encode with Minkowski signature, which is built on the Gauss-Bonnet theorem. Next, we transform our action into Euclidean signature via Wick rotation. We assume all discrete elements of our spacetime are the same equilateral triangles. By constructing the CDT action, we have converted the Feynman path integral into a partition function which can be handled via statistical methods.

8

CHAPTER 2. CAUSAL DYNAMICAL TRIANGULATIONS

In short, this distribution is the central part of the numerical approach to Causal Dynamical Triangulations.

2.3

Recovering the Einstein-Hilbert action

General Relativity assumes that spacetime is dynamic. This means that the metric of spacetime is itself a dynamical variable [28]. Thus, it wonderfully equates spacetime geometry with the energy and momentum contained therein. This geometry is fully understood as a Pseudo Riemannian manifold. In the variational principle approach to derive the Einstein field equations, we will assume that the Ricci scalar, which is the simplest curvature invariant available to us, completely defines the action. We will only study a matter-less universe where there is only spacetime. As Einstein said, curvature of spacetime changes in the presence of matter. You may ask: “Why are we so interested in an empty universe?”. There is only one reason we’re studying empty spacetime: It’s easier. Even empty spacetime exhibits a rich set of dynamics. Black holes are “empty spacetimes”, for example. And the expanding de-sitter universe, which resembles our own, can be well described by an empty spacetime with a positive cosmological constant. It’s a first step towards a more difficult and complicated calculations. For the variational principle, we need to find an action where it helps to define spacetime curvature. So, our first aim is: “To construct an action that, when extremized, yields the Einstein field equations in the same way that the action for a classical system yields the Euler-Lagrange equations.” We first define the action in (3+1) dimensions and later restrict to (1+1) dimensions. We define the action as [28] Z S = Ld(4) V (2.1) where, L and V refers to scalar Lagrange density and 4-volume respectively. Since, volume is a coordinate independent quantity, it can be written in a covariant way. 0 0 The 4-volume element of a locally Minkowskian co-ordinate system {xα }3α0 =00 is 0

0

0

0

d(4) V = dx0 dx1 dx2 dx3 . From the appendix A.3, equation 2.1 becomes Z √ S = L −gd4 x.

(2.2)

In principle, the Lagrange density contains contributions from both gravity and matter, L = LGeometry + LMatter . However, we are only interested in geometry. So, we set L = LGeometry .

(2.3)

We now guess the terms for LGeometry so that we recover the vacuum Einstein field equations. Suppose that LGeometry = k1 R + k2 (2.4)

2.3. RECOVERING THE EINSTEIN-HILBERT ACTION where k1 and k2 are constants and R is the Ricci scalar. Putting equation 2.4 in 2.3. We get, Z √ SEH = (k1 R + k2 ) −gd4 x.

9

(2.5)

If we vary equation 2.5, we find Z Z √ √ 4 4 δSEH = k1 δ −gd x R −gd x + k2 δ Z Z √ √ 4 µν = k1 δ g Rµν −gd x + k2 δ −gd4 x (∵ R = g µν Rµν ) Z Z Z √ √ √ 4 µν 4 µν 4 = k1 −gg δ(Rµν )d x + −gRµν δ(g )d x + Rδ −g d x Z √ + k2 δ( −g)d4 x ∴ δSEH = k1 δSEH(1) + δSEH(2) + δSEH(3) + k2 δSEH(4) (2.6) where Z

√ −gg µν δ(Rµν )d4 x,

Z

√ −gRµν δ(g µν )d4 x,

δSEH(1) = δSEH(2) = Z δSEH(3) = and,

Rδ Z

δSEH(4) =

√

−g d4 x,

√ δ( −g)d4 x.

Consider δSEH(1) : Z δSEH(1) =

√

−gg µν δRµν d4 x.

(2.7)

σ Now consider the variation of the Ricci Tensor i.e. δRµν = δRµσν . The Riemann-Christoffel σ curvature tensor R µλν is defined as [29]:

∂Γσµλ ∂Γσµν − + Γηµλ Γσνη − Γηµν Γσλη ∂xν ∂xλ = ∂ν Γσµλ − ∂λ Γσµν + Γηµλ Γσνη − Γηµν Γσλη .

Rσµλν =

(2.8)

If we contract σ and λ, we attain the Ricci tensor, Rµν = Rσµσν = ∂ν Γσµσ − ∂σ Γσµν + Γηµσ Γσνη − Γηµν Γσση .

(2.9)

The variation of the Ricci tensor is thus δRµν = ∂ν δΓσµσ − ∂σ δΓσµν + Γηµσ δΓσνη + Γσνη δΓηµσ − Γηµν δΓσση − Γσση δΓηµν = ∂ ν δΓσµσ + Γσνη δΓηµσ − Γηνµ δΓσση − Γηνσ δΓσµη − ∂σ δΓσµν + Γσση δΓηµν − Γηµσ δΓσνη − Γηνσ δΓσµη . (2.10)

10


We also know the covariant derivative of a tensor T σµν is [30], ∇ψ T σµν = ∂ψ T σµν + Γσψη T ηµν − Γηψµ T σνη − Γηψν T σµη . And the covariant derivative of the δ small change of T σµν is ∇ψ δT σµν = ∂ψ δT σµν + Γσψη δT ηµν − Γηψµ δT σνη − Γηψν δT σµη . Thus, we can say ∇σ δΓσµν = ∂σ δΓσµν + Γσση δΓηµν − Γηµσ δΓσνη − Γηνσ δΓσµη

(2.11)

∇ν δΓσµσ = ∂ν δΓσµσ + Γσνη δΓηµσ − Γηνµ δΓσση − Γηνσ δΓσµη .

(2.12)

and,

Using equation 2.11 and 2.12 in 2.10, we attain δRµν = ∇ν δΓσµσ − ∇σ δΓσµν . Using equation 2.13 in 2.7, we attain Z √ δSEH(1) = −gg µν ∇ν δΓσµσ − ∇σ δΓσµν d4 x.

(2.13)

(2.14)

The rules for covariant derivatives tells us that ∇ν (g µν δΓσµσ ) = δΓσµσ ∇ν g µν + g µν ∇ν δΓσµσ ∴ g µν ∇ν δΓσµσ = ∇ν (g µν δΓσµσ ) − δΓσµσ ∇ν g µν .

(2.15)

Similarly, ∇σ (g µν δΓσµν ) = δΓσµν ∇σ g µν + g µν ∇σ δΓσµν ∴ g µν ∇σ δΓσµν = ∇σ (g µν δΓσµν ) − δΓσµν ∇σ g µν .

(2.16)

By combining equations 2.15, 2.16 and 2.14, we attain, Z √ −g ∇ν (g µν δΓσµσ ) − δΓσµσ ∇ν g µν − ∇σ (g µν δΓσµν ) − δΓσµν ∇σ g µν d4 x. δSEH(1) = (2.17) Now, our job is to find ∇λ g µν . First, assert that the covariant derivative of the metric is zero. And this is true for all metric compatible connections. This is how the Christoffel symbols are defined. For simplicity, we will first go for ∇λ gµν and then, we will go towards our aim. Recall from [29] that ∇λ gµν = ∂λ gµν − Γηµλ gην − Γηλν gµη .

(2.18)

2.3. RECOVERING THE EINSTEIN-HILBERT ACTION

11

Also, we can relate between gµν and Γηµν as [29], ∂λ gµν = Γηµλ gην + Γηλν gµη .

(2.19)

So, combining equation 2.18 and 2.19, we can have ∇λ gµν = 0.

(2.20)

By definition, the contra-variant and covariant form of metrical tensor (i.e. g µν and gµν ) can be related as [29], g µν gων = δωµ ∴ g µν gµν = 1.

(2.21)

Taking covariant derivative of equation 2.21, we get g µν ∇λ gµν + gµν ∇λ g µν = 0.

(2.22)

Combining equation 2.22 and 2.20, we can have ∇λ g µν = 0 Thus, we proved our assertion. Combined with equation 2.17, this implies that Z √ δSEH(1) = −g ∇ν (g µν δΓσµσ ) − ∇σ (g µν δΓσµν ) d4 x Z √ = −g∇σ g µν δΓσµσ − g µν δΓσµν d4 x Z √ −g∇σ V σ d4 x ∴ δSEH(1) =

(2.23)

(2.24)

where we have introduced V σ = g µν δΓσµσ − g µν δΓσµν . Thus, equation 2.24 is a covariant divergence of a vector field V σ over a manifold M. If we apply the generalized Stokes’ theorem [29] to equation 2.24 so that the 4-volume becomes a hyper-surface with normal vector nσ , we attain Z Z √ √ σ 4 −g∇σ V d x = −gnσ V σ d3 x. (2.25) M

Boundary of M

The right-hand side of the equation 2.25 gives the boundary contribution at infinity, which we can set to zero by making the variation vanish at infinity [28]. Therefore, this term contributes nothing to the total variation i.e. δSEH(1) = 0. Let us now examine the variation of the metric tensor. We need to relate determinant of metric tensor gµν with it’s cofactor Aµν . From the appendix A.4, we can write, g = gµν Aµν .

(2.26)

12


If we differentiate equation 2.26 with respect to gµν , we find that ∂g = Aµν . ∂gµν

(2.27)

If we multiply both sides of equation 2.26 by the inverse metric g µν , we attain g µν g = (g µν gµν ) Aµν ∴ Aµν = g µν g (∵ from equation 2.21).

(2.28)

Let us examine the variation of g. We have that ∂g δg = δgµν (by the chain rule) ∂gµν = (Aµν ) δgµν (∵ from equation 2.27) µν ∴ δg = g gδgµν (∵ from equation 2.28). (2.29) √ The variation of −g is √ 1 δ −g = − √ δg 2 −g 1 (g µν gδgµν ) (∵ from equation 2.29) =− √ 2 −g √ √ −g µν ∴ δ −g = (g δgµν ) . (2.30) 2 Our next job is to convert δgµν to δg µν . This can be done by finding the variation of equation 2.21. i.e. δ (gµν g µν ) = 0 or, gµν δg µν + g µν δgµν = 0 ∴ g µν δgµν = −gµν δg µν

(2.31)

Substituting equation 2.31 into 2.30, we obtain √ √ −g gµν δ (g µν ) . (2.32) δ −g = − 2 After substitution of equation 2.32 in 2.6, the total variation of Einstein-Hilbert action becomes, δSEH = k1 δSEH(2) + δSEH(3) + k2 δSEH(4) Z Z Z √ √ 4 √ µν 4 = k1 −gRµν δ(g )d x + Rδ −g d x + k2 δ( −g)d4 x √ Z Z √ −g µν 4 µν 4 = k1 −gRµν δ(g )d x + R − gµν δ (g ) d x 2 Z √ −g µν + k2 − gµν δ (g ) d4 x 2 Z √ 1 k2 µν = −g k1 Rµν − gµν R − gµν δg d4 x 2 2 Z √ k2 = (2.33) −g k1 Gµν − gµν δg µν d4 x 2

2.3. RECOVERING THE EINSTEIN-HILBERT ACTION

13

where, Gµν = Rµν − 21 gµν R and Gµν is the Einstein tensor. The functional derivative of the action must satisfy [28], δS =

Z X δS δΦi

i

i

δΦ

d4 x

(2.34)

where, {Φi } is a complete set of fields being varied (in our case, it’s just g µν ). We can find the stationary points, if each δS/δΦi = 0. i.e. 1 δSEH k2 √ = k1 Gµν − gµν = 0 µν −g δg 2

(2.35)

In order to find the values of k1 and k2 , we must compare to the Einstein field equations. Before doing so, we first introduce units to the Einstein equations by multiplying both sides of our equation by 16πG [28]. Equation 2.35 then becomes (16πGk1 )Gµν + (−8πGk2 )gµν = 0

(2.36)

where, G is well-known Newton gravitational constant. Comparison of equation 2.36 with the vacuum Einstein field equations then yields the values of k1 and k2 as 1 16πG Λ and k2 = − . 8πG k1 =

We can substitute the values of k1 and k2 into equation 2.4 and the scalar Lagrange density 2.3 to attain L=

1 (R − 2Λ) . 16πG

Putting the value of L in equation 2.2, we can get the Einstein-Hilbert action. i.e. SEH

1 = 16πG

Z

√

−g (R − 2Λ) d4 x

(2.37)

Since, we can define the n-volume in-terms of the Jacobian (see appendix A.1). So, EinsteinHilbert action for n-dimensional manifold is Z √ 1 SEH = −g (R − 2Λ) dn x. (2.38) 16πG For our purpose, we use (1+1) dimensional manifold. So, Einstein-Hilbert action become SEH

1 = 16πG

Z

√

−g (R − 2Λ) d2 x.

(2.39)

14


2.4

Discretizing the Einstein-Hilbert action

Einstein gave us a description of gravitation in-terms of geometry. Now, we will approximate spacetime as a lattice of flat 4-dimensional simplices1 , which are the higher-dimensional generalization of triangles, (called pentachorons). Italian mathematician Tullio Regge first developed a formalism for approximating continuous manifolds as simplicial ones. This formalism, which we will make use of, is now called Regge calculus [31]. For obvious reasons, we sometimes call these approximate manifolds piecewise flat (see figure 2.3).

Figure 2.3: Illustration of Regge Calculus, taken from [32] Even-though we discretized our spacetime, Causal Dynamical Triangulations does not assume that the discreteness we impose is fundamental or physical. Rather it is an approximation technique. We are interested in studying quantum gravity in (1+1) dimensions, so we will use 2simplices (i.e. triangles) to triangulate our manifold. For simplicity, we will use equilateral triangles. One of the best examples of this type of approximation is the so-called geodesic dome, a 2-sphere approximated using triangles. For simplicity, we assume periodic boundary conditions in space and time. This means our spacetimes will be topologically equivalent to a torus. In two dimensions, the Riemannian curvature becomes the Gaussian curvature [33]. The Gauss-Bonnet formula tells us that in two dimensions, adding this boundary curvature term to the internal curvature term gives us exactly a topological constant [33]. I.e. Z

√ R −gd2 x = 2πχ(M)

(2.40)

where χ(M) is the Euler characteristics that depends only on the homotopy class of M. Combining equations 2.40 and 2.39, we attain two dimensional Regge action, 1

For example, 3-dimensional simplex is tetrahedron

2.4. DISCRETIZING THE EINSTEIN-HILBERT ACTION

∴ SR =

1 Λ X (2πχ (M)) − Vi 16πG 8πG i

15

(2.41)

where Vi is the volume of ith triangle. As we supposed earlier that our universe possesses toroidal topology, the genus is one. So, the Euler characteristics χ(M) will be, χ(M) = 2 − 2 × genus =2−2×1 ∴ χ(M) = 0. So, equation 2.41 becomes, ∴ SR = −

Λ X Vi . 8πG i

(2.42)

Since we are working with the Causal Dynamical Triangulations without preferred foliation, there are two group of triangles (i.e. triangle with sides time-like, time-like and spacelike (∆tts ) and, space-like, space-like and time-like (∆sst )) so that it will follow Minkowski’s recommended triangles with correct signature for triangulating two dimensional Lorentzian spacetime. In the figure 2.4, only two triangles which are at the center has the correct signatures and, the green lines inside these triangles indicate light rays through the corner vertices.

Figure 2.4: The four triangle types which can be constructed from using link lengths, time-like edges (red) and, space-like edges (blue), adapted from [21] When gluing one Lorentzian triangle with another, we glue space-like edges with space-like and time-like edges with time-like. If we randomly glue these Lorentzian triangles together, causality violations will occur at the vertices (see figure 2.5). In order to incorporate the local causality condition, we need to know the light cone structure in both Lorentzian triangles, especially at vertices. In order to get valid light cone at the vertices, we require that there are exactly four sectors crossings when circling around a vertex [21]. We create these gluing rules because we want a meaningful direction of time and follow causal structure. So, each triangle has a side (or vertex) which is in the future compare to the other sides (or vertices). We want the future of one triangle to connect to the past of its neighbor. That’s why there are several types of triangles and that’s why we need these gluing rules. This is illustrated in figure 2.6,

16


Figure 2.5: Violation of causality, adapted from [21]

Figure 2.6: Validation of causality, adapted from [21]

To understand the flow of time, we can assign two Lorentzian triangles with consistent of future-pointing arrows at each edge. Since the interior of every Minkowskian triangle has a well-defined light cone structure, a choice of which one is the past and which one is the future light cone induces an orientation on its time-like edges, which can be captured by drawing future-pointing arrow onto the edges. And for space-like edges which is drawn perpendicular to the edges [34]. As shown in the figure 2.7.

Figure 2.7: Time orienting Lorentzian triangles, adapted from [34] If local causality holds, we can at least locally extend the time-orientation as in the figure 2.8.

Figure 2.8: The time orientation of a given triangle determines the time orientation of its direct neighbors, adapted from [34]

2.5. WICK-ROTATING THE REGGE ACTION

17

It is clear that the time-orientations of every neighbor triangle is uniquely determined by consistency at the shared link. Depending on the chosen spacetime topology it may not be possible to extend the time-orientation globally. We therefore require that such a global extension always exists [34] by assuming toroidal topology.

Figure 2.9: ∆tts types, adapted from [21]

Figure 2.10: ∆sst types, adapted from [21]

These two groups (i.e. ∆tts and ∆sst ) can be oriented as in the figure 2.9 and 2.10. And, these triangles consist of light-cones so that, any test matter will be bounded these light-cones. So, equation 2.42 can be written as, SR = −

Λ {Ntts × V (∆tts ) + Nsst × V (∆sst )} 8πG

(2.43)

where, Ntts and Nsst are the number of triangles of respective types respectively. And, V (∆tts ) and V (∆sst ) are the volume of triangles of respective types respectively. From appendix A.6, if we substitute the volumes in equation 2.43, we attain ( ) p √ α(4 + α) 2 Λ 4α + 1 2 SR = − a + Nsst × a Ntts × 8πG 4 4 o p √ Λa2 n ∴ SR = − Ntts × 4α + 1 + Nsst × α(4 + α) (2.44) 32πG where, a is the length of space-like edges or, simply called the lattice spacing between two vertices of a Minkowski triangle on the spatial axis. α is the asymmetry parameter such that α > 0. This parameter appears because the time-like edges are of length −αa. We call Λ the cosmological constant and G the Newton’s gravitational constant.

2.5

Wick-rotating the Regge action

Wick rotation is an analytic continuation of a theory through the complex plane. In short, it map the time τ to an imaginary time iτ . Thus, this transforms a pseudo-Riemannian manifold into a Riemannian one. This makes the signature of metric positive definite and it also maps the path integral into a partition function. This transformation concept was first introduced by Gian Carlo Wick [35]. If we Wick rotate the Regge action, we map our quantum field theory problem into a statistical physics problem. We can then sample from a partition function, where the probability of each state is given by the Gibbs distribution. In short, the Causal Dynamical Triangulations is similar to the Ising model where spins configurations are determined by the same distribution (see figure 2.11).

18


Figure 2.11: Lattice illustration of an Ising 2D model where up and down arrows represents the spin up and spin down respectively, adapted from [36] If we Wick rotate the action 2.44 then it will change Minkowski signature of (−+) to Euclidean signature (++). This can be done by re-defining lt 2 = +αa2 i.e. α 7→ −α. See figure 2.12. This Wick rotation implies √

4α + 1 7→

√

−4α + 1 p p α(4 + α) 7→ −α(4 − α).

(2.45) (2.46)

For consistency, we define the range of α so that, the value under square root will always stay imaginary and non-vanishing. This can be done by demanding that 1 4 −α(4 − α) < 0 ⇒ 0 < α < 4 −4α + 1 < 0 ⇒ α >

(∵ α > 0).

Thus, the new bounds on α are 14 < α < 4. This is a much stronger restriction than the original value (i.e. α > 0). So we can rewrite the mapping 2.45 and 2.46 to p √ −1(4α − 1) 7→ i 4α − 1 (2.47) p p −α(4 − α) 7→ i α(4 − α). (2.48) In-order to get minus sign in-front of these mappings 2.47 and 2.48, we rotate them through the lower half of the complex plane. This is visualized as in figure 2.13,

2.6. MAPPING FEYNMAN PATH INTEGRAL INTO A PARTITION FUNCTION

19

7→

Figure 2.12: Rotating real time to imaginary time, inspired by [37].

Figure 2.13: Rotating the positive complex plane to its lower half, inspired by [37].

Finally mappings 2.47 and 2.48 become √ √ i 4α − 1 7→ −i 4α − 1 p p i α(4 − α) 7→ −i α(4 − α).

(2.49) (2.50)

Putting the required mappings 2.49 and 2.50 in equation 2.44, we attain o √ p Λa2 n −i 4α − 1 × Ntts + −i α(4 − α) × Nsst 32πG o p iΛa2 n√ 4α − 1 × Ntts + α(4 − α) × Nsst = 32πG = iSE ∴ iSR [T ] 7→ −SE [T ] SR = −

(2.51)

where, T is the triangulated piece-wise flat manifold. Thus, SE [T ] is the Euclidean version of Regge action and given by o p Λa2 n√ SE [T ] = 4α − 1 × Ntts + α(4 − α) × Nsst 32πG

2.6

such that,

1 < α < 4. 4

(2.52)

Mapping Feynman path integral into a partition function

Finally, we arrive at the point where we intuitively define our probability distribution as similar to the Gibbs distribution in statistical physics. This can be done using the concept famously known as the “Feynman picture of quantum mechanics”, introduced by Richard Phillips Feynman during his doctoral work on “The Principle of Least Action in Quantum Mechanics (1942)”, under the supervision of John Archibald Wheeler at Princeton University. Feynman’s main result was a modification of Dirac’s variational principle approach to quantum mechanics.

20


In 1933, Paul Adrien Maurice Dirac made the observation that action can play as a central role in quantum mechanics as in classical mechanics. Dirac considered the Lagrangian formulation to be more fundamental than the Hamiltonian [38]. However, Dirac was unable to perfect his approach. In 1942, Feynman successfully rectified Dirac’s approach in his dissertation and suggested the path propagator should correspond to probability phase factor eiS/~ where S is the classical action evaluated along the classical path [38]. Finally, in 1948, he developed the complete form of his approach where he suggested the path propagator K[x0 , x1 , t] can be written as a sum over all possible paths (in quantum field theory, it is called as sum over histories) between the initial and final points [38]. And, each path contributes eiS/~ to the propagator. The idea is to sum over all paths, weighting the value of each path by the appropriate phase. Feynman argued that all possible paths will contribute to the probability of a quantum particle transitioning from some initial position to some final position. Thus, this approach is equivalent to the Schrödinger picture (1926) of time-evolution |ψ(t)i for some initial state |ψ(to )i. Anyone who can understand Young’s double slit experiment in optics should understand the basic ideas behind path integrals. In figure 2.14, a particle that obeys classical mechanics follows one path between times t0 and t1 but, if it obeys quantum mechanics, then it follows infinitely many paths.

Figure 2.14: Solid line represents the path followed by a classical particle and dashed curve represents the paths followed by a quantum particle, inspired by [39] We need a formalism for the path integral for a one dimensional motion of an quantum particle evolution with time. This is given by [39]: Z x1 K[x0 , x1 , t] = D[x]eiS[x]/~ x0

where Z

t1

Ldt

S[x] = t0

is the action of the system, and L is the Lagrangian. D[x] does not mean the differential form

2.6. MAPPING FEYNMAN PATH INTEGRAL INTO A PARTITION FUNCTION

21

of the co-ordinates x. Rather it is the differential form of all possible paths between endpoints x0 and x1 (see figure 2.15).

Figure 2.15: Quantum paths, inspired by Figure 2.16: Quantum spacetime geometries, inspired by [37] [37] We now define the gravitational path integral analogous to the Feynman path integral. In our earlier discussion of the path propagator, we had two points: initial and final points. Analogously, the gravitational propagator has two spatial lines (see figure 2.16). These lines represent the geometry of space at some initial and final times. This can be symbolically written as 1 0 , t] and defined with natural units2 as [9], , gµν G[gµν 0 1 G[gµν , gµν , t]

Z =

D[gµν ]eiS[gµν ]

(2.53)

G

where gµν is the metric tensor , G represents the class of all possible geometries of the Lorentzian spacetime and D[gµν ] is the differential form of all possible spacetime geometries 0 1 between gµν and gµν . S[gµν ] is the classical action i.e. Einstein Hilbert action (see equation 2.39). Since, we discretized our spacetime geometry by applying Regge calculus our action is given by equation 2.52. So, the gravitational path integral (i.e. equation 2.53) will transform into the discretized gravitational path integral as [40] 0 1 G[gµν , gµν , t] =

X T

1 iSR [T ] e C(T )

(2.54)

where T is the class of inequivalent Lorentzian triangulations which is corresponds to inequivalent discretized spacetime geometries. We have chosen the measure factor m(T ) = 1/C(T ) which means the combinatorial weight of each geometry is not exactly 1, but the inverse of the order C(T ) of the symmetry (i.e. automorphism) group of the triangulation T [40]. In-short, while doing the monte-carlo moves in numerical simulation, we will have repeated number of triangulations. Thus, this measure factor m(T ) will keep track of the repeated number of triangulations. 2

In natural units, c = ~ = 1. In this report, unless otherwise stated, we will always use natural units.

22

2.7


Concluding remarks

Now, we have all the necessary recipes to study the full partition function for pure gravity in the causal dynamically triangulated model. That means, the discretized gravitational path integral (i.e. equation 2.54) will transform into the partition function as, Z=

X T

1 −SE [T ] e C(T )

(∵ from mapping2.51).

(2.55)

For the Euclidean action, we assume asymmetry parameter α = 1 and lattice spacing, a = 1. Thus, equation 2.52 becomes, √ 3Λ SE [T ] = (Ntts + Nsst ). (2.56) 32πG C(T ) can be neglected in Monte-Carlo simulations because it cancels out in the appropriate Monte-Carlo step [41]. For numerical point of view, the volume and curvature information of the spacetime can be extracted out by simple counting of the number of Minkowski triangles. In-short, our problem will reduced to combinatorics.

CHAPTER

3

IMPLEMENTATION DETAILS “Reality is a network of granular events; the dynamic that connects them is probabilistic; between one event and another, space, time, matter, and energy melt into a cloud of probability.” - Carlo Rovelli

In this chapter, we start the explanation from our data structure to Monte-Carlo moves. Before we proceed, there is a number of important concepts that need to be known very first. We briefly develop these ideas herein. These are the key ingredients for numerical simulation.

3.1

Blueprint

Figure 3.1: Beginning with a homogeneous (1+1) dimensional universe (left), large vacuum fluctuations in circumference arise (right), adapted from [37]. 23

24

CHAPTER 3. IMPLEMENTATION DETAILS

Causal Dynamical Triangulations (CDT) is a Monte-Carlo method that uses lattices to represent the structure of spacetime. This means we change our spacetime curvature by applying Monte-Carlo moves under Metropolis-Hastings algorithm1 . This ubiquitous sampling tool helps us to approximate the likely configurations from any arbitrary configurations. We initialize2 our universe (equivalent to say spacetime or geometry) by applying volume3 increasing move (i.e. Alexander move) without sampling our spacetime [42] and will make foliated structure exactly like in standard CDT (see the left figure 3.1). This is because we want to have our choice volume before we run the simulation. By making a loop in the code [43], one can also generate desire volume (refer appendix B.1). However, we have found the volume increasing approach to be simple and effective. After attaining the desired volume, we apply the full set of Monte Carlo moves to drive the spacetime towards a physically likely configuration, i.e., one that extremizes the action (see the right figure 3.1). We make it a non-foliated structure by applying pinching moves [22]. In above figure 3.1, the left one is the flat spacetime in which the number of equilateral triangles attached to a vertex are exactly equal to six. But in right one, there are more (or, less) than six triangles. When applying the volume decreasing moves which change the number of attached triangles on a vertex to less than six may eventually collapse our spacetime thus, we will need to be aware of this condition.

x

Figure 3.2: Topologically cylindrical spacetime cut from initial to final time slice and projected to a flat surface, adapted from [44] Our spacetime topology is that of a 2-torus. So, we run Monte-Carlo simulations without changing that topology to generate the ensembles of spacetimes. And, while running the simulation to calculate Spectral dimension under these ensembles, first we will specify the initial 1 We will explain this algorithm in the chapter 5 because we haven’t arrived at this point in writing our code. In chapter 3 and 4, we will discuss only the topic which we have completed in the time-frame. 2 In our implementation, we starts from 9 tts triangles(see figure 3.3) before applying volume increasing move 3 We mean volume as a total number of triangles in the triangulated spacetime.

3.2. DATA STRUCTURE

25

and final time then, we cut the 2-torus into a cylinder (see figure 3.2) then, a free particle will go for a random walk on the spacetime. But the spacetime points at the edges of cut remain same which means the boundary condition is unchanged [45]. This will avoid the global closed timelike curves (CTCs) but during simulation we may have local CTCs [22].

3.2

Data structure

Figure 3.3: (1+1)D Minkowski triangulated spacetime equivalent to topological 2-torus The most important thing about the data structure is, how the geometric objects (i.e. vertices, edges and triangles) are related to each other. One can think of our data structure for toroidal spacetime as a representation of every object that makes up the spacetime. It makes them unique from one another. Thus, every objects need to have their own data structure to remember this information. To build our spacetime structure, first we need to know how to represent a toroidal spacetime in this form. In figure 3.3, the left one is our spacetime consists of 9 vertices, 18 timelike (red) edges, 9 spacelike (blue) edges, 18 tts triangles and 0 sst triangles. And, the flat representation in left figure is equivalent to the skeleton torus in right. This figure is very important during the construction of data structure. Secondly, we need to give number for every vertices and at-last these numbers are their IDs. To make the toroidal spacetime like in figure 3.3, we need to have initial data. Our programming language preference is Python 3.x. version under the object oriented paradigm. The data representing the spacetime are of the form two torus data = [( timelike 1 , timelike 2 , spacelike 3) , ... , ( timelike x , timelike y , spacelike z )] where (timelike, timelike, spacelike) represents tts triangle. The edge of any type representation will be in the form of edge = s e t ( [ e n d p o i n t v e r t e x 1 , e n d p o i n t v e r t e x 2 ] ) .

26


where endpoint vertices represents their IDs. We prefer to use a set container to store endpoints of an edge because it makes two endpoints unique. For example, the set([1,1]) is {1} which is not an edge but a vertex only. We have 18 timelike edges so, we need 18 data to represent this edge type which is given below. # Timelike data : t = [ set ([1 ,2]) , set ([2 ,3]) , set ([3 ,1]) , set ([4 ,6]) , set ([6 ,8]) , set ([8 ,4]) , set ([5 ,7]) , set ([7 ,9]) , set ([9 ,5]) , set ([3 ,4]) , set ([2 ,8]) , set ([8 ,5]) , set ([1 ,6]) , set ([6 ,9]) , set ([9 ,1]) , set ([4 ,7]) , set ([7 ,3]) , set ([5 ,2])] We have used vertices number (or, IDs) from the figure 3.3. We have 9 spacelike edges so, we need 9 data to represent this edge type which is given below. # S pa c e l ik e edges s= [ set ( [ 1 , 4 ] ) , set ([4 ,5]) , set ([5 ,1]) , set ([2 ,6]) , set ([6 ,7]) , set ([7 ,2]) , set ([3 ,8]) , set ([8 ,9]) , set ([9 ,3])] These timelike and spacelike edges are stored in list container so, we can call the edge by its index. These infrastructure is used to make an entire triangulated spacetime which we call triangulations (or, sometime graph). Since we have 18 tts triangles and 0 sst triangles, we need 18 tts triangles to represent the prototype spacetime which is given below.

3.2. DATA STRUCTURE

two torus data = [( set ( t ( set ( t ( set ( t ( set ( t ( set ( t ( set ( t ( set ( t ( set ( t ( set ( t ( set ( t ( set ( t ( set ( t ( set ( t ( set ( t ( set ( t ( set ( t ( set ( t ( set ( t

27

[0]) , [12]) [15]) [15]) [6]) , [17]) [1]) , [10]) [4]) , [13]) [7]) , [16]) [2]) , [9]) , [5]) , [11]) [8]) , [14])

set ( t [12]) , set ( t [3]) , set ( t [3]) , set ( t [6]) set ( t [17]) , set ( t [0]) set ( t [10]) , set ( t [4]) set ( t [13]) , set ( t [7]) set ( t [16]) , set ( t [1]) set ( t [9]) , set ( t [5]) , set ( t [11]) , set ( t [8]) set ( t [14]) , set ( t [2])

, set ( s [3])) , , set ( s [0])) , , set ( s [4])) , , set ( s [1])) , , set ( s [5])) , , set ( s [2])) , , set ( s [6])) , , set ( s [3])) , , set ( s [7])) , , set ( s [4])) , , set ( s [8])) , , set ( s [5])) , set ( s [0])) , set ( s [6])) , , set ( s [1])) , , set ( s [7])) , , set ( s [2])) , , set ( s [8]))]

Up to this point, geometric objects does not know about which is its neighbors and also, does not know its ID. So, we need a function that can give all the features we need! Algorithm 3.1. The algorithm for this function goes like this: 1. First, we need to check the two torus data which says a triangle edges in tuple container is three. And also, we need to check that each edge has two unique points. 2. The vertices in existence will be the set union of the triangle’s edges which then provides the vertices for that triangle. The union of vertices in all triangles is all of the vertices in entire graph. 3. Once we make sure the set is what we want, we need to make sure no torus is currently initialized. 4. Then, we need a vertex class that generates a vertex object, a vertex ID and keeps track of the points such that these vertices always belong to a triangle. 5. We make tts triangle by using 3 vertices: 2 timelike and 1 spacelike. Thus we need two triangles classes (one for tts and another for sst) and two edge classes (one for timelike and another for spacelike). While making objects of any class the initial data are taken from two torus data. This is only at the initial time. 6. At this point, the triangle and vertex classes do not know about their neighbors so, we connect all the triangles and tell the vertices what triangles connect them. Thus, results our spacetime made from 18 tts triangles. This function is only useful during initialization of triangulated spacetime. Now, we know how we can initialize our spacetime but, we still must understand how vertex, edge and triangle classes are made.

28


3.2.1

ID creation

IDs are integers which are created during the creation of objects in the appropriate constructor method (i.e. def init () ). Algorithm 3.2. The algorithm for creating IDs goes like this: 1. We make three important global variables: last used id = 0 recycled ids = set ( [ ] ) i n s t a n c e s = {} The last used ID means get the maximum ID from a list of all active IDs. The recycled IDs means a list of IDs which were previously in use but are now unused. And, instances contains all the instances of the class in which the data are represented in the form of Python dictionary where key is the object’s ID and value is the object itself. We can easily call by class.instances gives {id: {, ...}. This means we call all class’s objects, and memory location means the memory address of the object. 2. When we make an object of any class that may be of type vertex, edge or triangle, we must create an ID. We first search for recycled IDs container, if there are any, we pick any integer and give to the newly created object. But, if there are no recycled IDs remaining, we create an integer by incrementing the last used ID by 1. So, its ID will be (last used id + 1). This is a very useful ingredient for every geometric class.

3.2.2

Destroying object

When we delete an object, we do not delete the ID. Rather we store it in the recycled IDs container for further use. This prevents object ID numbers from growing too quickly and avoids integer overflow during simulation. Algorithm 3.3. The algorithm for deleting object goes likes this: 1. First we store the ID of the object that going to be deleted in the recycled IDs container. 2. Then we delete the object and its ID from the instances container.

3.2.3

Vertex

Vertex are represented as a point in the spacetime. And its class keeps track of the points such that it is always the vertex of a triangle. This class vertex object stores its ID, number of tts triangles and sst triangles, number of timelike and spacelike edges attached to this vertex. Our vertex object constructor looks like this:

3.2. DATA STRUCTURE

def

init

29

( s e l f , new id = 0 , t t s t r i a n g l e l i s t = [ ] , ss t tr ian gle li st = [] , timelike edge list = [] , spacelike edge list = []):

where we have defined: • new id = ID number for this vertex object and it’s just an integer • tts triangle list = list of all tts type triangles • sst triangle list = list of all sst type triangles • timelike edge list = list of all timelike edges • spacelike edge list = list of all spacelike edges. All the empty lists will be filled during the initialization which is defined in Algorithm 3.1. The physical meaning of a vertex is an event in the spacetime.

3.2.4

Timelike edge

This timelike edge class specifies the timelike edges of a triangle. This class object stores ID and vertex pair. Our timelike edge object constructor looks like this: def

init

( s e l f , new id = 0 , v e r t e x p a i r = [ ] ) :

This type of edge has a specific direction and is either past-directed or future-directed. Future directed means test particle can move from past to future along this edge. Past directed means test particle can move from future to past along this edge. In our model, we avoid past directed because physical timelike particles can travel only into the future. Causal particles can move along edges of this type with speeds less than the speed of light. We define timelike edges as a tuple (a,b) such that a is pointing to b and, a is the past event and b is the future event, where a and b are the vertex object IDs. We can make an instance variable inside our constructor as # Need t o f i n d w h i c h e n d p o i n t i s p a s t and f u t u r e s o # t h a t t h e t i m e l i k e edge i s always f u t u r e d i r e c t e d . s e l f . past = tuple ( s e l f . v e r t i c e s ) [ 0 ] s e l f . future = tuple ( s e l f . v e r t i c e s ) [ 1 ]

3.2.5

Spacelike edge

This spacelike edge class specifies the spacelike edges of a triangle. This class object stores an ID and a vertex pair. Our spacelike edge object constructor looks like this: def

init

( s e l f , new id = 0 , v e r t e x p a i r = [ ] ) :

Spacelike edges do not have specific direction. If a test particle moves along this type of edge, it will travel faster than the speed of light, which is unphysical.

30

3.2.6


TTS triangle

This class defines the tts type of triangle object where two edges are timelikes and one is spacelike. Our tts triangle object constructor looks like this: def

init

3.2.7

SST triangle

( s e l f , new id = 0 , p o i n t l i s t = [ ] , spacelike edge list =[] , timelike edge list =[] , t t s t r i a n g l e l i s t =[] , s s t t r i a n g l e l i s t = [ ] ) :

This class define the sst type of triangle object where two edges are spacelike and one is timelike. Our sst triangle object constructor looks like this: def

3.3

init

( s e l f , new id = 0 , p o i n t l i s t = [ ] , spacelike edge list =[] , timelike edge list =[] , t t s t r i a n g l e l i s t =[] , s s t t r i a n g l e l i s t = [ ] ) :

Causality validation

Causality is a condition in which for an event there should be specific past, present and future. Thus, results in a light cone that vividly specifies this scenario.

Figure 3.4: Two dimensional lightcone where the bold point is an event In figure 3.4, all particles will be inside the up vertical and down vertical light cones. We call up vertical light cone as future light cone and down vertical means past light cone. The four lines as a boundary of light cone are null lines. The timelike edges are inside the light cone and the spacelike edges are outside the lightcone. The following definitions may not be used in our case but may be useful if we define here. Future timelike edges: This is a type of edges where a reference endpoint is at past vertex of a future-directed timelike edges. For our case, reference endpoint is a central vertex. Past timelike edges: This is a type of edges where a reference endpoint is at future vertex of a future-directed timelike edges.

3.3. CAUSALITY VALIDATION

31

ke

ke

Fu

i cel

pa

tur et

rs

im

eli

o ke

eli tim

ed ge

These definitions are illustrated in figure 3.5.

Central vertex

sp eli ke

or

e e

g ed

tim

ke

eli im

lin

ac

ll

eli

Nu

st t

Pa

ke

Spacelike edge

Figure 3.5: Understanding future and past timelike edges The violation and validation of local causality was discussed in chapter 2. But in this section, our aim is to experimentally verify whether or not a triangulation is causally valid. Algorithm 3.4. The algorithm for checking the local causality condition goes like this: 1. Choose a starting triangle and one of its edges then, start moving in a particular direction. 2. Count how many times we switch from timelike to spacelike before we come back to the edge we started out from. 3. If it is not equal to 4, local causality is violated. We use this procedure only for validation. During a simulation, we can enforce that our triangulation is causal by starting from a causal triangulation and allowing only Monte Carlo moves which map to a causal triangulation.

32


3.4

Monte-Carlo method

The term Monte-Carlo was coined in the 1940s by physicists Stanislaw Ulam, Enrico Fermi, John von Neumann, and Nicholas Metropolis (among others) working on the nuclear weapons project at Los Alamos National Laboratory. The name is inspired by the Monte Carlo Casino in Monaco because it is related to random numbers. This solve many nuclear weapon problems at the Manhattan Project [46]. They didn’t have computers during the Manhatten project. At the time, “computer” meant a person who did a calculation for them. The Manhattan project did have large adding machines but a Monte-Carlo simulations at the time involved quite a lot of manual pen and paper work. In fact, there were large booklets of pseudo-random numbers that had been pre-computed by algorithms much like the Mersenne Twister algorithm we’re using. The Monte-Carlo (MC) method in scientific computation is possibly one of the most important numerical approaches to study problems spanning all numerically solve able scientific disciplines. The main idea is to randomly sample a volume in any fixed dimensional space under sampling using some algorithm to obtain results at the price of statistical error. Generating random numbers on a computer is hard thing. Computers are deterministic, so random numbers are actually pseudo-random, and generated by an algorithm intended to mimic randomness. By default, Python uses the Mersenne Twister algorithm to generate random numbers. And it’s reliable. But this is a subtle thing. The sampling techniques used for MC are MetropolisHastings, simulated annealing, parallel tempering, Gibbs sampling, rejection sampling and so on. But in our case, we will be using well-known Metropolis-Hastings to sample our spacetimes which depends upon Boltzmann probability distribution. We have defined our probability distribution from equation 2.55. i.e. P (T ) =

1 e−SE [T ] C(T )

Z −SE [T ]

∴ P (T ) =

e C(T )Z

(3.1)

where, Z=

X T

and,

1 −SE [T ] e C(T )

√

SE [T ] =

3Λ (Ntts + Nsst ). 32πG

Now, we will be explaining about the important requirements for this algorithm.

3.4.1

Detailed balance

The Metropolis-Hastings algorithm only works if set of moves are ergodicity and detailed balance. We suppose that our moves are ergodic4 in nature. So, we proceed to discuss detailed balance condition. The sampling technique uses the idea of Markov chain of successive con4

We will discuss ergodicity in next section.

3.4. MONTE-CARLO METHOD

33

figurations5 Ti where each configuration Tj is constructed from a previous configuration Ti via a suitable transition probability P (Ti → Tj ). The main idea is that, when reaching equilibrium, it should satisfy the detailed balance condition. i.e. Peq (Ti )P (Ti → Tj ) = Peq (Tj )P (Tj → Ti )

(3.2)

where, we call Peq (Ti ) the probability of being in ith configuration of spacetime during equilibrium condition and this is equation 3.1. The equation 3.2 suggests that, at equilibrium there is an equal probability for the transitions Ti → Tj and Tj → Ti . Using equation 3.1 in 3.2. we get, e−SE [Tj ] e−SE [Ti ] × P (Ti → Tj ) = × P (Tj → Ti ) C(Ti )Z C(Tj )Z Canceling Z in both side and putting alone the ratio of transition probabilities in left side. At right side, we compare Boltzmann factors between two spacetimes which are different only by one Ergodic move. Thus we get, P (Ti → Tj ) (1/C(Tj ))e−SE [Tj ] = . P (Tj → Ti ) (1/C(Ti ))e−SE [Ti ] Because of Ti and Tj are different only by one ergodic move such that C(Ti ) can be approximated as equal to C(Tj ). And if these factors are equal, the measure cancels out. We get, e−SE [Tj ] P (Ti → Tj ) = −S [Ti ] . P (Tj → Ti ) e E P (Ti → Tj ) ∴ = e−(SE [Tj ]−SE [Ti ]) . P (Tj → Ti )

(3.3)

After putting the value of final action from equation 2.56, the right hand side of equation 3.3 will be ! √ 3Λ (∆Ntts + ∆Nsst ) (3.4) exp(−(SE [Tj ] − SE [Ti ])) = exp 32πG where ∆N = NTj − NTi describes the difference in triangle count after a move has been executed. We split the transition probability into two separate parts [22]. i.e. P (Ti → Tj ) = g(Ti → Tj )A(Ti → Tj )

(3.5)

where g(Ti → Tj ) is the trial probability that helps to pick the suitable region at random in the spacetime, and A(Ti → Tj ) is the probability of accepting the move. We splits transition 5 The terms “successive configurations” and “successive states” are interchangeable in our case because in a given configuration we try to change the curvature of small region in the spacetime using only one randomly picked move. We mean configuration as a triangulated spacetime.

34


probability is because it depends upon the region of spacetime we pick at random, and chances of having another valid triangulations after a move. In the Ising model with N spins in any dimensional space, the trial probability g(si → sj ) is 1/N because the probability of picking a spin at random is 1/N . Similary, that of inverse trial probability g(sj → si ) is 1/N is also same. These cancel out! So, we don’t need to worry of calculating them. But, this is not the case for CDT. So, we need to calculate the trail probability and its inverse for every Monte-Carlo moves. By rearranging equation 3.3 using 3.5, the condition of detailed balance can be put into the following form. i.e. g(Tj → Ti ) A(Ti → Tj ) = e−(SE [Tj ]−SE [Ti ]) × = x (say). A(Tj → Ti ) g(Ti → Tj )

(3.6)

The Metropolis choice then is to pick the following condition that satisfies the above equation 3.6: A(Ti → Tj ) = min(1, x).

(3.7)

This condition can be easily implemented6 by generating a random number r ∈ [0, 1] and by accepting the proposed move if r ≤ x .

3.5

Monte-Carlo moves

We defined Monte-Carlo moves as an ergodic set of moves. A set is ergodic if it is able to transform any valid triangulation into any other valid triangulations, by running an arbitrary number of moves. A triangulations is valid if [22] 1. The triangulation is a manifold. 2. Each triangle has type tts or sst. 3. Triangles are glued together only on links of the same type. 4. At each vertex there are four sectors such that causal structure is preserved, as discussed in section 3.3. For CDT in (1+1) dimensions ergodicity is proven but in LCDT there is no analytic proof [22]. So, we assume it does provide us the valid triangulations. We also check consistency throughout our code to ensure our triangulations are always valid. In this point, we need moves that can change the curvature of spacetime. While defining the move, it’s very important that we need its probability of selecting a specific region for the move and its inverse move on the same region. In our implementation, we have first check causality before executing the move during Metropolis-Hastings sampling.

6 We will explain the Metropolis-Hastings algorithm in Chapter 5. We have made this section to show that why we need transition probability for every moves.

3.5. MONTE-CARLO MOVES

3.5.1

35

Alexander move

Figure 3.6: Before Alexander move

Figure 3.7: After Alexander move

The Alexander move is a move that takes two tts triangles and transforms into four tts triangles. Thus, this is a volume increasing move on the spacetime. To have desire volume, we will be running this move a certain number of times. It is the only move that is always possible and never violates causality in any vertex in the region abcd of figure 3.6. In this move, a new vertex is created on the spacelike edge which is the shared edges of two tts triangles. In figure 3.6 and 3.7, one of the eight possible Alexander moves is shown. In order to calculate the trial probability of Alexander move, the steps goes like this: 1. Suppose, we have N number of vertices in our triangulated spacetime. We pick a vertex a(say, in figure 3.6) at random with the probability 1/N . 2. Within the probability of vertex a, we pick anyone neighbor vertex b at random with probability 1/Ne (a) where Ne (a) is the total number of neighbor vertices of a. Thus, we pick an edge. This edge must be spacelike. If we do not make this distinction, this edge can be timelike thus, there are two inverse Alexander moves possible and we need an additional factor of 1/2 in inverse trial probability. So in our implementation, we avoid the edge to be timelike. So, probability of picking the spacelike edge at random with reference to vertex a is N1 Ne1(a) . 3. Likewise we picked vertex a, we have also equal chances of being picked vertex b. And then, we get the same spacelike edge. So, probability of picking the spacelike edge at random with reference to vertex b is N1 Ne1(b) where Ne (b) is the total number of neighbor vertices of b. 4. Finally the trial probability of Alexander move by picking the spacelike edge is sum of the results from step 2 and 3 [22] because these probabilities are mutually exclusive. i.e. 1 1 1 1 + N Ne (a) N Ne (b) 1 1 ∴ g(Ti → Tj ) = + . N Ne (a) N Ne (b) g(Ti → Tj ) =

5. Now, we will take two tts triangles that shares this edge. Then, we perform Alexander move like in figure 3.7.

36


Now, we need to calculate the inverse trial probability g(Tj → Ti ). Since, the number of vertices is increased by one (i.e. N + 1) after running the Alexander move. So, the probability of picking the newly created vertex e is: g(Tj → Ti ) =

1 . N +1

Algorithm 3.5. The algorithm in-order to implement Alexander move like as in figure 3.7 goes like this: 1. Randomly pick anyone vertex and find the edges that are connected to it. 2. Check whether these edges are spacelike or not. 3. If there are more than one spacelike edges connected to it, randomly pick anyone spacelike edge. 4. Find two triangles that share this spacelike edge. If these triangles is of type tts then, go to the below step. If not, stop the process. 5. Create a new vertex e on the spacelike edge ab. Thus, we have two spacelike edges (i.e. ae and eb). 6. Create two timelike edges (i.e. ce and ed). 7. Create four tts triangles (i.e. aec, ebc, aed and ebd). 8. Delete tts triangles (i.e. abc and abd) and spacelike edge ab. 9. Say the deleted objects’ neighbors (i.e. vertices and triangles) that they are no any more. 10. Connect neighbors to newly created geometric objects and vice versa. The change in the volume of spacetime on tts triangle is 2 (i.e. ∆Ntts = 2) but, on sst triangle is 0 (i.e. ∆Nsst = 0). This means the spacetime volume is actually increased by 2. Thus, the Alexander move is a volume increasing move.

3.5.2

Collapse move

Figure 3.8: Before collapse move

Figure 3.9: After collapse move

3.5. MONTE-CARLO MOVES

37

The Collapse move is a move that takes two tts triangle and delete those triangles by collapsing the spacelike edge. In figure 3.9, the collapse move is shown. The trial probability for this move is same as that of Alexander move. i.e. ∴ g(Ti → Tj ) =

1 1 + . N Ne (a) N Ne (b)

But the inverse trial probability is a quite different. In order to calculate the inverse trial probability of collapse move, the steps goes like this: 1. We randomly pick a vertex (k say in figure 3.9) with probability 1/(N − 1). This means we have N -1 vertices because vertices a and b merged into k. Noting that this N is a number of vertices of the triangulation before collapse move is applied. 2. Within this probability of picking k, we need to randomly pick vertices c and d. 3. Let’s work with c. The total number of neighbor vertices of k is Ne (k) = (Ne (a) − 3) + (Ne (b) − 3) + 2 = Ne (a) + Ne (b) − 4. Thus, the probability of picking the neighbor vertex c is 1/Ne (k). 4. Since, we randomly pick one neighbor c of k. So, the remaining neighbor vertices of k is Ne (k) − 1 = (Ne (a) + Ne (b) − 4) − 1 = Ne (a) + Ne (b) − 5. Thus, the probability of picking the neighbor vertex d is 1/(Ne (k) − 1). 5. These two probabilities are dependent on the probability of picking the vertex k. So, they are mutually inclusive case. Thus, we need to multiply them as 1 1 1 . (N − 1) (Ne (a) + Ne (b) − 4) (Ne (a) + Ne (b) − 5) 6. We can also have another case, that is to randomly pick any one vertex k then, first pick d and then c which also give the same probability like in step 5. But in the former case, we picked c first then d. 7 These two probabilities are mutually exclusive to each other, so we need to add them which results the final form of inverse trial probability as [22] g(Tj → Ti ) =

2 . (N − 1)(Ne (a) + Ne (b) − 4)(Ne (a) + Ne (b) − 5)

In short, the inverse collapse move involves picking a vertex and then selecting two timelike edges from two different sector of the light cone of that vertex. This is the main reason we work for above 7 steps in-order to get inverse trial probability. Algorithm 3.6. The algorithm in-order to implement collapse move like as in figure 3.9 goes like this: 1. The initial steps upto 4 of algorithm 3.5. are same for implementation of collapse move. 2. We merge vertices a and b into a newly created vertex k which means we delete spacelike edge (i.e. ab), timelike edges (i.e. ac, bc, ad and bd) and two tts triangles (i.e. abc and abd).

38

CHAPTER 3. IMPLEMENTATION DETAILS 3. Create two timelike edges (i.e. kc and kd). 4. Say the deleted objects’ neighbors (i.e. vertices and triangles) that they are no any more. 5. Connect neighbors to newly created geometric objects and vice versa.

The change in the volume of spacetime on tts triangle -2 (i.e. ∆Ntts = −2) but on sst triangle is 0 (i.e. ∆Nsst = 0). This means the spacetime volume is actually decreased by 2. Thus, the collapse move is a volume decreasing move.

3.6

Concluding remarks

We have presented the detailed implementation of CDT upto where we ended up to start writing this report.

CHAPTER

4

RESULTS “Try, and try again, boy, you will succeed at last. ” - Udayaraj Khanal recites this poem line by James H. Fassett

In this chapter, we highlight our discussion on our implemented codes and showing they are working as we thought.

4.1

Usage

We haven’t fully completed our code so, we usually test our code using debugging.py module that we have created. This module contains the below codes: ””” P a t h : / F u l l c d t o n e p l u s o n e / d e b u g g i n g . py Time−s t a m p : A u t h o r : Damodar R a j b h a n d a r i ( 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 t h e p y t h o n 3 . x i n t e r p r e t e r and p l a y a r o u n d w i t h t h e 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 a s np import s c i p y a s s p import random # I m p o r t i n g c l a s s d a t a 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 a s t # I n o r d e r t o 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 s h o u l d be # i m p o r t e d f i r s t and t h e n , 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 a s s t import t r i a n g l e p r o p e r t y a s t p 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 a s sm import m o v e f a c t o r y a s m import u t i l i t i e s a s u t

39

40

CHAPTER 4. RESULTS In Python console, we call this module by:

>>> from d e b u g g i n g import * We can see the imported modules in our current Python environment by calling this command. i.e. >>> p r i n t ( d i r ( ) ) [ ’ builtins ’ , ’ doc ’ , ’ l o a d e r ’ , ’ name ’ , ’ p a c k a g e ’ , ’ s p e c ’ , ’ i n i t i a l i z a t i o n ’ , ’m’ , ’ np ’ , ’ random ’ , ’sm ’ , ’ s p ’ , ’ s t ’ , ’ t ’ , ’ t p ’ , ’ u t ’ ]

4.2

Initial data structure

We will be discussing the initial data structure using two torus data which was described in chapter. To initialize our spacetime, we need to give following commands in Python console then, gives the 18 tts triangles. >>> i n i t i a l i z a t i o n . b u i l d f i r s t t w o t o r u s ( ) [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18] The vertex, timelike, spacelike, tts triangle and sst triangle classes are written in sub triangle property.py module so, we can know whether or not all the required geometric objects are made by calling: >>> s t . c l a s s . i n s t a n c e s where st is short form of sub triangle property.py defined in debugging.py module. This means to know vertex objects, we need to give below command: >>> {1: 2: 3: 4: 5: 6: 7: 8: 9:

st . vertex . instances }

which results 9 vertices from triangulated spacetime. Now, we are going to show that every vertices has its own information by calling the vertex object key from vertex class instances:

4.2. INITIAL DATA STRUCTURE

41

>>> p r i n t ( s t . v e r t e x . i n s t a n c e s [ 1 ] ) VERTEX OBJECT => ID : 1 TRIANGLE TYPE : TTS t y p e : [ 1 , 2 , 1 7 , 1 8 , 6 , 1 3 ] SST t y p e : [ ] EDGES TYPE : Timelike edges : [ 1 , 2 , 18 , 13] Spacelike edges : [2 , 6] Likewise, we can call timelike instances as: >>> s t . t i m e l i k e . i n s t a n c e s {1: }

and calling its object information as: >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 1 ] ) Time−l i k e e d g e o b j e c t . ID : 1 V e r t i c e s : [1 , 2] We call spacelike instances as: >>> {1: 2: 3: 4: 5: 6: 7: 8: 9:

st . spacelike . }

42

CHAPTER 4. RESULTS

>>> p r i n t ( s t . s p a c e l i k e . i n s t a n c e s [ 1 ] ) Space−l i k e e d g e o b j e c t . ID : 1 V e r t i c e s : [2 , 6] We call tts triangle instances as: {1: }

and calling its object information as: >>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 1 ] ) tts triangle object . ID : 1 V e r t i c e s : [1 , 2 , 6] S p a c e l i k e Edge : [ 1 ] T i m e l i k e Edge : [ 1 , 2 ] Neighbor t t s t r i a n g l e s : [8 , 2 , 6] Neighbor s s t t r i a n g l e s : [ ] We call sst triangle instances as: >>> s t . s s t t r i a n g l e . i n s t a n c e s {} Since, our initial triangulations is of standard CDT so, there are no any sst type of triangles. We have listed all other vertices, timelikes, spacelikes and tts triangles informations in appendix C.

4.3

Causality code

We have presented our causality check code as a snippet and it is defined in vertex class. We used Algorithm 3.4. which is defined in chapter 3 to write get sector vertices() and check causality(). This looks like this:

4.3. CAUSALITY CODE

43

# Counts l i g h t c o n e s e c t o r s def g e t s e c t o r v e r t i c e s ( s e l f ) : ””” F i n d t h e number o f s e c t o r ( e q u a l s t o t r a n s i t i o n l i n e ) i n a v e r t e x . T r a n s i t i o n l i n e r e p r e s e n t s t h e n u l l l i n e o f l i g h t cone . R e s u l t s : T o t a l s e c t o r s == T o t a l t r a n s i t i o n l i n e s . ””” triangles = self . get sst triangles () + self . get tts triangles () e d g e s = lambda t r i a n g l e : t r i a n g l e . g e t t i m e l i k e e d g e s ( ) + \ triangle . get spacelike edges () # Pick a t r i a n g l e ! pick tri = triangles [0] # To c h e c k t r a n s i t i o n f r o m two s e c t o r , we n e e d : previous edge = 0 # i n i t i a l l y nothing t r a n s i t i o n l i n e = 0 # Boundary o f t h e l i g h t c o n e . # R e s u l t a n t t r i a n g l e s r e p r e s e n t s the t e s t e d t r i a n g l e s in 1 s t loop resultant triangles = [ triangles [0]] # 1 s t loop f o r i i n np . a r a n g e ( l e n ( t r i a n g l e s ) ) : for t in t r i a n g l e s : i f t not in r e s u l t a n t t r i a n g l e s : f o r e1 i n e d g e s ( p i c k t r i ) : f o r e2 i n e d g e s ( t ) : i f e1 == e2 : pick tri = t r e s u l t a n t t r i a n g l e s . append ( t ) i f p r e v i o u s e d g e == 0 : pass else : i f t y p e ( e1 ) == t y p e ( p r e v i o u s e d g e ) : pass else : transition line = transition line\ + 1 p r e v i o u s e d g e = e1 # 2 nd l o o p for t in r e s u l t a n t t r i a n g l e s [ 0 : 2 ] : f o r e1 i n e d g e s ( p i c k t r i ) : f o r e2 i n e d g e s ( t ) : i f e1 == e2 : pick tri = t i f p r e v i o u s e d g e == 0 : pass else : i f t y p e ( e1 ) == t y p e ( p r e v i o u s e d g e ) : pass else : transition line = transition line + 1 p r e v i o u s e d g e = e1 return t r a n s i t i o n l i n e

44

CHAPTER 4. RESULTS

# Checks s i n g l e v e r t e x c a u s a l i t y . def c h e c k c a u s a l i t y ( s e l f ) : ””” T h i s w i l l c h e c k w h e t h e r o r n o t t h e r e i s e x a c t l y one l i g h t c o n e on a v e r t e x . i . e . c h e c k s l o c a l c a u s a l i t y r e q u i r e m e n t f o r a g i v e n v e r t e x . R e t u r n True , i f v e r t e x i s c a u s a l , e l s e F a l s e . N o t e : I n s t a n d a r d CDT , t h e v e r t e x c a u s a l i t y v a l i d a t i o n i s always t r i v i a l . ””” i f s e l f . g e t s e c t o r v e r t i c e s ( ) == 4 : return True else : return False # Checks g l o b a l ( o v e r a l l ) c a u s a l i t y @classmethod def c h e c k c a u s a l i t y a l l ( c l s ) : ””” T h i s method w i l l c h e c k t h e g l o b a l c a u s a l i t y o f a l l t h e v e r t i c e s and t h e c l s s h o u l d a l w a y s n e e d t o be v e r t e x c l a s s . Need t o i m p o r t p a c k a g e : f u n c t o o l s CALL I T IN DEBUGGING AS : st . vertex . check causality all () I f a l l T r u e : r e t u r n s True , e l s e F a l s e . ””” r e t u r n r e d u c e ( lambda x , y : x and y , map ( lambda v : c l s . c h e c k c a u s a l i t y ( v ) , cls . instances . values ( ) ) )

Since, vertices of triangulated spacetime is itself a set of events. So, each event should have causal relation on other events that means each vertex should have exactly one light cone. To check any one of vertex causality, we run a test by calling this: >>> s t . v e r t e x . i n s t a n c e s [ 1 ] . c h e c k c a u s a l i t y ( ) True which means vertex of ID 1 has exactly one light cone. Now, we check all the vertices causality at once by calling this: >>> s t . v e r t e x . c h e c k c a u s a l i t y a l l ( ) True which means all the vertices has exactly one light cone each.

4.4. MONTE-CARLO MOVES WORKS

4.4

45

Monte-Carlo moves works

In-order to run Monte-Carlo moves, we need to call the MovesFactory class from the move factory.py. i.e. >>> a = m. M o v e s F a c t o r y ( ) >>> a Now, we need to select any one of the move from the set at random: l i s t o f a l l m o v e s = [ AlexanderMove , C o l l a p s e M o v e ] by calling this command: >>> b = a . t r y r a n d o m ( ) >>> b

4.4.1

Testing Alexander move

We are going to test the Alexander move class by running the below commands: >>> b . g e t r a t i o t r i a l p r o b a b i l i t y ( ) 2.7 which means the move selected a region with the ratio of trial probabilities 2.7. Now, we are going to apply this move to change the structure of spacetime by: >>> b . e x t r a c t a l l i n f o a n d u p d a t e ( ) Now, we can find the vertices of new spacetime to be: >>> l e n ( s t . v e r t e x . i n s t a n c e s ) 10 which was previously 9. Again, we can find the timelike edges to be: >>> l e n ( s t . t i m e l i k e . i n s t a n c e s ) 20 which was previously 18. We can find the spacelike edges to be: >>> l e n ( s t . s p a c e l i k e . i n s t a n c e s ) 10 which was previously 9. We can find the tts triangles to be: >>> l e n ( s t . t t s t r i a n g l e . i n s t a n c e s ) 20 which was previously 18. Now, we are going to check the global causality of this spacetime as:

46

CHAPTER 4. RESULTS

>>> s t . v e r t e x . c h e c k c a u s a l i t y a l l ( ) True Thus, we know that Alexander move is perfectly working. In the next step, we are again going to randomly select Collapse move from list of all moves and apply this move on the newly generated spacetime. This can be done by the same command: >>> b = a . t r y r a n d o m ( ) >>> b

4.4.2

Testing Collapse move

We are going to test the collapse move class by running the below commands: >>> b . g e t r a t i o t r i a l p r o b a b i l i t y ( ) 0.11904761904761904 which means the move selected a region with the ratio of trial probabilities 0.11904761904761904. Now, we are going to apply this move to change the structure of spacetime by: >>> b . e x t r a c t a l l i n f o a n d u p d a t e ( ) Now, we can find the vertices of new spacetime to be: >>> l e n ( s t . v e r t e x . i n s t a n c e s ) 9 which was previously 10. Again, we can find the timelike edges to be: >>> l e n ( s t . t i m e l i k e . i n s t a n c e s ) 18 which was previously 20. We can find the spacelike edges to be: >>> l e n ( s t . s p a c e l i k e . i n s t a n c e s ) 9 which was previously 10. We can find the tts triangles to be: >>> l e n ( s t . t t s t r i a n g l e . i n s t a n c e s ) 18 which was previously 20. Now, we are going to check the global causality of this spacetime as: >>> s t . v e r t e x . c h e c k c a u s a l i t y a l l ( ) True Thus, we know that collapse move is perfectly working.

4.5. DESIRE VOLUME

4.5

47

Desire volume

Before running the Monte-Carlo simulation, first we need to fixed the volume of spacetime to reduce the finite size effects. So, we run Alexander move for a number of times to increase our volume.

4.6

Concluding remarks

We were expecting that if we run the collapse move after the Alexander move then, we can get the same initial spacetime that we started using two torus data, which is proved by sections 4.4.1 and 4.4.2. We successfully finished our project in the sense that we wrote perfectly working codes described in the chapter 3.

CHAPTER

5

FUTURE WORK “It is possible to choose a discrete approximation of a problem and a discrete approximation to solutions of that problem which are incompatible. There is, in other words, a degeneracy of discretizations. ” - Jonah Maxwell Miller

In this chapter, we briefly describe our future works in order to complete our simulation code. Our new software toolkit for LCDT is freely available under GNU public license and contribute to the open source community.

5.1

Other moves

Due to the time frame of our project, we haven’t had a chance to implement the inverse Alexander move, inverse collapse move, inverse pinching move, inverse flip move, pinching move, or flip move. We hope to finish these in the near future. To introduce the last two moves in-short. The pinching move takes four tts triangles and transform them into two sst triangles. The flip move takes two sst triangle and changes the curvature at the vertices but does not change the number of triangles. Even if we pick flip move at random from a set of moves, there is a high chances of rejection because of the lack of the required combination of sst triangles. Thus, the pinching move has a higher probability of being accepted than the flip move. These two moves make the foliated structure to non-foliated. For more details, see the work of Ben Ruijl [22].

5.2

Metropolis-Hastings algorithm

The algorithm goes like this: 1. Randomly apply one of the ergodic moves to the spacetime.

48

5.3. VOLUME CONTROL

49

2. Check the causality of vertices in the region where move changes it. If it violate the causality then, apply its inverse move to go back to the previous spacetime. If not, proceed to the next step. 2. Calculate whether the new spacetime is more or less likely than the old spacetime, as given by its un-normalized weight in the path integral eiSE [T ] . 3. If the new spacetime is more likely than the previous spacetime, accept the change with probability P = 1. 4. If it is less likely, accept the change in probability P = P (new)/P (old). 5. Repeat until the spacetime is approximately saturated which means spacetime remains same even if we proceed to apply any moves.

5.3

Volume control

While running the simulation, most of the Monte-Carlo moves changes the volume of spacetime and therefore we may not keep track of the number of triangles. Thus, eventually collapse our spacetime or may change the spacetime topology. So, we need to add a potential to the action which forces the volume to fluctuate around a prescribed value. One best example may be a quadratic potential of the form [22] δS = (Vcur − V )2 where Vcur is the current volume and V is the desired volume and > 0 is a constant that controls how strict the volume fixing should be. The other parameter we can tune is the cosmological constant Λ. These are our future works.

5.4

Contribution to an open source scientific community

We provide our implementation, which is based on the Rajesh Kommu [42], in a public repository [47]. We haven’t implemented the above discussion and one can also contribute to help us in our project. We are grateful to the countless developers contributing to open source projects on which we relied in this work, including Python [48], NumPy [49], SciPy [50] and Matplotlib [51].

5.5

Motivation to start this project work

We started this project with an ambition of rewriting the (1+1) dimensional LCDT code1 from scratch, and this code will be further developed to prove our hypothesis that geodesic distance will provide a meaning improvement to “jumping distance” used by spectral and Hausdorff dimension calculations. This could be a tool for better understanding renormalization group flow in CDT. Heisenberg’s uncertainty principle tells us that on very short length scales (roughly Planck length), the uncertainty in momentum can be much larger than its average magnitude. This 1

(1+1) dimensional LCDT code was previously written by Ben Ruijl [22].

50

CHAPTER 5. FUTURE WORK

can drive violent fluctuations in spacetime geometry. Quantum field theories are naturally violent at short length scales [52]. What makes quantum gravity unique is that this violence can undermine the meaning of “short length scale”. After all, if the metric fluctuates violently, so does how we measure distance and time. We are still interested in the dimensional reduction in short distance. At large scales, spacetime behaves as a smooth four-dimensional Pseudo-Riemannian manifold. But roughly at the Planck scale, the dimensions are reduced to two [9]. This shows that quantum effects are very important and can lead to fractal geometry. Thus, the dimension is scale dependent [53, 54, 55]. Why two dimensions? One way we can improve our understanding is by studying the relation of the geodesics with spectral dimension. We state this way because the spectral dimension is studied in the function of diffusion time. This is shown in the figure 5.1,

Figure 5.1: The spectral dimension ds of the universe as a function of diffusion time σ, taken from [56]. where, the thick curve plots the average measured spectral dimension, while the highlighted area represents the error bars. In fact, diffusion on fractal structure is well studied in statistical physics [57]. In CDT, no one has studied and defined geodesic distance associated with the fluctuating metric of spacetime. But, one may find the definition of geodesic distance in Lioville quantum gravity which is associated to a Gaussian free field on a regular lattice [58]. Our future research will provide a modification of the usual perception of geodesic motion of classical general relativity and a better understanding of how the force works for very small scales.

5.6

Concluding remarks

If you have any questions or comments about this report or the source code, please contact us at [email protected]. Happy learning!

APPENDIX

A

SUPPLEMENT DERIVATIONS A.1

4-Volume in-terms of Jacobian 0

0

Let {xα }3α0 =00 be locally Minkowskian co-ordinate system for (3+1) dimensional spacetime. Suppose, we have another locally Minkowskian co-ordinate system {xβ }3α0 =0 . The co-ordinates in each system can be thought of as functions of the co-ordinates in the other system. i.e. 0

0

0

0

x0 = x0 (x0 , x1 , x2 , x3 ) .. . x3 = x3 (x0 , x1 , x2 , x3 ) To transform between our coordinate systems, we use the chain rule: 0

∂xα β dx . dx = ∂xβ α0

(A.1)

If we put this in the form of matrix expression then, we have:  00   0  0 0 dx ∂x0 ∂x0 dx · · · ∂x3 1 dx10   ∂x. 0 .  ..  dx    0  =  .. .  . . dx2  dx2  . 0 0 3 0 ∂x3 · · · ∂x dx3 dx3 ∂x0 ∂x3 We call this matrix the Jacobian:  ∂x00 ∂x0

 Jacobian matrix (J) =  ...

0

∂x3 ∂x0

This can be written in component form as 0

0

dxα = Jβα dxβ . The volume element in (1+1) dimensions is [59] 51

0

··· ...

∂x0 ∂x3

···

∂x3 ∂x3



..  . .  0

52

APPENDIX A. SUPPLEMENT DERIVATIONS d(2) V = |J|dx0 dx1

where |J| is the determinant of the Jacobian. Similarly, the volume element in (2+1) dimensions is d(3) V = |J|dx0 dx1 dx2 . And the volume element in (3+1) dimensions is d(4) V = |J|dx0 dx1 dx2 dx3 .

A.2

(A.2)

Relationship between the Jacobian and the metric

The metric tensor for arbitrary curvilinear coordinates on a flat spacetime is given by [29] 0

gµν

0

∂xγ ∂xδ = ηγ 0 δ 0 . ∂xµ ∂xν

In terms of Matrix Language, this can be written as g = J T ηJ where J T is the transpose of Jacobian matrix J. Taking determinant in both side. we get, |g| = g = |J T ηJ| = |J T ||η||J| = |J|2 |η|

(∵ |J T | = |J|).

(A.3)

We know −1 0 |η| = 0 0

0 1 0 0

0 0 1 0

0 0 = -1. 0 1

Thus, equation A.3 becomes g = −|J|2 or, |J|2 = −g √ ∴ |J| = −g.

A.3

Co-ordinate based 4-volume element

Combining equations A.2 and A.4, we attain d(4) V = where d4 x = dx0 dx1 dx2 dx3 .

√

−gd4 x

(A.4)

A.4. RELATE DETERMINANT OF THE METRIC TENSOR TO ITS COFACTOR

A.4

53

Relate determinant of the metric tensor to its cofactor

Recall that the inverse of a non-singular matrix M can be written as [60] M −1 =

1 T A . |M |

where A is the cofactor matrix of M and T is the transpose. In matrix language, gµν and g µν are g and g−1 . So, g−1 can be written as, 1 g−1 = AT g −1 ∴ g g = AT

(A.5)

where, g is non-singular. For now, A stands for cofactor of g. Now, multiply equation A.5 by g. We can have, gg−1 g = gAT ∴ Ig = gAT

(A.6)

∵ gg−1 = I, i.e. the identity matrix. Thus, this can be written in-terms of tensor notation as g = gµν (Aµν )T ∴ g = gµν Aνµ where, Aµν is the tensor notation of a cofactor of g. Because gµν and thus Aµν are symmetric, we can exchange the indices and attain g = gνµ Aµν ∴ g = gµν Aµν .

A.5

(A.7)

Relation of space-like interval and time-like interval

Considering the following situation. Suppose, a train is moving with a velocity v relative to the outside observer O such that v < c. Now, we setup an arrangement where we send a light ray at the velocity of c = 1 from the laser which is situated on the base and mirror is attached on the roof so that, it reflect after traveling Y distance as seen in figure A.1. There are two observers, one observer O0 sitting inside the train which means, his velocity relative to the light source is zero, and his worldline is X 0 = 0 . And, another observer O sitting outside the train, and his worldline is X = 0.

54

APPENDIX A. SUPPLEMENT DERIVATIONS

O'

Figure A.1: Light path observed by ob- Figure A.2: Light path observed by observer O relative to earth server O0 relative to the source of light Now, recall the expression for proper distance between two events from special relativity. We have [28], (A.8)

(4s)2 = −(c 4 t)2 + (4x)2 + (4y)2 + (4z)2 . For our case, equation A.8 will squeeze into,

(A.9)

(4s)2 = −(4t)2 + (4x)2 + (4y)2 .

Again, the proper distance for the light is zero. i.e. (4s)2 = 0. So, equation A.9 becomes (4t)2 = (4x)2 + (4y)2 .

(A.10)

We have two cases because of observations done by the two observers. First, let’s discuss for the case of observer O0 inside the train. Suppose the time noted down by the O0 using his clock for which, light will come back at it’s source is T 0 such that equation A.10 gives, (T 0 )2 = (2Y )2 .

(A.11)

Second, let’s discuss for the case of observer O outside the train. Suppose, the time noted down by O using his clock for which, light will come back at it’s source is T and the distance traveled by the light source at time T is X. So, equation A.10 gives, T 2 = Square of total distance (i.e. path BAC) traveled by the light ray " ( 2 )#2 X = 2 Y2+ 2 = (2Y )2 + X 2 = (T 0 )2 + X 2 ∴ −T 2 + X 2 = −(T 0 )2 .

(∵ from equation A.11)

This means, the interval in any frame is the same as the interval measured in the frame where events happened in the same place. So, −T 2 + X 2 is the proper distance observed by O and T 0 is the proper time observed by O0 . Thus, (Proper time)2 = −(Proper distance)2 .

(A.12)

A.6. AREA OF MINKOWSKIAN EQUILATERAL TRIANGLE

55

This means, in special relativity, time and space are necessarily inter-convertible. In CDT, to have more freedom when defining our lattice, we introduce an asymmetry parameter α . That means α is a free parameter and we can choose it to be whatever we like. This α keeps track of the ratio between proper time and proper distance. Mathematically, it can be defined as [9], (Proper time)2 = −α(Proper distance)2 .

A.6

Area of Minkowskian equilateral triangle

A.6.1

Area of tts triangle type

(A.13)

To calculate the area of Minkowski triangle of type tts [61], we draw a triangle (4ABC) of that type in figure A.3.

Figure A.3: Minkowski triangle of type tts Suppose, ls is the proper distance which is equal to lattice cut-off a between the two vertices of the Minkowski triangle on the spatial axis that means, ls2 = a2 . And, lt is the proper time. So, from equation A.13, we can relate proper time with proper distances as, lt2 = −αa2 . We can define the Pseudo-pythagorean theorem for the spacetime interval between two events as [28], AC 2 = −AD2 + DC 2 or, AD2 = −AC 2 + DC 2 2 ls 2 = − (lt ) + 2 a2 = −(−αa2 ) + 4 a√ ∴ AD = 4α + 1 2 Thus, volume of Minkowski triangle of type tts is, 1 V ol(4tts ) = AD × CB 2 1 a√ = × 4α + 1 × ls 2√ 2 4α + 1 2 ∴ V ol(4tts ) = a. 4

(A.14)

(A.15)

56

A.6.2

APPENDIX A. SUPPLEMENT DERIVATIONS

Area of sst triangle type

To calculate the area of Minkowski triangle of type sst [61], we draw a triangle (4ABC) of that type in figure A.4.

Figure A.4: Minkowski triangle of type sst We can define the Pseudo-pythagorean theorem for the spacetime interval between two events as [28], −AC 2 = AD2 − DC 2 or, AD2 = −AC 2 + DC 2 2 lt 2 = − (ls ) + 2 −αa2 2 = −a + 4 ap ∴ AD = −(4 + α) 2

(A.16)

(A.17)

In equation A.16, it’s little bit different than we discussed in the sub-section A.6.1. This is because, space-like edge AC actual lies inside the light cone due to the orientation of it. So, the spacetime interval between two vertices A and C should convert into time-like by using equation A.12 i.e. −AC 2 . Thus, volume of Minkowski triangle of type sst is, 1 V ol(4sst ) = AD × CB 2 1 ap = × −(4 + α) × lt 2 2 √ 1 ap = × −(4 + α) × ( −αa) 2p 2 α(4 + α) 2 ∴ V ol(4sst ) = a. 4

(A.18)

APPENDIX

B

SUPPLEMENT CODES B.1

Another approach to initialize data structure

We can also use loop method to generate a Minkowski triangulated spacetime using 100 tts triangles in below code (which we suppose the desired volume). Since we need to work with data structure of geometric objects, this approach needs furthermore constructions before we run it. The code looks like this: ””” Time−s t a m p : A u t h o r : Damodar R a j b h a n d a r i ( 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 : I n CDT ( 1 + 1 ) D, we h a v e two t y p e s o f t r i a n g l e s − up p o i n t i n g ( t y p e 1 ) and down p o i n t i n g ( t y p e 2 ) . For s i m p l i c i t y , I t w i l l be e a s y t o v i s u a l i z e i n Minkowski s p a c e t i m e . i . e . ˆ | | | | | | | | | | | time | | | | −−|−−−−−−−−−−−−> axis | | | | | | | | | | −|−−−−|−−−−|−−−−|−−−−|−− > space a x i s ””” import numpy a s np import m a t p l o t l i b . p y p l o t a s p l t n = 10 N = np . a r a n g e ( n )

57

. . . . . .. . . . Type 2 . . . . . . . . . . Type 1 . . . . . . . . . .

58

APPENDIX B. SUPPLEMENT CODES

f or j in N: f o r i in N: # For Type 1 t r i a n g l e v e r t i c e s 1 = np . a r r a y ( [ [ i , j ] , [ i +1 , j ] , [ i , j + 1 ] ] ) # For Type 2 t r i a n g l e v e r t i c e s 2 = np . a r r a y ( [ [ i +1 , j ] , [ i , j + 1 ] , [ i +1 , j + 1 ] ] ) t r i a n g l e 1 = p l t . Polygon ( v e r t i c e s 1 , c o l o r = ’ r ’ ) t r i a n l g e 2 = p l t . Polygon ( v e r t i c e s 2 , c o l o r = ’b ’ ) # gca ( ) s t a n d s f o r g e t c u r r e n t a x i s . i . e . # I t returns the current axis instance . p l t . gca ( ) . a d d l i n e ( t r i a n g l e 1 ) p l t . gca ( ) . a d d l i n e ( t r i a n l g e 2 ) # I t w i l l put a l l the t r i a n g l e s properly . plt . axis ( ’ scaled ’ ) p l t . x l a b e l ( ” Space a x i s ” ) p l t . y l a b e l ( ” Time a x i s ” ) x i n t , y i n t = np . a p p e n d (N+ 1 , 0 ) , np . a p p e n d (N+ 1 , 0 ) plt . xticks ( xint ) plt . yticks ( yint ) p l t . show ( t r i a n g l e 1 ) p l t . show ( t r i a n l g e 2 )

And the output of this code gives:

10 9 8

Time axis

7 6 5 4 3 2 1 0 0

1

2

3

4 5 6 Space axis

7

8

9

10

Figure B.1: Triangulation of (1+1)D spacetime before any move or anti-move was made

APPENDIX

C

SUPPLEMENT CODE OUTPUTS We have listed all the remaining vertices, timelikes, spacelikes and tts triangles information which was left behind in Chapter 4.

C.1

Vertices

>>> p r i n t ( s t . v e r t e x . i n s t a n c e s [ 2 ] ) VERTEX OBJECT => ID : 2 TRIANGLE TYPE : TTS t y p e : [ 1 , 5 , 6 , 7 , 8 , 1 2 ] SST t y p e : [ ] EDGES TYPE : Timelike edges : [8 , 1 , 6 , 7] Spacelike edges : [1 , 5] >>> p r i n t ( s t . v e r t e x . i n s t a n c e s [ 3 ] ) VERTEX OBJECT => ID : 3 TRIANGLE TYPE : TTS t y p e : [ 1 8 , 7 , 1 1 , 1 2 , 1 3 , 1 4 ] SST t y p e : [ ] EDGES TYPE : Timelike edges : [12 , 13 , 14 , 7] Spacelike edges : [9 , 7]

59

60

APPENDIX C. SUPPLEMENT CODE OUTPUTS

>>> p r i n t ( s t . v e r t e x . i n s t a n c e s [ 4 ] ) VERTEX OBJECT => ID : 4 TRIANGLE TYPE : TTS t y p e : [ 2 , 3 , 4 , 1 3 , 1 4 , 1 5 ] SST t y p e : [ ] EDGES TYPE : Timelike edges : [ 3 , 4 , 14 , 15] Spacelike edges : [2 , 4] >>> p r i n t ( s t . v e r t e x . i n s t a n c e s [ 5 ] ) VERTEX OBJECT => ID : 5 TRIANGLE TYPE : TTS t y p e : [ 1 6 , 1 7 , 4 , 5 , 6 , 1 5 ] SST t y p e : [ ] EDGES TYPE : Timelike edges : [16 , 17 , 5 , 6] Spacelike edges : [4 , 6] >>> p r i n t ( s t . v e r t e x . i n s t a n c e s [ 6 ] ) VERTEX OBJECT => ID : 6 TRIANGLE TYPE : TTS t y p e : [ 1 , 2 , 3 , 8 , 9 , 1 0 ] SST t y p e : [ ] EDGES TYPE : Timelike edges : [10 , 9 , 2 , 3] Spacelike edges : [1 , 3] >>> p r i n t ( s t . v e r t e x . i n s t a n c e s [ 7 ] ) VERTEX OBJECT => ID : 7 TRIANGLE TYPE : TTS t y p e : [ 3 , 4 , 5 , 1 0 , 1 1 , 1 2 ] SST t y p e : [ ] EDGES TYPE : Timelike edges : [12 , 11 , 4 , 5] Spacelike edges : [3 , 5]

C.2. TIMELIKE EDGES

>>> p r i n t ( s t . v e r t e x . i n s t a n c e s [ 8 ] ) VERTEX OBJECT => ID : 8 TRIANGLE TYPE : TTS t y p e : [ 1 6 , 7 , 8 , 9 , 1 4 , 1 5 ] SST t y p e : [ ] EDGES TYPE : Timelike edges : [ 8 , 9 , 16 , 15] Spacelike edges : [8 , 7] >>> p r i n t ( s t . v e r t e x . i n s t a n c e s [ 9 ] ) VERTEX OBJECT => ID : 9 TRIANGLE TYPE : TTS t y p e : [ 1 6 , 1 7 , 1 8 , 9 , 1 0 , 1 1 ] SST t y p e : [ ] EDGES TYPE : Timelike edges : [18 , 17 , 10 , 11] Spacelike edges : [8 , 9]

C.2

Timelike edges

>>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 2 ] ) Time−l i k e e d g e o b j e c t . ID : 2 V e r t i c e s : [1 , 6] >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 3 ] ) Time−l i k e e d g e o b j e c t . ID : 3 V e r t i c e s : [4 , 6] >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 4 ] ) Time−l i k e e d g e o b j e c t . ID : 4 V e r t i c e s : [4 , 7] >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 5 ] ) Time−l i k e e d g e o b j e c t . ID : 5 V e r t i c e s : [5 , 7]

61

62


>>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 6 ] ) Time−l i k e e d g e o b j e c t . ID : 6 V e r t i c e s : [2 , 5] >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 7 ] ) Time−l i k e e d g e o b j e c t . ID : 7 V e r t i c e s : [2 , 3] >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 8 ] ) Time−l i k e e d g e o b j e c t . ID : 8 V e r t i c e s : [8 , 2] >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 9 ] ) Time−l i k e e d g e o b j e c t . ID : 9 V e r t i c e s : [8 , 6] >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 1 0 ] ) Time−l i k e e d g e o b j e c t . ID : 10 V e r t i c e s : [9 , 6] >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 1 1 ] ) Time−l i k e e d g e o b j e c t . ID : 11 V e r t i c e s : [9 , 7] >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 1 2 ] ) Time−l i k e e d g e o b j e c t . ID : 12 V e r t i c e s : [3 , 7] >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 1 3 ] ) Time−l i k e e d g e o b j e c t . ID : 13 V e r t i c e s : [1 , 3] >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 1 4 ] ) Time−l i k e e d g e o b j e c t . ID : 14 V e r t i c e s : [3 , 4]

C.3. SPACELIKE EDGES

>>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 1 5 ] ) Time−l i k e e d g e o b j e c t . ID : 15 V e r t i c e s : [8 , 4] >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 1 6 ] ) Time−l i k e e d g e o b j e c t . ID : 16 V e r t i c e s : [8 , 5] >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 1 7 ] ) Time−l i k e e d g e o b j e c t . ID : 17 V e r t i c e s : [9 , 5] >>> p r i n t ( s t . t i m e l i k e . i n s t a n c e s [ 1 8 ] ) Time−l i k e e d g e o b j e c t . ID : 18 V e r t i c e s : [9 , 1]

C.3

Spacelike edges

>>> p r i n t ( s t . s p a c e l i k e . i n s t a n c e s [ 2 ] ) Space−l i k e e d g e o b j e c t . ID : 2 V e r t i c e s : [1 , 4] >>> p r i n t ( s t . s p a c e l i k e . i n s t a n c e s [ 3 ] ) Space−l i k e e d g e o b j e c t . ID : 3 V e r t i c e s : [6 , 7] >>> p r i n t ( s t . s p a c e l i k e . i n s t a n c e s [ 4 ] ) Space−l i k e e d g e o b j e c t . ID : 4 V e r t i c e s : [4 , 5] >>> p r i n t ( s t . s p a c e l i k e . i n s t a n c e s [ 5 ] ) Space−l i k e e d g e o b j e c t . ID : 5 V e r t i c e s : [2 , 7]

63

64


>>> p r i n t ( s t . s p a c e l i k e . i n s t a n c e s [ 6 ] ) Space−l i k e e d g e o b j e c t . ID : 6 V e r t i c e s : [1 , 5] >>> p r i n t ( s t . s p a c e l i k e . i n s t a n c e s [ 7 ] ) Space−l i k e e d g e o b j e c t . ID : 7 V e r t i c e s : [8 , 3] >>> p r i n t ( s t . s p a c e l i k e . i n s t a n c e s [ 8 ] ) Space−l i k e e d g e o b j e c t . ID : 8 V e r t i c e s : [8 , 9] >>> p r i n t ( s t . s p a c e l i k e . i n s t a n c e s [ 9 ] ) Space−l i k e e d g e o b j e c t . ID : 9 V e r t i c e s : [9 , 3]

C.4

TTS triangles

>>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 2 ] ) tts triangle object . ID : 2 V e r t i c e s : [1 , 4 , 6] S p a c e l i k e Edge : [ 2 ] T i m e l i k e Edge : [ 2 , 3 ] Neighbor t t s t r i a n g l e s : [1 , 3 , 13] Neighbor s s t t r i a n g l e s : [ ] >>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 3 ] ) tts triangle object . ID : 3 V e r t i c e s : [4 , 6 , 7] S p a c e l i k e Edge : [ 3 ] T i m e l i k e Edge : [ 3 , 4 ] Neighbor t t s t r i a n g l e s : [ 2 , 10 , 4] Neighbor s s t t r i a n g l e s : [ ]

C.4. TTS TRIANGLES

>>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 4 ] ) tts triangle object . ID : 4 V e r t i c e s : [4 , 5 , 7] S p a c e l i k e Edge : [ 4 ] T i m e l i k e Edge : [ 4 , 5 ] Neighbor t t s t r i a n g l e s : [3 , 5 , 15] Neighbor s s t t r i a n g l e s : [ ] >>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 5 ] ) tts triangle object . ID : 5 V e r t i c e s : [2 , 5 , 7] S p a c e l i k e Edge : [ 5 ] T i m e l i k e Edge : [ 5 , 6 ] Neighbor t t s t r i a n g l e s : [12 , 4 , 6] Neighbor s s t t r i a n g l e s : [ ] >>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 6 ] ) tts triangle object . ID : 6 V e r t i c e s : [1 , 2 , 5] S p a c e l i k e Edge : [ 6 ] T i m e l i k e Edge : [ 1 , 6 ] Neighbor t t s t r i a n g l e s : [1 , 5 , 17] Neighbor s s t t r i a n g l e s : [ ] >>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 7 ] ) tts triangle object . ID : 7 V e r t i c e s : [8 , 2 , 3] S p a c e l i k e Edge : [ 7 ] T i m e l i k e Edge : [ 8 , 7 ] Neighbor t t s t r i a n g l e s : [ 8 , 12 , 14] Neighbor s s t t r i a n g l e s : [ ] >>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 8 ] ) tts triangle object . ID : 8 V e r t i c e s : [8 , 2 , 6] S p a c e l i k e Edge : [ 1 ] T i m e l i k e Edge : [ 8 , 9 ] Neighbor t t s t r i a n g l e s : [1 , 9 , 7] Neighbor s s t t r i a n g l e s : [ ]

65

66


>>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 9 ] ) tts triangle object . ID : 9 V e r t i c e s : [8 , 9 , 6] S p a c e l i k e Edge : [ 8 ] T i m e l i k e Edge : [ 9 , 1 0 ] Neighbor t t s t r i a n g l e s : [ 8 , 16 , 10] Neighbor s s t t r i a n g l e s : [ ] >>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 1 0 ] ) tts triangle object . ID : 10 V e r t i c e s : [9 , 6 , 7] S p a c e l i k e Edge : [ 3 ] T i m e l i k e Edge : [ 1 0 , 1 1 ] Neighbor t t s t r i a n g l e s : [11 , 9 , 3] Neighbor s s t t r i a n g l e s : [ ] >>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 1 1 ] ) tts triangle object . ID : 11 V e r t i c e s : [9 , 3 , 7] S p a c e l i k e Edge : [ 9 ] T i m e l i k e Edge : [ 1 1 , 1 2 ] Neighbor t t s t r i a n g l e s : [18 , 10 , 12] Neighbor s s t t r i a n g l e s : [ ] >>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 1 2 ] ) tts triangle object . ID : 12 V e r t i c e s : [2 , 3 , 7] S p a c e l i k e Edge : [ 5 ] T i m e l i k e Edge : [ 1 2 , 7 ] Neighbor t t s t r i a n g l e s : [11 , 5 , 7] Neighbor s s t t r i a n g l e s : [ ] >>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 1 3 ] ) tts triangle object . ID : 13 V e r t i c e s : [1 , 3 , 4] S p a c e l i k e Edge : [ 2 ] T i m e l i k e Edge : [ 1 3 , 1 4 ] Neighbor t t s t r i a n g l e s : [ 2 , 18 , 14] Neighbor s s t t r i a n g l e s : [ ]

C.4. TTS TRIANGLES

>>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 1 4 ] ) tts triangle object . ID : 14 V e r t i c e s : [8 , 3 , 4] S p a c e l i k e Edge : [ 7 ] T i m e l i k e Edge : [ 1 4 , 1 5 ] Neighbor t t s t r i a n g l e s : [15 , 13 , 7] Neighbor s s t t r i a n g l e s : [ ] >>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 1 5 ] ) tts triangle object . ID : 15 V e r t i c e s : [8 , 4 , 5] S p a c e l i k e Edge : [ 4 ] T i m e l i k e Edge : [ 1 6 , 1 5 ] Neighbor t t s t r i a n g l e s : [16 , 4 , 14] Neighbor s s t t r i a n g l e s : [ ] >>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 1 6 ] ) tts triangle object . ID : 16 V e r t i c e s : [8 , 9 , 5] S p a c e l i k e Edge : [ 8 ] T i m e l i k e Edge : [ 1 6 , 1 7 ] Neighbor t t s t r i a n g l e s : [17 , 9 , 15] Neighbor s s t t r i a n g l e s : [ ] >>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 1 7 ] ) tts triangle object . ID : 17 V e r t i c e s : [9 , 5 , 1] S p a c e l i k e Edge : [ 6 ] T i m e l i k e Edge : [ 1 7 , 1 8 ] Neighbor t t s t r i a n g l e s : [16 , 18 , 6] Neighbor s s t t r i a n g l e s : [ ] >>> p r i n t ( s t . t t s t r i a n g l e . i n s t a n c e s [ 1 8 ] ) tts triangle object . ID : 18 V e r t i c e s : [9 , 3 , 1] S p a c e l i k e Edge : [ 9 ] T i m e l i k e Edge : [ 1 8 , 1 3 ] Neighbor t t s t r i a n g l e s : [17 , 11 , 13] Neighbor s s t t r i a n g l e s : [ ]

67

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