C O N D I T I O N A L P R O B A B I L I T I E S IN T H E Q U A N T U M C O S M O L O G Y OF P O N Z A N O - R E G G E T H E O R Y By Roman J. W. Petryk B. Sc., University of Manitoba, 1996 Certificate of Advanced Study in Mathematics, University of Cambridge, 1997 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E O F M A S T E R OF S C I E N C E in T H E FACULTY OF G R A D U A T E STUDIES D E P A R T M E N T OF PHYSICS A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA December 1999 © Roman J. W. Petryk, 1999 In presenting this thesis i n partial fulfillment of the requirements for an advanced degree at the University of B r i t i s h C o l u m b i a , I agree that the L i b r a r y shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and A s t r o n o m y T h e University of B r i t i s h C o l u m b i a 6224 A g r i c u l t u r a l R o a d Vancouver, B . C . , Canada V 6 T 1Z1 Date: Abstract We examine the discrete Ponzano-Regge formulation of (2+l)-dimensional gravity i n the context of a consistent histories approach to quantum cosmology. We consider 2dimensional boundaries of a 3-dimensional spacetime. T h e 2-dimensional boundaries are tessellated as the surface of a single tetrahedron. T w o classes of the tetrahedral tes- sellation are .considered—the completely isotropic tetrahedron and the two-parameter anisotropic tetrahedron. Using Ponzano-Regge wavefunctions, we calculate expectation values and uncertainties for the edge lengths of these tetrahedra. In doing so, we ob- serve finite size effects i n the expectation values and uncertainties when the calculations fail to constrain the space of histories accessible to the system. There is, however, an indication that the geometries of the tetrahedra (as quantified by the ratios of their edge lengths) freeze out w i t h increasing cutoff. Conversely, cutoff invariance is observed i n our calculations provided the space of histories is constrained by an appropriate fixing of the tetrahedral edge lengths. It is thus suggested that physically meaningful results regarding the early state of our universe can be obtained providing we formulate the problem in a careful manner. A few of the difficulties inherent in quantum cosmology are thereby addressed i n this study of an exactly calculable theory. ii Table of Contents Abstract ii Table of Contents iii List of Tables vi List of Figures vii Acknowledgements x 1 Introduction 1 1.1 Gentle Immersion—An Answer to the Question: "Why Should Anyone Study Ponzano-Regge Theory?" 1 1.2 A Brief Overview and Relevant Background 5 1.3 The Structure of the What Remains to be Said 5 2 Mathematical and Physical Preliminaries 2.1 2.2 7 The 6j-symbol 7 2.1.1 Defining and Representing the 6j-symbol 7 2.1.2 Quantum Gravity and the Semiclassical Limit of the 6j-symbols . The Ponzano-Regge Partition Function and Wavefunction 12 14 2.2.1 Tessellating Manifolds M 2.2.2 The Ponzano-Regge Partition Function 16 2.2.3 Ponzano-Regge Wavefunction (for Manifolds with Boundary) . . . 17 B and Their Boundaries dM B iii 14 3 The Isotropic Class of Tetrahedron 20 3.1 The Isotropic Wavefunction 20 3.2 Isotropic Expectation Values < lx > and Uncertainties Alx 23 3.3 Semiclassical Isotropic Analysis 31 3.3.1 Semiclassical Isotropic Expectation Values < Ix > sc and Uncer- tainties Alx c 31 S 4 Subclass A Two-parameter Anisotropic Calculations 4.1 37 Subclass A 38 4.1.1 The Subclass A Wavefunction 38 4.1.2 Subclass A Expectation Values < Ix > and Uncertainties Alx • • 46 4.1.3 Subclass A Expectation Values < l • • 47 4.1.4 Subclass A < l 4.1.5 Curvature Expectation Values < 0 > 4.1.6 Subclass A < l > and Uncertainties Al Y x Y > and Al Y along contours of l x and K Y > and Al 67 A l o n g Contours of l Y 55 x and K 80 5 Subclass E Two-parameter Anisotropic Calculations 5.1 5.2 90 The Subclass E Wavefunction 91 5.1.1 Subclass E Expectation Values < l 5.1.2 Subclass E Expectation Values < l 5.1.3 Subclass E < lx > and Alx along contours of l 5.1.4 Subclass E < l x Y > and Uncertainties Al x • • 94 > and Uncertainties Al Y . . 95 and K Y Y > and Al Y A l o n g Contours of l x Comparison of Subclass A and E Results and K 99 113 118 6 Conclusion 123 Bibliography 126 iv Appendix 128 v List o f Tables Two-parameter anisotropic subclasses vi L i s t of F i g u r e s 2.1 A representation of the 6j-symbol—right- and left-handed labelling of the tetrahedron t • 10 2.2 T h e tetrahedron and semiclassical 6j-symbol 13 2.3 Tessellating the boundary of M 15 2.4 C o m p o s i t i o n of manifolds M 3.1 Tessellation of the single completely isotropic boundary 22 3.2 Isotropic * and | * | as functions of K and X 24 3.3 Isotropic ^ and | ^ | as functions of X (A-dependence suppressed). 3.4 Isotropic < l\- > and Al 3.5 < l 3.6 C o m p a r i n g exact and semiclassical isotropic < l B B and N B w i t h common boundary dM B = dN . B 2 2 as functions of K x x > sc and Al Xsc . . . 19 25 28 as functions of K 33 x > and Al x otK : as functions 35 4.1 Tessellation of the single ellipsoidal anisotropic boundary 39 4.2 Subclass A * and | * | as functions of X and Y for K=20 42 4.3 Subclass A * and | * | as functions of K and X for y = 2 0 43 4.4 Subclass A \& and |vl>| as functions of K and Y for A = 2 0 44 4.5 Subclass A ^ as a function of X and Y for # = 2 0 . 45 4.6 Subclass A < lx > and Al 48 4.7 Subclass A 2 2 2 r x as a functions of K as a function of A" 49 > and Aly as functions of i f 51 <lx> 4.8 Subclass A <l Y vii 4.9 Subclass A - g * - as a function of K 4.10 Subclass A 52 as a function of K 53 4.11 Subclass A %*- as a function of K 54 4.12 < l 58 > \i as a function of K for contours of fixed ly. . . x 4.13 Alx\i as a function of K for contours of fixed ly 59 4.14 as a function of K for Zy=100.5 60 Y 4.15 < lx >\K as a function of Zy 63 4.16 Al \ as a function of ly 64 4.17 Al \ as a function of ly 66 4.18 - ^ ^ f - as a function of ly for AT=100 68 4.19 < / x > and Al 69 x K x K as functions of ly and x 4.20 Angles at the vertices of the tetrahedron 70 4.21 < 9 y > \ K as a function of ly 72 4.22 < Q y > \ K as a function of l 73 Y X Y 4.23 < Oyy > and < 9 y > as functions of l X 4.24 A9 \ YY K 4.25 A6 y\ X K y as a function of ly 77 K 4.28 < Qyy >max\K 4.29 < >max\K 4.30 <l Y and < Q y >\, a a n < n ^ > ^ < 6AT K 79 functions of ly for K=100 as functions of l Y 78 for 7C=100 81 82 84 ^ ^ ' " as a function of 7T 85 as functions of lx 86 A 4.33 Relative uncertainty <l*>\ 4.34 < ly > \ >mmlic S for 7^=100 as functions of K < Y K a rnin\K < and Al \ functions of ly for K=100 s Y and A / y l , K a K as functions of l K 4.31 Relative uncertainty 4.32 < ly > \ > \ X X QXY 74 76 | ^ and AQ y\ y and K as a function of ly 4.26 < Qyy > \ 4.27 A 0 Y a s a m n c ti° and A / y | , as functions of l A x viii n °f lx and K 88 89 5.1 Tessellation of the single anisotropic ovoidal boundary 5.2 Subclass E and A two-parameter anisotropic 6j-symbols as functions of X and Y 92 93 5.3 Subclass E < l > and Alx as a functions of i f 96 5.4 Subclass E as a function oi K 97 5.5 Subclass E < l > and Al 98 5.6 Subclass E x Y as functions of K Y < J y > < « y > as a function oi K 100 as a function of i f 101 5.7 Subclass E 5.8 Subclass E 5.9 < lx > \i as a function of i f for contours of fixed l as a function of i f 102 Y 5.10 Al \i x a s a Y 5.11 function of i f for contours of fixed l Y as a function of i f for Zy=100.5 104 105 106 <lX>\ly 5.12 < lx >\K as a function of l 108 5.13 Alx\ as a function of l 109 5.14 as a function of l Y K Y for i f = 1 0 0 Y 5.15 < l x > and Alx as functions of l Y > I, 5.16 < l 1 Y and Al \, Y Y 5.19 Al \ Y K > \ K as functions of i f 114 ^'| 115 H < / y and as a function of lx 116 as a function oi lx j^ Jf " < Y>\ Y as a function of i f X 5.20 Relative uncertaintv 5.21 < l > \ K 112 x 'lx 5.17 Relative uncertainty 5.18 < l and i f Ill and Al \ Y K 117 fy as a function of lx 119 as functions of lx and K 120 l K ix Acknowledgements I send out my deepest thanks to everyone without whose aid, patience, and caring I could not have achieved a l l that I d i d . In particular, I thank my thesis advisor K r i s t i n Schleich for a l l her assistance, helpful criticism, and insightful commentary. W i t h o u t her editorial critique of its many drafts, this thesis would have been far less than what it has become. O u r many hours of conversation were b o t h pleasant and enlightening. I also thank her for introducing me to some of the computational techniques useful to the study physical systems (Dennis Ritchie would be proud of the progress I've made). I also thank D o n W i t t and M a t t h e w C h o p t u i k for their assistance and advice on many issues, particularly computer hardware and programming. A s well, I owe much gratitude to G o r d o n Semenoff for reading and critiquing this thesis (especially on such short notice). Likewise, I pass on my very warm thanks to friends, colleagues, and staff at the Department of Physics and A s t r o n o m y of the University of B r i t i s h C o l u m b i a . It was very pleasant interacting and working w i t h all of them. A n d not least, I owe much to my loving wife and family. I thank my wife A n d r e a for her friendship, encouragement, patience, sacrificed hours of sleep, and editorial advice and commentary (we practically wrote this thesis together). There is a good chance I would not have survived this challenge without her. Finally, I thank my mother M a r y (and M i h a j l o ) , father B i l l , brother Michael (also known as C . G . ) , and grandmother M a r t h a for a l l their support. Chapter 1 Introduction I been fightin' gravity since I was t w o . © Les Claypool [1] 1.1 Gentle Immersion—An Answer to the Question: "Why Should Anyone Study Ponzano-Regge Theory?" To answer the posed question, we need to consider the meaning of physical law as it pertains to the study of physical cosmology. In [2] Hartle presents one of the most elegant discussions on this subject to appear i n electronic print. Let us summarize the ideas presented in his introduction: • a physical law is something which explains and predicts the properties of all physical systems "without exception, without qualification, without approximation"; • we have recently begun to expand our understanding of physical law—where we once understood a physical law to be one explaining the dynamics of the universe , 1 we now understand that physical law must be allowed to include theories of the initial conditions of the universe ; 2 • if we follow the current trends i n physics, we would conclude that the present state The Schrodinger wave equation and classical Einstein equation are two such dynamical physical laws. We have the recent slew of cosmological observations (such as cosmic microwave background isotropy and its likes) to thank for spurring on movement towards this modern usage. 1 2 1 2 Chapter 1. Introduction of the universe should be explicated i n terms of both its i n i t i a l quantum mechanical state, and its quantum mechanical evolution; • thus, physical law has been broadened to allow inclusion of the quantum mechanics of the universe as a whole; • quantum cosmology is the name we now apply to the field of study attempting to formulate laws predicting and explaining the initial quantum state of the physical universe, as well as its evolution. Clearly, this broader notion of physical law should encapsulate both the older notions governing the classical dynamics of our observed universe, as well as the newer notions introduced by quantum cosmology. So we now understand that physical law means something more than it used to, and that quantum cosmology is the field which studies the initial state of the universe. T h i s thesis is, however, not a study of etymology and the semantics of modern science. Let us address the larger issue at hand—how to obtain a classical spacetime from the quantum wavefunction which describes the universe as a whole. T h i s is one of the central problems of quantum cosmology. It has been suggested (cf. [2]) that the resolution of this quantum to classical cosmological problem w i l l likely involve answers to three questions: 1) W h a t is the true theory of gravitational dynamics? 2) W h a t are the initial conditions for the wavefunction of the universe? 3) How does one extract classical or quasi-classical behaviour from a quantum mechanical system without external observers—in fact, is it even meaningful to pose 3 T h i s problem is in some sense tantamount to the philosophical question of whether one's own mind is capable of understanding itself. 3 Chapter 1. Introduction 3 this question? A particularly insightful discussion i n response to the t h i r d question is to be found i n [3]. Furthermore, Hartle [4], G e l l - M a n n [5], Griffiths [6] and Ormnes [7] have proposed an interpretation of quantum mechanics applicable to closed systems without external observers. T h i s interpretation is based i n the consistent histories approach to quantum mechanics. The consistent histories (or decoherent histories) approach can be understood as follows. A history consists of a sequence of measurements of quantum observables. Such a sequence can be formulated as a series of quantum mechanical projection operators acting on the initial quantum state of the system. Each such sequence or history is associated with the probability of the resulting state. 4 If the overlap i n probabilities of the set of final states is small, they are said to be decoherent or consistent. However, not a l l sets of histories w i l l decohere—whether they do or don't w i l l depend on the initial state of the system, the evolution of the system, and the observables being measured along each history. If they do decohere they are said to behave i n a classical manner. 3 T h e consistent histories approach thereby provides a solution to the third question of our list (i.e., how classical behaviour arises w i t h i n a closed quantum mechanical system), allowing us to focus on the solution to the problems posed by the first two questions (i.e., finding the true theory of gravitational dynamics arid determining the initial state conditions for our universe). Now, the consistent histories approach is has been successfully applied to the study of quantum systems as it relates to gravity (cf. [8], [9], [10], [11]). Most work of this This probability corresponds to the modulus squared of the resulting state amplitude. Remember that if they decohere, their final state amplitudes have very little overlap. If this is so, then the final states can be measured in the classical sense. That is, the elements of the set of final states no longer quantum mechanically interfere the same way that do, say, the momentum and position of an elementary particle. 4 5 Chapter 1. Introduction 4 manner was, however, performed on the simplest of models and assuming high degrees of symmetry. It would certainly be interesting to apply such interpretations and techniques to the study of gravity itself while relaxing the constraints of symmetry. Ponzano-Regge theory is one point from which we can begin to address the issues involved i n the quantum to classical cosmological problem i n 3-dimensional gravity. Ultimately, our goal is to formulate 4-dimensional gravity. However, many issues i n defining the quantities of the theory are not resolved. We are not even sure of which theory of gravitational dynamics to use. T h e Ponzano-Regge theory of gravity is 3dimensional not 4-dimensional. However, unlike 4-dimensional gravity, Ponzano-Regge is a completely specified theory. T h e issues of the formulation of i n i t i a l conditions and dynamics are thus solved. Furthermore, the theory provides additional degrees of freedom over those of the simple models studied before us (cf. [8], [9], [10], [11])- Ponzano-Regge theory may thus provide a useful testing ground for the formulation and interpretation of the consistent histories approach to quantum mechanics. T h i s thesis will study these issues. Firstly, however, we note that Ponzano-Regge gravity is formulated i n terms of the l i m i t of a cutoff theory—this distinguishes it from the T u r a e v - V i r o formulation of 3manifold invariants [14]. Clearly, then, one must try to understand the nature of this l i m i t i n the computation of quantum amplitudes. We w i l l attempt to do so by asking the question: "How does the cutoff affect quantities such as expectation values and conditional probabilities computed to implement the consistent histories approach? In particular, are all quantities cutoff dependent, or can one find quantities that are cutoff independent?" Chapter 1. 1.2 Introduction 5 A B r i e f Overview and Relevant Background In 1968 Ponzano and Regge[12] noted a connection between the 6j-symbol relating spins and the 3-dimensional Regge action (cf. [13]). They also formulated the partition function for calculating amplitudes on 3-manifolds without boundary (cf. [12]), thereby providing us with a well defined, exact, calculable theory of 3-dimensional quantum gravity based on the sum over histories approach to quantum mechanics. 6 Many studies of the Ponzano-Regge theory have taken place since its introduction. Specifically, in 1991 Turaev and Viro [14] defined a quantum 6j-symbol and developed a topological quantum field theory analogue to the Ponzano-Regge theory for 3-manifolds. In 1992 Ooguri [15] demonstrated that the Ponzano-Regge partition function is equivalent to Witten's 2+1 formulation of gravity on closed orientable manifolds. Also in 1992, Iwasaki [16] showed how the Ponzano-Regge partition function can be written as a sum over surfaces of the 3-dimensional spacetime and attaches a geometrical meaning to the regulated divergences of the theory. As well, in 1997 Barrett and Crane [17] demonstrated that the Ponzano-Regge partition function satisfies a discrete version of the Wheelerde Witt equation, thereby giving further credibility to the notion that Ponzano-Regge theory is truly a theory of gravity. These listings are, however, merely an introduction to the complete body of papers in print dealing with the various aspects of Ponzano-Regge theory. 1.3 T h e S t r u c t u r e of the W h a t R e m a i n s to be Said In Chapter 2 we begin this study with the definition of the 6j-symbol and go on to state the Ponzano-Regge partition function (wavefunction on a manifold without boundary), Their study also revealed a tessellation independent property of the system. The Ponzano-Regge theory is, however, not rigorously a topological field theory. 6 Chapter 1. Z[M], Introduction 6 for (2+1)-dimensional gravity, and provide a definition of the Ponzano-Regge wavefunction, \ & [ M B , { J J } ] for (2+l)-dimensional manifolds with boundary. In C h a p - ter 3 we evaluate the wavefunction on the simplest 2-dimensional completely isotropic boundary tessellation—a single, completely isotropic tetrahedron. There we also find cutoff dependence i n two measured quantities: the expectation value of tetrahedral edge lengths < lx > and the uncertainty of the tetrahedral edge lengths Al X Chapter 2 we also evaluate the expectation values and uncertainties i n the semiclassical limit and compare the results to those using the exact 6j-symbol. In Chapters 4 and 5 we evaluate the wavefunction for two of the simplest cases of anisotropic boundary tessellation—the single, two-parameter anisotropic tetrahedra. There we find that the tetrahedral edge lengths expectation values < lx > and uncertainties Al x are also cutoff dependent, but that cutoff invariance can be observed provided the conditional probability amplitude constrains the number of allowed geometries entering the calculations. Chapter 6 contains a summary of our results, and in Chapter 7 we find the C programming language code of the function developed for this thesis to evaluate the 6j-symbol. Every effort has been made to present the relevant material i n as lucid yet succinct a manner as possible. To this end, equations have been presented using the most appropriate notation. A l t h o u g h the notational conventions of no particular work referenced in the study w i l l be strictly adhered to, the reader w i l l find our presentation most similar to those of Ponzano and Regge [12], Iwasaki [16], Ooguri [15], and to a lesser extent that of Turaev and V i r o [14]. Chapter 2 Mathematical and Physical Preliminaries Of course, we must first address the mathematical and physical formalism associated with the theory. A n d so we begin... 2.1 2.1.1 The 6j-symbol Defining and Representing the 6j-symbol Let ji, J2) h i Hi J5i J6 be non-negative integers or half integers. A n unordered 3-tuple of this set of jf-values is written as (ja,jb,jc) where a,b,c = 1,2, ...,6 and a^b^c. (2.1) The unordered 3-tuple (2.1) is then said to be admissible if the triangular inequalities life - J c l < 3 a < (2.2) jb+jc are met and the sum over j-values 3a + 3b + (2.3) 3c is an integer. Furthermore, an ordered 6-tuple expressed as (2.4) is said to be admissible if all the unordered 3-tuples h,32,h), {.33,3A, 35), {kiki3i)i 7 { h , k , k ) (2.5) Chapter 2. Mathematical and Physical Preliminaries 8 are admissible. We are automatically guaranteed integer values for the sums « 5 = fa + fa + fa + fa, 6 32 = n + fa + 35 + fa, ™7 = fa + fa + fa + (2.6) fa if the four 3-tuples of equation (2.5) are admissible. A d m i s s i b i l i t y furthermore guarantees that n <n , g 0 = 1,2,3,4, h h = 5,6,7, (2.7) where n are given by g ni = fa + fa + fa n = fa + fa + fa, n = fa + fa + fa, n = fa + fa + 2 4 3 fa, (2.8) and n-h are given by equation (2.6). We can now associate a 6j-symbol f fa fa fa ) I fa 35 (2-9) J fa of SU (2) with the ordered 6-tuple of equation (2.4). For admissible 6-tuples (j\, fa, fa, fa, fa, fa) the 6j-symbol is explicitly given (cf. [12], [14], [18]) as fa fa fa ) _ \=AY B{z) v (2.10) J 34 fa fa ) where A = [A(fa,fa,fa)A(fa,fa,fa)A{fa,fa,fa)A(j ,fa,fa)}K (2.11) 2 B{z) = ( - i r ( z + l ) ! [ ( z - n ) ! ( ^ - n ) ! ^ - n ) ! ( z - n 4 ) ! ( n 5 - z ) ! ( n - ^ ! ( n - ) ! ] - , 1 1 2 3 6 7 2 (2.12) and the sum is over a l l non-negative integer values of z resulting i n non-negative factorial arguments. A(ji,jj,jk) is here denned according to /- • A ( *''"^ A •\ (fa + 3j ~ Jk)KJi + fa ~ jjV-tij + 3k ~ = ^T^Ti)! , (2 . - 13) Chapter 2. Mathematical and Physical Preliminaries 9 (2.14) for inadmissible (ji.32,J3,34,35,3e)Many detailed properties and symmetries of the 6j-symbol are discussed in [12], [14], and [18]. We will especially make use of (2.15) in discussions that follow. There is a natural geometric representation for the 6j-symbol—it is the standard 3dimensional tetrahedron t (see Figure 2.1). If we consider a 3-dimensional tetrahedron t we will observe that it has four vertices, six edges, and four triangular faces. There is thus a 1-to-l correspondence between the number of edges of t and the number of arguments of the 6j-symbol. In particular, we may label the edges of t as follows. Choose a face on t, and then choose a particular edge in that face. Label that edge with the first j-value—j\. Then respectively label the other two edges in that face j 2 which shares no vertices with ji and label it j\. vertices with j 2 and j . Next find the edge 3 Then find the edges which share no and jz and respectively label them j 5 and j . 6 One could at this point ask which edge to label j and which edge to label j once we have chosen j\—there are 2 3 clearly two distinct choices for this procedure. Figure 2.1 shows both choices for the case where the base of t is chosen as the starting point of our labelling. We can (somewhat arbitrarily) choose to call the two labellings right- and left-handed. However, whether we chose right- handed or left-handed labelling for a given tetrahedron makes no difference to any calculations since the symmetry given by equation (2.15) states that these two labellings are functionally equivalent. 1 *It is for this reason that we don't concern ourselves with the handedness of our labellings in the analysis which follows. Chapter 2. Mathematical and Physical Preliminaries 10 Figure 2.1: A representation of the 6j-symbol—right- and left-handed labelling of the tetrahedron t. Chapter 2. Mathematical and Physical Preliminaries 11 A l t h o u g h all diagrams w i l l show the edges of tetrahedra labelled i n terms of the jvalues, it is important to note that the lengths of these edges associated w i t h the assigned j-values are given b y 2 k=3i + \, i = l,2,...,6. (2.16) It is useful to study the implications of the triangular inequalities i n relation to t and its edge lengths. E x a m i n i n g the geometry of t, we see that equation (2.2) simply guarantees that the edges li, l , Is form a closed triangle of non-zero surface area (i.e., if \ji — j \ 2 2 < 33 5- ji + 3 2 then |/i — l \ < l < l\ 4-1 ). Similarly, the triangular inequalities of 3-tuples 2 3 2 (J3, ji, 3 s ) , ( J 5 , 3 6 , 3 i ) , and (j ,J4,je) 2 respectively guarantee the remaining three faces form triangles of non-zero surface area. T h a t is, admissibility guarantees that the edges ji, 3 2 , •••) je form a closed tetrahedron of non-zero volume V. T h e triangular'inequalities do not, however, guarantee that the associated tetrahedron has real positive volume (see Subsection 2.1.2 and Chapter 4). W i t h o u t additional restriction, it is possible to construct hyperflat (V 2 < 0) tetrahedra. T h i s occurs when the sum of angles between the three edges forming a vertex is greater than 2-7T. Clearly, the V 2 be embedded in a 3-dimensional Euclidean space, while the V 2 There is, however, an interpretation by which the V 2 > 0 tetrahedra can < 0 tetrahedra cannot. < 0 tetrahedra can be thought of as being embeddable in a 3-dimensional Lorentzian space (cf. [19], [20]). We will therefore respectively refer to the V 2 > 0 and V 2 < 0 tetrahedra as Euclidean and Lorentzian. Occasionally, we w i l l i n this sense refer to the Euclidean or Lorentzian regimes of the tetrahedra. The length of the edge is chosen to be U = ji + \ because in this case U approaches the length of the angular momentum vector yjiJji + V) in the semiclassical limit ji » 1. 2 12 Chapter 2. Mathematical and Physical Preliminaries Quantum Gravity and the Semiclassical Limit of the 6j-symbols 2.1.2 There is a much deeper connection than the 1-to-l correspondence between the edges in t and the arguments of the 6j-symbol. It is revealed i n the semiclassical form of the 6j-symbol. Consider the case where a particular tetrahedron i n the given triangulation has edge lengths given by equations (2.1.1), as displayed i n Figure 2.2. For sufficiently large ji, j, 2 j , Ponzano and Regge [12] demonstrate the associated 6j-symbol is approximated 6 by 3\ 32 33 34 3b J6 7 = - (go, + \)0, , s :7T (2.17) c where 0$ is the interior angle between the outward normals of the two tetrahedral faces sharing the lf h edge, and where the square of the tetrahedral volume V is given by (cf. [12], [21]) 0 h V 2 / 2 4 l l i 1 l 2 2 1 li 1 l7l i 2 0 1 2 5 0 Z ii ii 0 li l2 i 2 1 b 2 3 l 11 1 1 1 (2.18) 0 In order to obtain a meaningful (non-imaginary) result from equation (2.17) the tetrahedral volume V must be real. The region of validity for approximation (2.17) is thus restricted to values of Ik (k = 1,2, ...,6) for which equation (2.18) yields positive values for V . 2 We now note that the Regge action of the tetrahedron i n Figure 2.2 is given by [13] SRegge = '^J-i&i i=l — ^2Ui i=l + ^)^i1 (2.19) Chapter 2. Mathematical and Physical Preliminaries Figure 2.2: The tetrahedron and semiclassical 6j-symbol. 13 Chapter 2. Mathematical and Physical Preliminaries 14 —that is, the gravitational contribution at each edge of the tetrahedron is liQ{ = (ji + For a complex of tetrahedra w i t h n internal edges SR egge S e = Reg9 J2Ui i=x \)0i. would be written + lWi. (2.20) z The semiclassical approximation (2.17) is therefore equal to the cosine of the Regge action for a single tetrahedron up to a constant factor and a phase shift (cf. [12]). We also note that Ponzano and Regge [12] have provided valid equations (derived v i a the W K B method) for the semiclassical approximation of the 6 j - s y m b o l in the Lorentzian regime (V 2 < 0). T h e Ponzano-Regge P a r t i t i o n F u n c t i o n and Wavefunction 2.2 2.2.1 Tessellating Manifolds M B and T h e i r Boundaries dM . B Before we begin discussion of Ponzano-Regge partition functions and wavefunctions, it is crucial to understand that these theories are formulated i n terms of discrete tessellations of 3-manifolds and their boundaries. For this purpose, consider maps T which tessellate 2-manifolds or 3-manifolds as a mesh of tetrahedra. T h i s tessellation is not unique—different T can map the manifold to a single tetrahedron or to many tetrahedra. A s well, the connectivity of tetrahedral vertices is not in general unique. T h a t is, we can choose the appropriate T to map the manifold as crudely or intricately as we desire, and the connectivity of the many tetrahedral vertices w i l l also depend on the particular choice of T. Since we w i l l be primarily concerned w i t h Ponzano-Regge wavefunctions, consider the tessellation of the 2-manifold boundary dM B as shown i n Figure 2.3. The figure displays two distinct mappings of the closed manifold boundary 8M . B Ti is a mapping which leads to the crudest tessellation is mapped to a single tetrahedron t w i t h edges labelled ji, j , 2 je- possible—8M The map B T 2 15 Chapter 2. Mathematical and Physical Preliminaries results i n a much finer (and more precise) tetrahedral modeling of 8MB- W h i c h mapping T we choose w i l l depend on how closely we want to approximate the manifold w i t h a tetrahedral m e s h . 3 Figure 2.3: Tessellating the boundary of M . B It is clear that the finer we make the mesh, the greater the number of tetrahedral edges we will have to include in our formulation. Since each of the edges must be associated with the appropriate 6,7-symbol, a very fine mesh will greatly increase the number of 6j-symbols involved. This will, in turn, greatly increase the degree of difficulty involved in the associated calculations. Computational power will thereby limit the type of problem we will be able to solve. 3 Chapter 2. Mathematical and Physical Preliminaries 2.2.2 16 The Ponzano-Regge Partition Function Now consider a closed 3-manifold M. Let T[M] be a (tetrahedral) tessellation of the manifold M with fixed connectivity and number of tetrahedra. Denote the sets of vertices, edges, faces and tetrahedra in T as So, S i , S and S 3 , respectively. Also, let s 2 number of objects contained in set S . m be the (For example, SQ is the number of vertices in m the set So of T[M], and si is the number of edges in the set S i of T[M].) Furthermore, let K (the cutoff) be a non-negative integer or half-integer, and let 4> be an admissible assignment of a non-negative integer or half-integer j\ < K to the i th edge in S i . The assignment <f> is said to be admissible if all the 6-tuples given by the assignment are admissible. Then: Definition 1. The Ponzano-Regge partition function for a manifold M (without boundary) is (cf. [12]) Z[M] = lim E A - ° f t ( - l ) ^ ( 2 j + l) f[[tn] <h i=l n=l i (2.21) where [t ] is given by n [t ] = n (_l)-0"l+j2+i3+i4+j5+j ) 6 I Jl U4 H J5 ^\ (2.22) k where \ J4 is the 6j-symbol for the n th Jb (2-23) 36 tetrahedron, and where the divergence regulating term A is given by A= £ (2p + l ) 2 (2.24) Since non-admissible 6-tuples yield vanishing 6j-symbols we could just sum over all 6tuples instead of restricting ourselves to only admissible 6-tuples in Z[M]. Also notice Chapter 2. Mathematical and Physical Preliminaries that since l i m ^ - ^ o o A is 0(K ), 3 17 Z[M] w i l l decay rapidly i n K unless we eliminate A by normalizing the partition function. Regge [13] demonstrated that the sum of contributions to SR egge from a l l tetrahedra in a tessellation approaches a value proportional to the action of Einstein gravity, X ( M ) , provided the number of edges and vertices i n the tessellation becomes very large. T h a t is, (2.25) where R is the R i e m a n n curvature scalar of M and dV is the volume element on M. C o m p a r i n g (2.21) to (2.25) reveals that the partition function Z[M] can be interpreted as the path integral formulation of gravity on a lattice. Specifically, a given 6j-symbol is proportional to the path integral amplitude for the associated tetrahedron, so the product of of the 6j-symbols is equivalent the path integral amplitude for a given simplicial geometry. T h e Ponzano-Regge partition function thereby provides a precise (exact) formulation of 3-d gravity. 2.2.3 Ponzano-Regge Wavefunction (for Manifolds with B o u n d a r y ) We have just seen that the Ponzano-Regge partition function provides a means of performing calculations on a 3-dimensional manifold, so one may ask " W h y should we bother to consider Ponzano-Regge wavefunctions?". T h e reasons are obvious. Suppose we begin w i t h a (2+l)-dimensional spacetime manifold. If we perform a slicing in the time dimension on this manifold we w i l l end up w i t h a foliation of spacelike surfaces. These spacelike surfaces w i l l be 2-manifold boundaries of the original (2+l)-dimensional spacetime manifold. If we were to then evaluate physical quantities on a few of these spacelike slices we would gain knowledge of how the properties of the boundary evolve in time—a very useful result! Chapter 2. Mathematical and Physical Preliminaries For this purpose, we now consider a 3-manifold M w i t h a closed 2-manifold boundary B 8MB- 18 A s before, we tessellate the manifold and respectively denote the sets of vertices, edges, faces and tetrahedra o/f the boundary of T[M ] let s be the number of objects i n the set S . m m and S 3 . We again as So, Si, S B 2 B y analogy, T[8M ] B is specifically the tetrahedral tessellation of the manifold boundary 8MB w i t h fixed connectivity and number of tetrahedra. T[8M ] B set B m We denote the sets of vertices, edges, faces and tetrahedra in as B , Bi, B 0 and B , respectively. We define b 2 3 m to be the number of objects in (i.e., bo is the number of vertices i n the manifold boundary, and 61 is the number of edges i n the boundary). Now, we again let K (the cutoff) be a non-negative integer or half-integer. Furthermore, we let (j) integer or half-integer h+l, bi+2, D e a < K (i = 1, 2, Si) to the i th edge i n Si. n admissible assignment of a non-negative bi) to the i ih edge i n B , x and ji < K (i — (Notice that we begin our labelling of edges at the boundary w i t h i = 1 and work our way to the interior once a l l boundary edges are labelled.) We now state the proposed definition for the wavefunction of a manifold with boundary: Definition 2. The Ponzano-Regge wavefunction for a manifold MB with boundary 8M B is given by *[M ,{j }]=^m 5:A-^)n(-i) (f> i=l B l j i o (2j +i)^ l n (- ) i=bi+l i 2 j ! (2j,+i) n n=l (2.26) wnere A and [t ] are defined as in (2.21) trough (2.24). n T h i s definition of ^ [ M , {Ji}] is chosen i n order to satisfy the composition law B Z[M#N] = lim Yl *K[M , b {Ji}]*K[N , B {Ji}], (2.27) °°{{Ji}\Ji<K} where WK[M , B {Ji}] and ^K[NB, {Ji}] are respectively the wavefunctions on manifolds Chapter 2. Mathematical and Physical Preliminaries M B and N B T[dM ] B w i t h common boundary dM B = T[dN ] B = dN B 19 and identical boundary tessellation before the limit K —> oo is taken. T h e action of such a composition is represented i n Figure 2.4. Figure 2.4: C o m p o s i t i o n of manifolds M B and N B w i t h common boundary dM B = dN . B The wavefunction ^[M , {J,}] is the starting point of our study. We w i l l want to B begin by addressing the issue of how to find good physical characterizations of our spacetime. In order to perform meaningful calculations on our (2+l)-dimensional spacetime manifold, we w i l l need to search for quantities which are invariant i n K. If the quantities show dependence on K (otherwise known as finite size effects) they w i l l be physically meaningless due to the infinite limit i n the formulation of \&. The gauntlet has been cast... so let us engage ourselves. Chapter 3 The Isotropic Class of Tetrahedron Perhaps the simplest class of calculations one can perform w i t h the Ponzano-Regge wavefunction is to evaluate expectation values and uncertainties for a 3-manifold M B single, completely isotropic, spacelike, 2-manifold boundary dM B with a (a 2-sphere). We w i l l refer to this as the isotropic class of problem. A s discussed in Chapter 2, we want to find physically meaningful (i.e., cutoff independent) characterizations of our spacetime. We will thus perform the simplest calculations on this, our simplest of systems, i n order to gain general insight on the issues at hand. Perhaps then we w i l l be able to successfully address the problem for a far less t r i v i a l set of systems. 3.1 The The Isotropic Wavefunction single, completely isotropic boundary can be tessellated by the action of T as a single tetrahedron whose edges are all of identical length. T h a t is, the faces of this tetrahedron are a l l equilateral triangles. Figure 3.1 provides a representation of this isotropic boundary and its tessellation. Notice that all the edges and vertices of the tetrahedral tessellation are on the boundary. We have appropriately labelled the edges of the tessellation w i t h 6 J - v a l u e s — J i , J , J3, J4, J5 and J — a c c o r d i n g to the conventions 2 6 outlined i n Section 2.2.3. Since the tetrahedron is to be made completely isotropic we fix J = J = J = J = J = J = X. 1 2 3 A 20 5 6 (3.1) Chapter 3. The Isotropic Class of Tetrahedron 21 T h e n , according to equation (2.26), the non-normalized wavefunction ( * [ M , {X, X, X, X, X, X}} = l i m V £ B (2p + l ) 2 j V>=o,i,..,K iis X X X ( 2 X + 1) 2 1 3 X X X (3.2) where X = 0, | , 1 , X . We now note that the 6-tuple ( J i , J 2 , ^ 3 , <7i> J5, Je) associated w i t h the 6j-symbol is inadmissible unless X is integer for Ji = J 2 X. = J3 — J4 = J5 = Je = Since the 6j-symbol vanishes for inadmissible 6-tuples, we can effectively restrict our values of X to be integer without losing any information in subsequent calculations. T h a t is, it makes no difference whether we choose to include or exclude vanishing probability amplitudes |\&| from the pending expectation value calculations. 2 We could now investigate to see what cutoff invariant information is given by wavefunction \I> or its corresponding probability amplitude by considering behaviour for many fixed, increasing values of K. F r o m equation(3.2) it is clear that the first term of \I> is a rapidly decreasing function of K (it is 0(K~ )), 6 the t h i r d term is an oscillating function of X. the second term increases as X , and 3 Overall, \ f is therefore expected to behave as an increasing amplitude oscillating function of X along contours of fixed K, and as a rapidly decreasing function of K (asymptotically approaching 0) along contours of fixed X. E x p l i c i t calculation precisely exhibits these characteristics for the wavefunction for all ranges of K and X. Figure 3.2 displays * for the range 0 < K < 20, 0 < X < 20. The 2 probability amplitude | \ l / | exhibits similar behaviour (see Figure 3.2)—the exceptions 2 are that has twice the frequency of oscillation and it is a strictly positive function of K and X while ^ can take on both positive and negative values. In itself, this analysis F o r clarity, we have explicitly written the boundary J-values as arguments of the wavefunction. In future, the arguments will be excluded when writing the wavefunction for the sake of brevity. Strictly speaking, X is restricted to the range 0 < X < K. However, the figure shows values X > K for the purpose of demonstrating the behaviours of $ and | * | over extended (disallowed) ranges of X. Also note that many values of * and |\t| exceed the range of the plot—these regions appear as holes in the plotted surfaces. 1 2 2 2 22 Chapter 3. The Isotropic Class of Tetrahedron Je=X J1=X Figure 3.1: Tessellation of the single completely isotropic boundary. Chapter 3. The Isotropic Class of Tetrahedron 23 gives little information on the system under scrutiny. T h i s is why the expectation values of l x = X + | and corresponding uncertainties must be investigated. In order to isolate the dependence i n X we calculate \I/ and \^f\ without the renor2 malization factor A . T h e results for 0 < K < 20, 0 < X < 20 are shown i n Figure 3 3.3. We here note that since A is factored out by the process of normalization when calculating expectation values, et cetera, it is logical to consider the A-suppressed \& and |^/| 2 3.2 as well. Isotropic E x p e c t a t i o n Values < lx > a n d Uncertainties Alx Since a l l of its J-values are equal to X, the edge lengths of the isotropic tetrahedron are lx h = = h = ^3 = U= ^5 = ^6 = X + —. (3-3) The edge length expectation values w i l l therefore be given by . Ef Q + ^M [X,X,X,X,X,X}\ 2 =0 S^K X where the explicit form of ^M B hi, B f P ' ^ ' is given by equation (3.2) and the sum i n x runs over a l l integer values less than or equal to K. Note the term i n the denominator—this is the 4 normalization factor for the wavefunction. Since K must be allowed to take on a l l integer and half-integer values up to infinity, it is computationally impossible to evaluate equation (3.4) as it is written. For this reason, we must investigate the behaviour of < lx > through the evaluation of partial expectation values. We find the partial expectation value by fixing the value of K i n the calculation. B y computing this partial expectation value for many increasing fixed-# values, we w i l l be able to refine our fixed-# approximation to the exact Ponzano-Regge wavefunction. 3 A s above. 4 Rigorously speaking, we should sum x over all integer and half-integer values. However, as previously mentioned, the amplitude vanishes for half-integer x, so we are free to simplify the calculation by restricting the sum to integer x. er 3. The Isotropic Class of Tetrahedron 2 Figure 3.2: Isotropic ^ and \^\ as functions of K and X. Chapter 3. The Isotropic Class of Tetrahedron Chapter 3. The Isotropic Class of Tetrahedron 26 It is difficult to see how < lx > w i l l behave w i t h change i n K by mere inspection. We i 12 have already found that the probability amplitude |\&| is a rapidly oscillating function of X and a rapidly decreasing function of K. However, the probability amplitude appears not only i n the numerator, but also i n the normalization factor. A s well, the factor (x +1) is an increasing function of x which adds more weight to each consecutive amplitude in the sum of the numerator. One could guess that < l x > would oscillate or perhaps increase with K, but this would be very speculative. A back-of-the-envelope calculation can be performed i n the following manner. If we assume the 6j-symbol is constant when averaged over a sufficiently large range of X, we find <lx>~ K ~ 7 f&dx £ ~ K (3.5) K l 2 o in the limit of large K. T h a t is, < l x > w i l l likely exhibit linear dependence i n K. B y performing the required computer-assisted calculations, we find that < l x does i n fact exhibit linearity in K. > Figure 3.4 displays the results of calculation for 180 < K < 200. T h e solid line represents the central value of < l x > = (0.8003 ± 0 . 0 0 0 4 ) # + (0.75 ± 0 . 0 4 ) , (3.6) —the result of an unweighted linear least-squares fit to the generated 0 < K < 200 data. The goodness of fit is measured from the coefficient of determination (R ) which is found 2 to be R 2 = .99995, indicating a firm linear correlation and suggesting that the choice of fitting function was appropriate. 5 Additionally, least-squares fits to 2 n d and 3 r d order polynomials reveals that not only are the higher order coefficients at least four orders of magnitude smaller than those of linear order, but also that there is no significant T h e probability that the observed data with v = 199 degrees of freedom and R have come from an uncorrelated parent population is of the order 1 0 . 5 2 - 4 3 0 — .99995 could Chapter 3. The Isotropic Class of Tetrahedron improvement i n R —this 2 27 indicates that fits to polynomials of quadratic and higher order are unsuitable. It is thus revealed that the edge length expectation values of our isotropic tetrahedron are linearly dependent i n K. Since an increase i n K allows more 6j-values to enter the sum of the expectation value, a larger K-v&lue amounts to a larger number of allowed configurations for our system. T h e amplitude of these extra configurations w i l l i n general be non-vanishing. T h e expectation value will therefore increase unless we limit the number of configurations entering the calculation. There is a way of understanding this behaviour by simple physical analogy. Since (2+l)-dimensional spacetimes are flat we can consider the normalized radius expectation value < r > of a sphere i n flat 3-dimensional space. If we integrate, not over all space, but up to some m a x i m u m radius R we obtain R R J rdV J 4irr dr „ = °~ir- = = !*• 3 <r> R JdV J 4-nr dr 2 o o T h a t is, the expectation value < r > varies linearly i n the radius R. If we fail to fix this m a x i m u m radius, the scale of the sphere w i l l grow without bound as we integrate out to infinity. Unless we constrain the size of our structure i n some other way, its expected size w i l l be determined by the volume of space, so any measurements we make on its geometry w i l l lack invariance in R. Furthermore, simple geometric analysis reveals the m a x i m u m edge length l m a x isotropic tetrahedron we can contain inside a sphere of radius < r > is fs lmax = y g < r > . (3.8) (3.7) into (3.6) yields a value lmax — 1.2272. (3.9) Chapter 3. The Isotropic Class of Tetrahedron Chapter 3. The Isotropic Class of Tetrahedron 29 Interestingly, we notice that the ratio between l and R is of the same order of mag- max nitude as the ratio between < l > and K i n equation (3.6). x We now understand the relationship between < l x attention to the calculation of uncertainty > and K, so let us t u r n our Al . x The uncertainty of edge length expectation value is given by the root-mean-square deviation from the mean Al x = yj< (l x < l >Y > . x (3.10) Just as for < lx >, it is impossible to evaluate equation (3.10) as it is written due to the infinite l i m i t i n K. W e must again evaluate the partial form of the given function for a set of many increasing values of K. From the behaviour of < l x >, we would predict a strong correlation between Alx and K. We would also expect Alx to approximate an increasing function of K of the form Al x oc K. (3.11) Let us elaborate on why this is so. A g a i n assuming the 6j-symbol is constant when averaged over a sufficiently large range of X we perform a second back-of-the-envelope calculation to obtain K <i\>- 8 % - K\ J-gidx ^ o 1 2 (3.i2) J and therefore Al x in the l i m i t of large K. T h a t is, Al x ~ K (3.13) should exhibit linear dependence i n K. Performing the required computer-assisted computations, we find that the relationship between the uncertainty and cutoff is of the expected form (3.11). T h e result of an Chapter 3. The Isotropic Class of Tetrahedron 30 unweighted linear least-squares fit to the generated Al values for 0 < K < 200 is x Al x = (0.16356 ± 0.00008)# + (0.12 ± 0.01). (3.14) The goodness of fit is again measured from the coefficient of determination, which is found to be R = 0.99995, indicating a firm linear correlation for the 0 < K < 200 data 2 set. Least-squares fits to 2 n d and 3 r d order polynomials can also be performed. T h e higher order coefficients are again found to be at least four orders of magnitude smaller than those of linear order, and there is no significant improvement i n R 2 for the higher order fits. T h e linear fit is thus found to be appropriate. We again return to our simple physical analogy of a sphere i n flat 3-dimensional space. Integrating up to the m a x i m u m radius R we obtain R R JYW J47rr dr < r * > = ° — l = fdV °- = 'fl?, R (3.15) fiirr dr 2 o o for the normalized expectation value of r . T h e n (3.7) and (3.15) yield the result 2 Jlo - A r = - R (3 16) T h a t is, the uncertainty A r varies linearly i n R. A g a i n , failure to fix the m a x i m u m radius in effect allows a greater volume of integration to determine the scale of measurements on the system. T h e uncertainty of the m a x i m u m edge length tetrahedron contained w i t h i n the sphere of radius < r > is therefore Al max = ^ A r ~ 0.316i?. We also observe that the ratio between Al max (3.17) and R i n our analogous calculation (3.17) is of the same order of magnitude as the ratio between Al x and K i n equation (3.14). In this sense, the results for the isotropic tetrahedron fit well w i t h interpreting ^ as the distribution of topologically spherical structures i n flat space. Chapter 3. The Isotropic Class of Tetrahedron 3.3 31 Semiclassical Isotropic Analysis Now that we have determined the behaviours of < l > and Al x we can ask how well x the semiclassical versions of the functions reproduce the data. After a l l , the evaluation of equation (2.17) should be much quicker and less intensive than equation (2.10) w i t h 6 its many factorials—so why not use it? 3.3.1 Semiclassical Isotropic Expectation Values < l > x sc and Uncertainties Analysis identical to that of the Section 3.2 can be performed on the isotropic tetrahedron using the semiclassical 6j-symbol approximation (2.17). T h e semiclassical isotropic wavefunction and expectation values are respectively given by V [M , {X, X, X, X, X, X}} = l i m SC / £ B (2p+l) 2 K—>oo [P=O,I,...,K V 2 (2X + 1) , (XXX 3 ) (XXX (3.18) and Ex=0 ( + \)\^sc,M \ i i i i i x < X >sc ~ l ^ x] x x x x x ,, , K ^2x=0 r T \^SC,MB B ii X I X X I X 1.2 I X I > I" " ) 3 19 X where equation (2.17) gives 2 i ( c o ( 6 ( 7 r - a r c c o s ( i ) ) ( X + i ) + f)) X X X ) X X 5 X) sc 0 ^ ( 2 ^ + 1)2 and we choose the principal value of arccos(|). (3.20) Since the 6 j - s y m b o l should vanish for non-integer x, the sum i n x again runs over a l l integers up to or equal to K. Investigation of < l x exact values < l x 3 > sc reveals that it exhibits linear dependence i n K just as do the > (see Figure 3.5). T h e result of an unweighted linear least-squares fit T h i s will be especially important in more complex classes of calculation. Chapter 3. The Isotropic Class of Tetrahedron 32 to the 0 < K < 200 data is found to be identical to equation (3.6) w i t h i n uncertainties. T h e coefficient of determination is again found to be R 2 = .99995 for the 0 < K < 200 data, and fitting to quadratic and cubic polynomials again demonstrates the unsuitability of non-linear descriptions. T h e degree to which the semiclassically approximated < lx >sc estimates the exact < lx > is further studied from the difference >sc <lx>~<lx at given K. (3.21) For the range 0 < K < 200 it is generally found that < l x > — < lx >sc becomes smaller w i t h increasing K (see Figure 3.6), indicating that the semiclassical approximation improves w i t h increasing cutoff. O n its own, this result is not unexpected since Ponzano and Regge [12] showed that the semiclassical 6j-symbol (2.17) is a good approximation for sufficiently large j k , and (on average) improves w i t h increasing ] \ . The apparent three-fold periodicity of the difference is, however, unanticipated. Considering the difference for every t h i r d value of K reveals that, at least from values K ~ 100 to K = 200, the quantity behaves as a damped sinusoidally oscillating function. Unweighted least-squares fits reveal that the three-fold periodic pattern for 99 < K < 200 is well approximated 7 by the function < lx > - < lx > sc = ° ^ l C 6 ( l + c cos(c 7s: + c )), 2 3 4 (3.22) where C! = 4.240, c = - 0 . 9 4 8 3 , c = 0.1104, c = 0.6983, c = 106.3, c = 1.639, 2 3 4 5 6 (3.23) for K = 99,102,105,...,198, Ci = 885.5, c = 0.9539, c = - 0 . 1 1 0 3 , c = 0.3442, c = 210.7, c = 2.451, 2 7 3 4 5 6 (3.24) Since the purpose of finding a functional form was simply to allow order-of-magnitude estimates of < lx > - < lx >sc for a higher range of K, the goodness of approximation was determined by visual inspection. Uncertainties in the fit parameters are therefore omitted. Chapter 3. The Isotropic Class of Tetrahedron Chapter 3. The Isotropic Class of Tetrahedron 34 for K = 1 0 0 , 1 0 3 , 1 0 6 , 1 9 9 and ci = 0.02794, c = 0.9528, c = 0.1106, c = 1.717, c = - 2 1 . 9 8 , c = 0.8400, 2 for K - 3 4 5 6 (3.25) 101,104,107, ...,200. T h e utility of equation (3.22) predominantly lies i n its ability to provide an order-of-magnitude estimate of < lx > — < lx >sc for 200 < K < 300. Similarly, the study of Al Xsc reveals a linear relationship i n K (see Figure 3.5). T h e result of a linear least-squares fit to Al values for 0 < K < 200 is found to be the Xsc same as that of equation (3.14) within uncertainties. T h e coefficient of determination is again found to be R 2 = .99995 for the 0 < K < 200 data, and fitting to quadratic and cubic polynomials once again demonstrates the unsuitability of non-linear descriptions. The degree of agreement between the semiclassical Alx and exact Alx is further sc studied from the difference A / x - Al (3.26) Xsc at specified K. For the range 0 < K < 200 it is generally found that the difference becomes smaller w i t h increasing K (see Figure 3.6). A s w i t h the difference < l x > — < lx > s this result is not unexpected i n light of [12]. W h a t is again unanticipated is the threefold periodicity of the difference. Considering Alx — Alx f ° every t h i r d value of K r sc reveals that, from values K ~ 100 to K = 200, the quantity behaves as a damped sinusoidally oscillating function. Unweighted least-squares fits reveal that the three-fold periodic pattern for 99 < K < 200 is reasonably approximated Al -Alxsc= x ( v , 0 7 , C l 2 [is. + Cn) 8 by the function ( l + c c o s ( c K + c )), 8 9 10 (3.27) A s in the case of equation (3.22), (3.27) was fit to Alx — Alx c to provide a means of estimating the order-of-magnitude difference for larger ranges K. The goodness of fit was again determined by visual inspection. Uncertainties in the fit parameter are therefore omitted. 8 S Chapter 3. The Isotropic Class of Tetrahedron 35 0.03 0.02 0.01 A X V A v X' -0.01 -0.02 -0.0025 -0.005 -0.0075 X < -0.01 -0.0125 -0.015 100 120 140 160 180 -0.0175 50 150 100 200 200 K Figure 3.6: C o m p a r i n g exact and semiclassical isotropic < l K. x > and Al x as functions of Chapter 3. The Isotropic Class of Tetrahedron 36 where c = -0.05588, c = 0.1790, c = 0.1091, c 7 8 9 = 0.9668, c w n = -7.141, c 12 = 0.9474, (3.28) for K = 99,102,105, ...,198, c = - 0 . 3 8 6 3 , c = 0.1803, c = 0.1114, c 7 8 9 = 2.693, c w n = 40.69, c 12 = 1.263, (3.29) = 1.724, (3.30) for K = 1 0 0 , 1 0 3 , 1 0 6 , 1 9 9 and c = - 7 . 4 6 8 , c = -0.1853, c = 0.1089, c i = 2.050, c 7 8 for K = 101,104,107, ...,200. 9 0 n = 110.9, c l2 A g a i n , the utility of (3.27) is its ability to provide an order-of-magnitude estimate of Alx — Al f ° 200 < K < 300. r Xsc Chapter 4 Subclass A Two-parameter Anisotropic Calculations In Chapter 2 we saw that the edge length expectation values and uncertainties of the completely isotropic, single tetrahedral tessellation are highly cutoff dependent. We interpreted this effect i n terms of a natural description of the distribution of spherical objects in flat space—essentially, the infinite l i m i t i n the cutoff increases the configurations accessible to the system. We have thus discovered that finite size effects w i l l be inevitable in the absence of a scale by which to compute physical quantities. We therefore need to find a means of isolating such a physical scale from the scale of discretization (as determined by the cutoff). Since the dynamics of Ponzano-Regge theory doesn't yield a physical scale on its own, we need to introduce one ourselves. We will do so by fixing tetrahedral edge lengths, and then proceed w i t h the analysis by evaluating conditional probabilities for our system. T h a t is, we w i l l ask: "Given some fixed edge length A, are the expectation values < B > \ A and uncertainties AB\ A of quantity B cutoff independent?" In terms of increasing complexity, the next class of calculations involves the evaluation of wavefunctions, probability densities, expectation values and uncertainties for a 3-manifold MB with a single, anisotropic, 2-manifold boundary 8MB w i t h 2-sphere topology, where the anisotropy is completely parameterized by 2 independent variables. We w i l l refer to this class of problem as the two-parameter anisotropic class. There are five subclasses of the two-parameter anisotropic type—they are characterized by the ordering of j-values and symmetries of the 6j-symbol. Table 4.1 lists a l l five subclasses. We w i l l 37 Chapter 4. Subclass A Two-parameter Anisotropic investigate subclasses A and E . Calculations 38 1 subclass A B Ji X X h X X h X Y h Y X h Y Y h Y Y C D E X X X X Y Y Y Y Y Y X Y Y Y Y Y Y Y Table 4.1: Two-parameter anisotropic subclasses. 4.1 Subclass A 4.1.1 The Subclass A Wavefunction A single, ellipsoidal anisotropic boundary can be tessellated by T as a single tetrahedron w i t h two independent edge lengths given by J\ = J2 = J3 = AT, J T h i s tetrahedral representation 2 4 = J = J = Y. 5 (4.1) 6 w i l l thus belong to subclass A . See Figure 4.1 for the graphical representation. Notice that, as in the isotropic class of calculations performed in Chapter 3, the tessellation T[dM ] B is chosen such that a l l vertices, edges and faces of the single tetrahedron t are i n the boundary of the tessellation. N o t e that there is only one subclass of the isotropic type—the isotropic class itself. Since this chapter discusses analysis of the tetrahedron with li = I2 = h = X + | , I4 — Z = IQ = Y + | , it is useful to note that in such a case V < 0 when \/3(Y + |) < X + | . For example, the smallest admissible tetrahedron of this type is l\ = I2 = h = \, h = h = ^6 = f, for which equation (2.18) yields V = - f f f i . 1 2 5 2 2 Chapter 4. Subclass A Two-parameter Anisotropic Calculations 39 T Je=Y J =Y 4 Js=Y J =X 2 J =X 3 Ji=X Figure 4.1: Tessellation of the single ellipsoidal anisotropic boundary. Chapter 4. Subclass A Two-parameter Anisotropic Calculations T h e n , according to equation (2.26), the wavefunction *[M ,{X,X,X,Y,Y,Y}] \ lim £ (2p + i ) K—>oo\p=0,l...,K J 2 where X = 0, | , 1 , K , is = B I 3 (XXX 2 ( 2 X + 1 ) 2 ( 2 F + 1)2 { Y and Y = 0, | , 1 , K . 40 Y }, (4.2) Y A g a i n , as i n the isotropic calculations, not all values X lead to an admissible 6j-symbol—admissible X are again integral. In these cases the 6j-symbol w i l l vanish. In contrast, a l l integer and half-integer values of Y w i l l result i n an admissible 6j-symbol (provided X is integer). A s i n the previous chapter, we could ask questions regarding the behaviour of \I> as a function of the variables X, Y, and K. F r o m equation (4.2) we see that the first term is again a rapidly decreasing function of K (it is 0(K~ )). 6 3 The second and t h i r d terms 3 are 0(X*) and 0(Y*) increasing functions of X and Y respectively. T h e final term is again a rapidly oscillating function of its arguments. We therefore expect \& and |\I>| to 2 decrease rapidly (asymptotically approaching O) as K increases along contours of fixed X and Y. T h e y are also expected to resemble increasing amplitude oscillating function of X and Y for fixed values of K. Figure 4.2 shows \ f and |\&| as functions of X and Y for fixed K=20 for the range 0 < X < 20, 0 < Y < 20. A g a i n notice that the behaviour 4 I 12 I 12 of \& and |\&| are very similar—the predominant difference between them is that \^>\ again a strictly positive function w i t h twice the frequency of oscillation of is Figures 4,3 and 4.4 respectively display ^/ and |\I/| as functions of X and K (fixed Y), and Y 2 andF oKr clarity, (fixed the X) . boundary NoticeJ-values that ^ are and |\I>|explicitly vanish listed for Xas <arguments 2Y. T h iof s occurs because For the again the wavefunction. 3 5 2 brevity, this explicitness will be dropped as we continue the study. A g a i n note that many values of * and | $ | exceed the displayed range of the plot—these regions appear as holes in the plotted surfaces. A s with Figure 3.2, X and Y are, strictly speaking, restricted to be less than or equal to K. However, values X > K and Y > K are again shown for demonstrative purposes. As well, apparent holes in the data are again the result of values which exceed the range of the plot. 4 5 Chapter 4. Subclass A Two-parameter Anisotropic Calculations 41 inadmissibility of the 6-tuple gives gives rise to a vanishing 6 j - s y m b o l . A l s o note that Figures 4.3 and 4.4 reveal that ^ and | ^ | Y than along contours of fixed X. decrease more rapidly along contours of fixed T h e behaviour displayed by a l l three sets of plots extends to a l l larger ranges of variables X, Y and K. We now observe that there is a behaviour i n the subclass A wavefunction and probability densities which was not present in the isotropic functions. We observe this new feature i n Figure 4.5—a rotated plot of the data for \I> from Figure 4.2. Here we see regions where ^ becomes a constant (it vanishes) i n Y for given K. Lorentzian regime of the tetrahedron 6 T h i s occurs i n the where the ratio y is large enough to cause inad- missibility i n the ordered set of 6j-values. T h i s vanishing ratio turns out to be 2. T h a t is, y > 2 is in the inadmissible regime of the ordered 6-tuple of jf-values. Since the set of 6j-values becomes inadmissible, the 6 j - s y m b o l itself vanishes. T h i s , in turn, results i n a vanishing wavefunction and probability density. However, no such relationship holds for sufficiently large ^ . T h i s is easy to understand using a geometric analogy—if we imagine the tetrahedron t we see that the three edges forming the base do not fix the m a x i m u m length of the three edges forming its peak. However, i f we fix the lengths of the edges forming the peak of the tetrahedron there is a m a x i m u m length the base edges can be before a larger base edge would require the tetrahedron to either deform or become disconnected . 7 In the subsection 4.1.4 we w i l l see that the vanishing of the wavefunction and probability amplitude for y give rise to cutoff invariants. 6 I.e., where V3(Y + \) < X + \ leads to V 2 < 0. T h i s analogy is not exact if we consider only 3-dimensional Euclidean space: Some of the X edge lengths large enough to deform or disconnect the tetrahedron are allowed—they merely result in the imaginary volume tetrahedra which are embeddable in a 3-dimensional Lorentzian space. 7 Chapter 4. Subclass A Two-parameter Anisotropic Calculations Chapter 4. Subclass A Two-parameter Anisotropic Calculations 20 0 43 Chapter 4. Subclass A Two-parameter Anisotropic Calculations 11 Chapter 4. Subclass A Two-parameter Anisotropic Calculations 5- 10 -5- 10 -1 • 10 Figure 4.5: Subclass A \I> as a function of X and Y for K Chapter 4. Subclass A Two-parameter Anisotropic 4.1.2 Subclass A E x p e c t a t i o n Values < l x Calculations > and Uncertainties 46 Al x Just as w i t h isotropic wavefunctions and probability densities, we must study subclass 2 A \I> and | ^ | for fixed values of K. We now evaluate all given functions for sets of increasing K to study whether we can isolate cutoff independent behaviour for certain sets of conditional probabilities similar to those already discussed. Now, the normalized expectation value of lx is given by S£ =o (x + %)\*M [x,x,x,y,y,y]\ 2 < l x > = y B 52x, =o ^ ^ \^M [x,x,x,y,y,y}\ 2 y B where the sums over variables x and y are carried out for all half-integer values less than or equal to K. 8 Since both x and y are summands we expect this formulation of < lx > to qualitatively behave like the isotropic expectation values of Chapter 2—there is nothing that qualitatively changes in this expression. We furthermore expect that < lx > should exhibit linear dependence i n K <l >otK. (4.4) x Similarly, we predict the uncertainty (root-mean-squared deviation from the mean) Alx to behave i n K as did the isotropic uncertainty—Alx should be linearly dependent on K Al x oc K. Performing the necessary calculations we find that (4.4) and (4.5) hold true. (4.5) In particular, performing the appropriate least-squares fits to the 0 < K < 50 data set (see Figure 4.6) reveals that < l x > = (0.688 ± 0.001) A" + (0.50 ± 0.04) (4.6) Rigorously speaking, the admissibility conditions will cause the associated probability densities to vanish for non-integer x, so we are free to limit our summation to integer values of x. 8 Chapter 4. Subclass A Two-parameter Anisotropic Calculations w i t h corresponding coefficient of determination R = 0.99961, and 9 Al = (0.2312 ± 0.0005)# + (0.11 ± 0.01) x w i t h coefficient 10 R 2 2 47 (4.7) = 0.99957 (see Figure 7). In subsection 4.1.3 we w i l l observe similar results for the analogous calculations of < ly > and Al . Y Study also reveals that the rate of change of < lx > w i t h respect to K is smaller than that of the isotropic calculation since there are a greater number of probability density terms contributing to the sum for a given value of K. of change of Alx We also observe that the rate is less than that of the isotropic calculation. T h i s is again due to the greater contribution from terms at given K. A s well, taking the ratio (4.8) < lx > for 0 < K < 50 shows that the relative uncertainty approaches a value i n the range 0-33 < < 0.34 (see Figure 4.7). 4.1.3 Subclass A Expectation Values < ly > and Uncertainties Al Y Similarly, the normalized expectation value of l Y is given by E£ =o (V + l)\^M [x,x,x,y,y,y}\ ^ 2 < l y > = y B Ex, =o \^M [x,x,x,y,y,y}\ 2 y B A s i n preceding discussions, one would predict < l Y > and Al Y to behave according to < l > oc K (4.10) Al oc K, (4.11) Y and Y 9 T h e probability that this coefficient of determination is the result of a v = 99 uncorrelated data set is of the order 10~ . 170 1 0 T h e corresponding probability for an uncorrelated data set is of the order I O - 1 6 8 . Chapter 4. Subclass A Two-parameter Anisotropic Calculations Figure 4.6: Subclass A < lx > and Al x as a functions of K. 48 Chapter 4. Subclass A Two-parameter Anisotropic Calculations • 0.36 A/ x • •• • • • 0.35 </,> 0.34 • 10 ' 20 ' Figure 4.7: Subclass A 30 40 K as a function of K. 50 Chapter 4. Subclass A Two-parameter Anisotropic Calculations 50 since the sum i n equation (4.9) runs over both x and y. (I.e., this case is again similar to that of the isotropic expectation values and uncertainties.) Performing the appropriate calculations we find (4.10) and (4.11) hold true. Leastsquares fits to the 0 < K < 50 data and reveals that < l Y > = (0.7514 ± 0.0005)if + (0.53 ± 0.01) w i t h corresponding coefficient of d e t e r m i n a t i o n Al Y w i t h coefficient 12 R 11 R 2 (4.12) = 0.99996, and = (0.1746 ± 0.0002)# + (0.119 ± 0.007) (4.13) = 0.99981 (see Figure 4.8). 2 Taking the ratio Al Y (4.14) < ly > for 0 < K < 50 furthermore shows that the relative uncertainty approaches a value in the range 0.23 < < 0.24 (see Figure 4.9). Furthermore, the ratio consistently yields values between 1.001 and 0.555 for 0 < K < 50, and displays convergence towards ~0.916 w i t h increasing K (see Figure 4.10). Meanwhile the ratio yields values between 0 and 1.604 for the 0.5 < K < 50 data set, and converges towards ~1.318 for increasing K (see Figure 4.11). Probability that R = 0.99996 is the result of a v — 99 uncorrelated data set is of the order 10 The corresponding probability for an uncorrelated data set is of the order I O . 1 1 12 2 -186 2 1 9 . er 4. Subclass A Two-parameter Anisotropic Calculations Figure 4.8: Subclass A < l > and Al Y Y as functions of K. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 0.26 • 0.255 AI Y <I > Y 0.25 •• • • • • • • • • 0.245 0.24 • • • • • • • . • • . • • • • • • .*• . ••*..• 0.235 10 20 30 •••••• • • • • • • *»• • 40 K Figure 4.9: Subclass A -=r*- as a function of K. 50 Chapter 4. Subclass A Two-parameter Anisotropic Calculations 0.96 • • 0.94 <lx> • • • • 0.92 • • » • • • • <I > Y •* W 20 30 40 • • • 0.88 K Figure 4.10: Subclass A as a function of K. 50 Chapter 4. Subclass A Two-parameter Anisotropic Calculations 1.4 m • • 1.35 • M x A. 1 Aly t ^ • • • •. • 1.25 • t K • Figure 4.11: Subclass A -rr^ as a function of K. > > Chapter 4. Subclass A Two-parameter Anisotropic 4.1.4 Subclass A < lx > and Al along contours of ly and K x A l t h o u g h < lx >, < ly >, Al and Al x 55 Calculations Y are K dependent, we have just seen that J ^ , c j ^ - and ^ j ^ - seem to asymptotically approach constant values w i t h increasing K. In some sense the geometry of the tessellated boundary seems to freeze out i n K. There are other instances where we can observe invariants i n K. These situations arise when we fix (constrain) certain parameters of the theory and investigate contours of the unconstrained variables. T h e first of these w i l l be i n < lx > and Al X There are two types of contour to consider: (1) fixed ly', and (2) fixed K. We w i l l direct our attention, in turn, at each of these contours. Distinguishing Terminology: Classically Allowed versus Physical Before we continue, it is important to understand the terminology we w i l l be using in the sections which follow. We have already encountered the use of Euclidean and Lorentzian as it pertains to the geometry of a tetrahedron. Euclidean, while those w i t h V 2 Recall that tetrahedra w i t h V 2 > 0 are referred to as < 0 are called Lorentzian. T h e distinction merely refers to the type of 3-dimensional space into which we can embed the tetrahedra. We w i l l continue to use Euclidean and Lorentzian i n this sense. We could alternatively refer to the V 2 > 0 as the classically allowed regime and V 2 <0 as the classically forbidden regime. T h i s is because we associate a classically allowed wavefunction with a boundary exhibiting characteristics of the Euclidean tetrahedron, and a classically forbidden wavefunction w i t h boundaries exhibiting the characteristics of the Lorentzian tetrahedron. T h e analogy is the wavefunction i n a potential barrier—the wavefunction is classically allowed outside the barrier, but classically forbidden inside the barrier. Classically allowed and classically forbidden is thus the terminology we Chapter 4. Subclass A Two-parameter Anisotropic Calculations 56 respectively apply to wavefunctions corresponding to Euclidean and Lorentzian boundary tetrahedra. There is, however, a distinction between classically allowed a n d physical as well as classically forbidden and unphysical. We have already found a number of quantities which exhibit dependence on the cutoff—that is, they w i l l exhibit finite size effects. We w i l l now find quantities which are invariant i n the cutoff K. T h a t is, they w i l l show no finite size effects. T h e calculation of these quantities, w i l l however involve contributions from both classically allowed and classically forbidden tetrahedra. W e w i l l refer to the quantities showing no finite size effects as physical since they are what we can meaningfully measure. Conversely, those quantities exhibiting finite size effects w i l l be referred to as unphysical. Contours of Fixed l Y — (Y + | ) the expectation value of lx is given by For a given value of l Y Ef < l X > \ l Y = = 0 {x + ±)\q [x,x,x,Y,Y,Y} f MB sr^K UT, r Y:? \* [X,X,X,Y,Y,Y]\* =0 ^ v v l • , 2 I 4 ' 1 7 ) MB T h a t is, there is no summation over parameter Y for this case of c o n s t r a i n e d - ^ tetrahedra. In section 4.1.1 we saw that ^ vanishes for a l l y > 2. We would thus expect a l l probability densities and (x + |)-values to contribute to < lx > \ i y until y exceeds this ratio. T h a t is, for a l l K probing the region y > 2 we expect < lx > \i to be a constant. However, until this y > 2 ratio condition is met, < lx > \i w i l l be dependent on Y (i.e., controlled by) K. T h a t is, i n this classically allowed regime, a l l 6j-values w i l l form admissible 6-tuples, so the corresponding 6j-symbols w i l l be non-vanishing. There w i l l therefore be non-zero contributions to the sum from a l l x < K i n this region. In particular, we may guess that <l x > \ t w i l l be a non-decreasing function of K. Chapter 4. Subclass A Two-parameter Anisotropic We similarly expect the uncertainty Alx\W Calculations 57 to exhibit strong dependence on K until we probe the region y > 2. T h e uncertainty is expected to be constant for all K allowing terms where y > 2. Performing the necessary calculations we clearly observe the expected results. < lx > \i is a non-decreasing function of K before the y > 2 ratio is exceeded (see Figure 4.12 for the ly= 20.5, 40.5, 60.5, 80.5, and 100.5 contours). Specifically, note that < lx > \ t does not clearly display direct proportionality to the cutoff for values of K probing y < 2. Also as expected, < lx > \i approaches a constant value for sufficiently large values of Y K. Also as expected, calculations reveal that the functional behavior of Al \i x tively similar to that of < l x >\i Y (see Figure 4.13 for the l = Y Y is qualita- 20.5, 40.5, 60.5, 80.5, and 100.5 contours). A g a i n , the function exhibits cutoff invariance when K is large enough to allow y > 2. There is, however, an interesting feature that we d i d not predict—lx\i show dramatically little variation in the Lorentzian regime. Y and Alx\ ly T h a t is, it appears the functions are nearly constant for (K + \) > \f2>ly where values (x + \) > ^/3ly enter the calculations. It is thus apparent that Lorentzian tetrahedra contribute remarkably little to the uncertainties and expectation values. Furthermore, taking the ratio A l x l <lx> l Y h (4.18) for the ly = 100.5 contour (see Figure 4.14) reveals that the relative uncertainty ap&lxh l x proaches a value i n the range 0.333 < ^ , ^ <lx>\i Y Y <- 0.335. < Chapter 4. Subclass A Two-parameter Anisotropic Calculations Figure 4.12: < lx > \i as a function of K for contours of fixed ly- 58 Chapter 4. Subclass A Two-parameter Anisotropic Figure 4.13: Alx\i a s Y a Calculations function of K for contours of fixed ly- Chapter 4. Subclass A Two-parameter Anisotropic Calculations Chapter 4. Subclass A Two-parameter Anisotropic Calculations 61 Contours of Fixed K Once we fix K to evaluate expectation values < l x > \K and uncertainties Al \ as a x K function oily, we begin to directly probe the geometry of our tessellated boundary. T h a t is, we expect to start observing whether or not the associated tetrahedron is isotropic. We w i l l therefore see whether or not < l x <IX>\K >\K observes (4.19) = W, or simply <lx >\K only. We have already seen that < l x afterwards. > \ l y (4.20) is regulated by K u n t i l y < 2, and is constant We have also seen that there is very little contribution from geometries probing the Lorentzian regime. However, when we fix K and sum over both X and Y to evaluate < l x >\K, a l l admissible 6j-values (both Euclidean and Lorentzian tetrahedra) w i l l contribute to the result provided the terms w i t h y > 2 enter the sum. T h a t is, provided, ly = Y + | < 7}(K + 1), a l l tetrahedra w i l l contribute. However, i f we allow ly = Y + | > ^(K + 1) our sums w i l l involve a l l Euclidean tetrahedra but not a l l Lorentzian tetrahedra. That is, the cutoff K w i l l restrict our sum and effectively omit contributions from geometries where y ~ 2. We therefore expect to see finite size (i.e., cutoff) regulation for sufficiently large ly at some fixed value of K. T h a t is, for sufficiently large ^> ^ tetrahedra will be a Euclidean, so \1> w i l l not vanish—the expectation values (space of histories) w i l l therefore be limited not by l Y (the dynamics), but by the cutoff K. We similarly expect ly to regulate the behaviour of Al \ x K regions with l Y i n the classically forbidden < ^(K + 1), and expect to see cutoff dependence for l It is again difficult to predict the exact behaviour of < l Y x > ^(K + 1). >\K and Al \ as functions x K of ly, but we do know that | ^ | is expected to contribute very little to < l x >\K i n the Chapter 4. Subclass A Two-parameter Anisotropic Calculations Lorentzian region where \fZl 62 > lx is allowed by the condition Y V3l >K + ^. Y (4.21) If contributions i n this Lorentzian region are small enough we may expect to observe a suppression of finite size effects. Calculations reveal that < lx >\K and Alx\ K 4.15 and 4.16 respectively display < l exhibit the predicted features. Figures >\K and Al \ x x K for the K = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100 contours. In detail, we observe the following. < l x < l x >\ K >\K varies linearly i n l y according to = (1.360 ± 0.001)/ + (0.05 ± 0.03), y when ly = Y + ^ < ^(K + 1). In fact, the < lx >\ K (4.22) values are identical between data sets provided ly < \{K + 1). E q u a t i o n (4.22) is the result of an unweighted linear leastsquares fit to the l Y < \(K + l) data, and holds true for each of the K = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100 contours . T h e coefficient of determination for the 7^=100 data 13 set of v = 99 degrees of freedom is R = 0.99994, indicating a firm linear correlation. 2 Additionally, least-squares fits to 2 n d and 3 r d 14 order polynomials again reveal that fits to quadratic and higher order are unsuitable. Furthermore, the contours reveal that < lx >\K oscillates i n regions where \fZl Y > K + ^—the region where K limits the number of classically allowed tetrahedra i n the sum. A s predicted, there is also a suppression of the finite size effects from classically forbidden geometries—< lx >\K exhibits very little K dependence when ^(K + |) > ly>±(K+l). All data sets observe this basic function. The given uncertainties, however, are for the K = 100 data set. Obviously, the data sets with smaller K have larger fit parameter uncertainties since there are fewer data points to fit the function to. T h e probability that a fit with R = 0.99994 and = 99 degrees of freedom results from an uncorrelated parent population is of the order 1 0 ~ . 13 1 4 2 211 Chapter 4. Subclass A Two-parameter Anisotropic Calculations Figure 4.15: < lx >\K as a function of ly- Chapter 4. Subclass A Two-parameter Anisotropic AA* Calculations 64 IK K=30 K=20 K= 15 K= 10 K=5 /y AA* IK K= 100 K=80 K=60 K=50 K=40 Figure 4.16: A / x | ^ as a function of Zy. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 65 It also appears that the m a x i m u m value < lx >max\K occurs at the smallest ly satisfying (4.21). W h e n we analyze these m a x i m u m values < lx >max\K as functions of K for the K = 5,10,15,20,30,40, 50,60,80, and 100 data sets we find < lx >max\K = (0.780 ± 0.001)7^ + (0.68 ± 0.07) — the result of a linear least-squares fit. R 2 (4.23) — 0.99998 for this fit of v = 8 degrees of freedom, indicating a strong linear c o r r e l a t i o n . A g a i n , least-squares 15 fitting to higher orders i n K demonstrates that non-linear descriptions are inappropriate. We thus observe that the ratio of < lx > \K to ly is Tf-invariant when l Y < ^(K + 1) and that finite size effects become apparent when the cutoff begins to exclude nonvanishing geometries from our calculations. In other words, we must be careful when we ask questions regarding the physics of our system—if we ask about behaviour i n regions where ly < ^(K + 1) we may get cutoff invariant (i.e., physically meaningful) results, but we w i l l likely not i f we ask questions elsewhere. We now turn our attention to AIX\K- Calculations reveal that AI \K resembles the superposition of a linear and a small- X amplitude oscillating function i n ly for ly < \(K + 1). Furthermore, AI \K is cutoff X independent provided we consider regions where ly < \{K + 1). Figure 4.17 isolates and displays the K = 100 contour. T h e result of unweighted linear least-squares fitting to l < \(K + 1) for the K = 100 data set reveals that Y AI \K X w i t h R = 0.99124. 2 = (0.388 ± 0.004)Z + (0.0 ± 0.1), (4.24) y 16 T h e probability of this fit resulting from an uncorrelated data set is of the order of 1 0 . T h e likelihood of an uncorrelated data set with v = 99 degrees of freedom to yield this R value is of the order I O . - 2 0 1 5 1 6 2 - 1 0 3 Chapter 4. Subclass A Two-parameter Anisotropic Calculations F i gure 4.17: A Z ^ I ^ as a function of7y. 66 Chapter 4. Subclass A Two-parameter Anisotropic Calculations 67 Furthermore, calculating AI \K (4.25) X < lx >\K for the K = 100 contour reveals that the relative uncertainty converges towards a value in the range 0.26 < '4*l* < 0.31 in the cutoff independent region of our system (see K Figure 4.18). In summary, Figure 4.19 displays the surfaces 17 both l and K for 0.5 <ly < 20.5, 0 < K < 20. 4.1.5 Curvature Expectation Values < 0 > Y of < l x > and Al x as functions of To place our observations in a different context, we may consider the curvature of the tessellated boundary at some vertex. Figure 4.20 shows the tetrahedral tessellation. There are two undetermined angles in this tetrahedron—9yy and 9 y. X It is easy to determine that ^ = - - ' ( i - | ^ § ) . <«•*) and ^ c o s " ^ 1 . (4.27) Note that if l\2 2' we get imaginary results for the angles. However, in this case X Y > 2, (4.29) so we are in the region disallowed by the admissibility conditions. 17 A s with our analysis of the wavefunctions, our theory is strictly defined only for the regions l Y < Chapter 4. Subclass A Two-parameter Anisotropic Calculations <lx • • • 0.35 ...» . . v / v * 0.325 • 0.3 • ••. • . . 0.275 K=100 •• • • • • • • • • • •.• # • •• 0.25 • 0.225 20.5 40.5 60.5 80.5 0.175 ly Figure 4.18: ^ f - as a function of l for AT=100. Y 100.5 Chapter 4. Subclass A Two-parameter Anisotropic Calculations 20.5 0 Al X 2 10.r* 5 20.5 o Figure 4.19: < lx > and Al x as functions of l Y and K Chapter 4. Subclass A Two-parameter Anisotropic Calculations Figure 4.20: Angles at the vertices of the tetrahedron. 70 Chapter 4. Subclass A Two-parameter Anisotropic Calculations 71 •K - We w i l l consider expectation values along contours of fixed K because we know that we w i l l be able to extract cutoff independent information from our system provided we restrict our calculations to l < \(K Y + 1). The expectation values of 9 and 9 Y YY for X fixed K are therefore given by 2(y+>)» <e >\ = YY v , K )'' 0 , -2 E^ \^M [X,X,X,Y,Y,Y}\' (4.30) B and Ef^cos< >\K = ( ^ y ) 1 \y [x,x,x,Y,Y Y\\ MB t _ _ _ , , ^ , 2 Y%= \*M [x,x,x,Y,Y,Ytf • r Q (4-31) B A s expected, computations reveal that < 0 > \ K YY > \ YY K provided ly < \{K + 1). Figure 4.21 shows < 6 and < 9XY > \K K a r e constant in for K = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100, while Figure 4.22 displays the results of < 9 y X > \ K for K = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100. Least-squares fitting to constants determines < Qy >\ Y K = (1.555 ± 0.006), (4.32) = (°- (4-33) and < B Y >\ X for l Y K 7 9 4 ± °- 0 0 3 ), < \{K -r 1) along the K = 100 contours. Figure 4.23 summarizes the data for K < 20. Since < 9 y Y > \ K > § and < 9XY > \K < f the the peak of the tetrahedron is some- what flattened. If the tetrahedron were isotropic we would have observed < 9yy > \ < &XY > \ K = K = f • Calculations and least-squares fitting also revealed that /\9 \ YY K = (0.500 ± 0.006) (4.34) Chapter 4. Subclass A Two-parameter Anisotropic Calculations < ®YY> K • 1. 8 • 1. 6 . 'mm' -.» • 1 .4 • • • 1.2 • 1 : * •• 0. 8 A • • K = 5 0 .5 5.5 K = 7 5 10.5 A *• * K = 2 0 15.5 20.5 • K=30 25.5 30 . 5 K=80 K=100 W < Q > YY \K 1 . 8 mr * -_ v. • "4 •A * **** " ^ /<=40 /<=50 /<=60 'Y Figure 4.21: < 9 Y > \K SLS a function of lyY Chapter 4. Subclass A Two-parameter Anisotropic K=5 0.7 K=10 K=15 Calculations K=20 K=30 \ 0.5 0.5 5.5 20.5 10.5 15.5 40.5 20.5 60.5 Figure 4.22: < 9 y X > \K a 25.5 80.5 s a function of ly. 30.5 100.5 Chapter 4. Subclass A Two-parameter Anisotropic Calculations <e > XY Figure 4.23: < Qyy > and < QXY > as functions of ly and K Chapter 4. Subclass A Two-parameter Anisotropic Calculations A9 \ XY for l Y 75 = (0.250 ± 0.003) K (4.35) < | ( # + 1) along the K = 100 contour. See Figure 4.24 for < 9 > \ YY along the K # = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100 contours, and Figure 4.25 for A9 \ XY the K = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100 contours. cutoff independent results for the region l along K We have again observed < | ( # + 1). Y Furthermore, the curvature at the vertex formed by the three length l Y edges is given by 6 y y = 2ir - 39 , (4.36) YY while the curvature at the vertex formed by two length l edges and one length l x Y edge is e XY = 2ir-(26 + ^). XY (4.37) Inserting the results from (4.32), (4.33), (4.34), and (4.35) into (4.36) and (4.37) yields the cutoff independent results < &YY >\ K < &XY >\ K = (1-62 ± 0.02), (4.38) = (3.649 ± 0.006), (4.39) and A 6 y y | ^ = (1.50 ± 0 . 0 2 ) , AQ \ = (0.500 ± 0.006), XY K for l Y (4.40) < T;(K + 1). Figure 4.26 shows the < Q YY > \ K (4.41) and < Q # = 1 0 0 contour, and Figure 4.27 shows the A 0 y y | ^ and AQ \ X Y > \ K results for the results for the # = 1 0 0 XY K contour. B y evaluating < 6yy >max\ = < YY Q K > \ K + A Q YY\ , K (4.42) er 4. Subclass A Two-parameter Anisotropic Calculations A 0 YY\ K 0.6 0.5 0.4 0.3 K=5 K=10 K=30 K=20 K=15 o.i 20.5 25.5 /f=£<? /f=<?<? 15.5 10.5 0.5 30.5 /,Y A 0 YY K 0.6 • • 0.5 • .... 9 m ., a . 4 . v> 0.4 0.3 A=-#7 0.2 /fe5Z? K=100 o.i 0.5 20.5 40.5 60.5 80.5 100.5 'Y Figure 4.24: A9YY\K a s a function of l . Y Chapter 4. Subclass A Two-parameter Anisotropic Calculations AG'XY\ K 1* 0.2 0.1 K=5 K=W K=15 K=20 K=30 0.05 20.5 25.5 Y AG'XY\ K 0.3 0.25 0.2 0.15 0.1 K=40 K=50 K=60 K=80 K=100 0 . 05 60 . 5 Y Figure 4.25: A9 y\ X K as a function of ly- 100.5 Chapter 4. Subclass A Two-parameter Anisotropic Calculations Chapter 4. Subclass A Two-parameter Anisotropic Calculations AOyy 2 K 1 1.5 ^ 1 K=100 0.5 0 40 . 5 20.5 5 60 . 5 80.5 100.5 IY ®XY A K 0.6 0. 5 K=100 0.4 0.3 0.2 0 .1 0 5 20.5 60 . 5 40 . 5 80.5 100.5 IY Figure 4.27: AQ Y\ Y K and A 6 x y | ^ as functions of lyfor#=100. Chapter 4. Subclass A Two-parameter Anisotropic Calculations < 0 y y > \ = <e >\ min K YY K 80 AQ \ , (4-43) + &QXY\ , (4-44) ~ &QXY\ > (4.45) YY K and < ®XY >max\ = < XY >\ < &XY >min\ = < XY > \ & K Q K K K K K we determine the m a x i m u m and m i n i m u m observable curvatures of < QYY > \K " ANA < QXY > \K t the vertices of the tetrahedron. Figure 4.28 displays the m a x i m u m and a m i n i m u m values of < O y y > 1^- for K=100, while Figure 4.29 displays the m a x i m u m and m i n i m u m values of < QXY > \K f ° 7f=100. r to observe negative curvatures for small l Y Interestingly, we find that it is possible at the vertex formed by the three length l Y edges. These occur at very small ly where classically forbidden \I> dominates. However, the vertex formed by two edges of length lx and one of length l w i l l always be positively y curved. In particular, least-squares fitting to the region l Y < ^(K + 1) yields < 0 y y > max \K (3.12 ± 0 . 0 4 ) , (4.46) (0.12 ± 0.04), (4.47) < QXY >max\n (4.15 ± 0 . 0 1 ) , (4.48) < QXY >TTUTIK (4.49) < 0 y y >min\K and 4.1.6 (3.15 ± 0 . 0 1 ) . Subclass A < l > and Al Along Contours of l Y Y x and K We w i l l continue the investigation of our A-subclass two-parameter anisotropic tetrahedron. Specifically, we w i l l evaluate the expectation values < l Y l x and K. > along contours of fixed Chapter 4. Subclass A Two-parameter Anisotropic Calculations Chapter 4. Subclass A Two-parameter Anisotropic Calculations Figure 4.29: < QXY >max IK- a n d < ®XY >mml/c a s functions of ly for #=100. 82 Chapter 4. Subclass A Two-parameter Anisotropic Calculations 83 Contours of Fixed l x For a constrained value of lx the expectation value of ly is given by ^, Ef < Ly > i (y+l)\*M [X,X,X,y,y,y}\ 2 =0 B — r; : —, . (4.0U) ZU\*"B[X,X,X,V,V,V]\ UX Calculations for < ly >\i and Al \i x Y reveal that they are strongly dependent non- x linear functions i n K. T h i s dependence persists for all values of K. Figure 4.30 displays < l Y > \ l x and Al \i Y for the l x = 10.5 data set. A l t h o u g h it is difficult to predict x the exact behaviour of < ly > | ; f r o m (4.50) alone, it is easy to understand the cutoff x dependence of < ly >\i and Al \i . x Y tetrahedral geometries w i t h l Y Notice that the sum involves summing over a l l x > l. A l l 6j-symbols are non-vanishing for such geome- x tries. Therefore, when we increase K we increase the admissible configurations included in our sum. F i n i t e size effects w i l l therefore be observed for all such calculations. F r o m Figure 4.30 it also appears that < l Y K » l. x >\i x and Al \i Y x become linear i n K for Then, taking the ratio < W» (4.51) A I y 1 i >\lx for the lx = 10.5 contour shows that the relative uncertainty seems to converge to a value i n the range 0.2 < < 0.3 (see Figure 4.31). Contours of Fixed K If we now evaluate < l Y > and Al Y along contours of fixed K we again find cutoff dependent results. Figure 4.32 shows the results for K = 10, 20, 30, 40, 50, 60, 80, and 100. W e again observe cutoff dependent behaviour for < l Y > \ K and A l Y K . However, we also see lx dependence i n the results. Additionally, because Al \ Y K tends to decrease w i t h l x while <l Y > \ K does not, Chapter 4. Subclass A Two-parameter Anisotropic Calculations Chapter 4. Subclass A Two-parameter Anisotropic Calculations Chapter 4. Subclass A Two-parameter Anisotropic Calculations < l > # Y 80 -•"*'*'*V-'v--w-^^ 1 K=10 ° 6O1 • * * m • * * * * * m * * **V V 40 . A A V / A V A ^ A A ^ ^ W * ^ 20 • K=60 #=30 ">Vl*W # = 2 0 #=yo • 20.5 40.5 60.5 80.5 100 . 5 Ix AI Y K 25 20 --,->v- **-v**^ v 15 i * *» • * * « • • ' W + * ^ * •** ** **r . * . • * . * f* * J • * ** *• / . * * * • • • . • 20.5 40.5 Figure 4.32: < l Y ~ K ****, K=S0 - K=60 K=50 60.5 > \ K=100 ****, A /C=40 * **A. K=10 *^ ** * * * * * * * * * * * * and Al \ Y K 8 0 .'5 100.5 as functions of l X Chapter 4. Subclass A Two-parameter Anisotropic Calculations 87 also tends to be a decreasing function in lx- The K = 100 contour shows the relative uncertainty decreasing to its minimum value of ~ 0.1742 (see Figure 4.33). Also note that the maximum value is ~ 0.455. Figure 4.34 displays the surfaces of < ly > and Aly as functions of both l and K x for 0.5 < l x < 20.5, 0 < K < 20. Chapter 4. Subclass A Two-parameter Anisotropic Calculations My K <ly> K 0.4 r 0.35 - 0.3 • 0.25 0.2 • • • •• • • • • • • • ••*•.... K=100 .... . • •... 0.15 20.5 40.5 60.5 80.5 100.5 Ix Figure 4.33: Relative uncertainty , }? as a function of l Y X Chapter 4. Subclass A Two-parameter Anisotropic Calculations 20.5 Figure 4.34: < ly > and Aly as functions of lx and K. 89 Chapter 5 Subclass E Two-parameter Anisotropic Calculations We know any expectation values we measure on the single-tetrahedral tessellation w i l l be cutoff dependent i f conditional probabilities fail to restrict the number of allowed configurations the system can adopt. T h a t is, the lengths and volume of our tessellated boundary w i l l show finite size dependence. We may, however, see the geometry of our boundary tessellation (as quantified by <\ >, X 7 ^ , and ^L) asymptotically freeze out with increasing cutoff. In contrast, measurements w i l l be cutoff invariant, provided the conditional probabilities restrict the number of possible configurations by, say, fixing an appropriate set of edge lengths before the observation is made. In Chapter 4 we classified the two-parameter anisotropic tetrahedra. We also found that we could understand the behaviour of the subclass A by considering how geometric restrictions limit the total number of configurations the tetrahedron could adopt. We believe the behaviour of other two-parameter anisotropic subclasses can be predicted and understood i n terms of configurational restrictions as well. We w i l l now demonstrate the validity of this configurational interpretation by examining the behaviour of a second subclass. 90 Chapter 5. Subclass E Two-parameter Anisotropic Calculations 91 T h e Subclass E Wavefunction 5.1 Consider the tessellation of a 2-dimensional ovoidal boundary to a single anisotropic tetrahedron. Figure (5.1) shows the tessellation for the case where the tetrahedral J- values are J i = X, J = J = J = J = J = Y. 2 3 5 4 (5.1) 6 A s mentioned i n Table 4.1, this is the subclass E two-parameter anisotropic system. According to equation (2.26), the wavefunction for this system is given as V[M ,{X,Y,Y Y,Y,Y}} B lim K—>oo t £ \p=0,l,...,K where X = 0 , \ , 1 , K , (2p + i ) 2 V : = 2 (2X + l)*(2y + l)$ { J and Y = 0,\,1,..., X Y Y Y Y Y \, (5.2) K. Now let us compare the behaviours of the 6j-symbols (X Y Y ( Y Y Y X X (5.3) and X) (5.4) Y Y Y J Figure (5.2) displays equations (5.3) and (5.4) as functions of X and Y. There are also quantitative similarities between the two 6j-symbols. T h e triangular inequalities presented i n section 2.1.1 guarantee that (5.3) and (5.4) both vanish when y > 2. However, while both integer and half integer values of Y are admissible for (5.4), only integer Y yield non-vanishing values for (5.3). Since the 6j-symbol is the object that fundamentally differentiates wavefunction (5.2) of subclass E from the wavefunction (4.2) of subclass A , any fundamental qualitative Chapter 5. Subclass E Two-parameter Anisotropic Calculations Figure 5.1: Tessellation of the single anisotropic ovoidal boundary. 92 Chapter 5. Subclass E Two-parameter Anisotropic Calculations 93 Figure 5.2: Subclass E and A two-parameter anisotropic 6j-symbols as functions of X and Y. Chapter 5. Subclass E Two-parameter Anisotropic Calculations 94 differences i n observed behaviour w i l l be due to the 6j-symbols alone. qualitative and quantitative similarities between the two 6j-symbols, Based on the we would therefore expect the subclass A and E wavefunctions to exhibit the same qualitative behaviour. We therefore predict the following properties for the subclass E system: 1) finite size effects w i l l be observed for expectation values < lx > and < l Y uncertainties Al x >, and and A / y ; 2) relative uncertainties a d n and ratios ^ j ^ - and ^ should asymptoti- cally approach constant values w i t h increasing K; 3) cutoff invariance w i l l be observed for < l x Y > \ l x t , <l x l x > \ K and Al \ , x K (2l — lx) < \ terms enter the calculations; and, when y > 2 4) < l > \ , Al \ Y , Al \i Y , <l Y > \ K and Aly\ K w i l l exhibit finite size effects. So let us now review the computational results i n brief. 5.1.1 Subclass E Expectation Values < l x > and Uncertainties Al x Now, the normalized expectation value of lx is given by T^x, =o ( + x < lx >= y T:x, =o y ^)\^M [x,y,y,y,y,y]\ 2 B \^M [x,y,y,y,y,y]\ (5.5) 2 B Note the sum is over both integer and half-integer values for variables x and y. Performing the necessary calculations we find a linear relationship between < l x > and K. In particular, performing the appropriate least-squares fits to the 0 < K < 50 data set (see Figure 5.3) reveals that < i x > = (0.533 ± O.OOIJA" + (0.32 ± 0.03) (5.6) Chapter 5. Subclass E Two-parameter Anisotropic Calculations w i t h corresponding coefficient of determination R = 0.9995, and 1 Al = (0.2916 ± 0.0007)K + (0.22 ± 0.02) x w i t h coefficient R 2 2 95 (5.7) = 0.99946 (see Figure 5.3). 2 Furthermore, taking the ratio (5.8) < lx > for 0 < K < 50 shows that the relative uncertainty approaches a value i n the range 0.546 < < 0.552 (see Figure 5.4). T h i s value is larger than - 0.335—the value obtained for the subclass A tetrahedron. 5.1.2 Subclass E Expectation Values < l > and Uncertainties Al Y Similarly, the normalized expectation value of l Y , ^ Ex, (y y=Q + Y is given by l)\^M [x,y,y,y,y,y]\ 2 B where x and y are again summed over all integer and half-integer values. Performing the appropriate calculations we find a linearity between < l Y > and K. Least-squares fits to the 0 < K < 50 data and reveals that < l Y >= (0.794 ± 0 . 0 0 1 ) 7 T + (0.53 ± 0 . 0 4 ) w i t h corresponding coefficient of determination R 3 Al Y w i t h coefficient R 4 2 (5.10) = 0.9997, and = (0.1632 ± 0.0003)TsT + (0.07 ± 0 . 0 1 ) (5.11) = 0.9995 (see Figure 5.5). 2 T h e probability that this coefficient of determination is the result of a v = 99 uncorrelated data set is of the order I O . T h e corresponding probability for an uncorrelated data set is of the order 1 0 ~ . P r o b a b i l i t y that R = 0.9997 is the result of a v — 99 uncorrelated data set is of the order 1 0 . T h e corresponding probability for an uncorrelated data set is of the order I O . 1 - 1 6 7 2 163 2 4 - 1 7 5 - 1 6 8 Chapter 5. Subclass E Two-parameter Anisotropic Calculations Chapter 5. Subclass E Two-parameter Anisotropic Calculations 0.57 Al x <lx> 0.56 0.55 0.54 Figure 5.4: Subclass E jV* as a function of K Chapter 5. Subclass E Two-parameter Anisotropic Calculations 0 10 20 30 40 K Figure 5.5: Subclass E < l > and Al Y Y as functions of K. 50 Chapter 5. Subclass E Two-parameter Anisotropic Calculations 99 Taking the ratio ^ih (512) for 0 < K < 50 furthermore shows that the relative uncertainty approaches a value i n the range 0.203 < < 0.205 (see Figure 5.6). T h i s is smaller than ~ 235—the value obtained for the subclass A tetrahedron. Furthermore, the ratio < h > , ^ < 5 1 3 > for 0 < K < 50 displays convergence towards a value in the range 0.665 < f ^ f < 0.675 w i t h increasing K (see Figure 5.7). T h i s shows that the tetrahedron tends to be somewhat egg-shaped. Also, it is smaller than ~ 0.916—the value for the subclass A tetrahedron. Meanwhile the ratio (5.14) Al Y for the 0.5 < K < 50 data set converges towards a value i n the range 1-775 < < 1.825 for increasing K (see Figure 5.8). 5.1.3 Subclass E < l x > and Al We have thus verified that < l x x Tjf"' 7^> ~w l a n < ^ along contours of l Y >, < l >, Al Y x and Al Y and K are K dependent, and that seem to asymptotically approach constant values with increasing K. We w i l l now continue searching for cutoff invariant measures on our tessellated boundary of the subclass E system by considering two methods of constraint: (1) fixed l ; and Y (2) fixed K. Let us begin w i t h the fixed l Y constraint. Chapter 5. Subclass E Two-parameter Anisotropic Calculations 0.2 0.18 Aly 0.16 <lv> 0.14 0.12 0.1 Figure 5.6: Subclass E as a function of K. 100 Chapter 5. Subclass E Two-parameter Anisotropic Calculations 101 0.9 <lx> < \ y > 0.8 0.7 0.6 0 10 20 30 K Figure 5.7: Subclass E as a function of K. 40 50 Chapter 5. Subclass E Two-parameter Anisotropic Calculations 102 3.5 Al x Al Y 2.5 0 10 20 30 K Figure 5.8: Subclass E ^f- as a function of K. 40 50 Chapter 5. Subclass E Two-parameter Anisotropic Calculations 103 Contours of Fixed l Y For a given value of ly = (Y + | ) the expectation value of lx is given by £* 0 < l x > 'V = (x + ±)\* [x,Y,Y,Y,Y,Y]f MB v ^ t f ivr, r _ ^ ,r w ,^1,2 E^ 0 l*M [x,y,y,r,y,y]r • ( 5 1 5 ) S T h a t is, while x is summed over all integer and half-integer values up to K, there is no summation over parameter Y for this case of c o n s t r a i n e d l y tetrahedra. Performing the necessary calculations we clearly observe the following results. < l >\ x is a non-decreasing function of K for all values of Y and K (see Figure 5.9 for the l = Y 20.5, 40.5, 60.5, 80.5, and 100.5 contours), and i t is a constant for ^(K + 1) > ly. Calculations also reveal that the functional behavior of Alx\iY to that of < l x is qualitatively similar >\i (see Figure 5.10 for the l = 20.5, 40.5, 60.5, 80.5, and 100.5 contours). Y Y A g a i n , the function exhibits cutoff invariance when ^(K + 1) > ly. A s i n the study of subclass A we note that lx\iY and A.lx\ly show dramatically little variation i n the Lorentzian regime. T h a t is, it appears the functions are nearly constant for (K + | ) > \/3l where values (x + | ) > \/3l Y Y enter the calculations. Lorentzian tetrahedra again contribute remarkably little to the uncertainties and expectation values. Furthermore, taking the ratio M x W < IX >\ly for the l Y (5.16) = 100.5 contour (see Figure 5.11) shows the relative uncertainty decay to a constant value ^, )\ LX Y ~ 0.4835. T h i s is larger than the subclass A result of ~ 0.2785. Contours of Fixed K A g a i n , when we fix K and sum over both x and y to evaluate < lx >\K or Alx\K, our calculations w i l l be regulated by one of two effects. Either K will be large enough to allow a l l Euclidean and Lorentzian tetrahedra to contribute to the sum (ly w i l l regulate l y Chapter 5. Subclass E Two-parameter Anisotropic Calculations <l > x 'Y 20 25 30 35 40 40 45 50 K 60 70 80 90 K <l > <l > x x 'Y 65 60 55 50 45 40 35 K= 60.5 60 70 80 90 100110120130 80 100 120 140 160 K K <l > x Y 100 120 140 160 180 200 220 K Figure 5.9: < l x >\ ly as a function of K for contours of fixed l Y Chapter 5. Subclass E Two-parameter Anisotropic Calculations 105 Chapter 5. Subclass E Two-parameter Anisotropic Calculations 107 < lx > \ ), or the space of histories w i l l be restricted to exclude some tetrahedral ampliK tudes when y is sufficiently small (K w i l l regulate < l x w i l l contribute to the amplitude if l > \ ). K Specifically, a l l tetrahedra = Y + | < \{K + 1), but our set of histories w i l l Y exclude at least one tetrahedral amplitude i f ly = Y + \ > | ( # + 1). Calculations reveal that <l >\K and Al \ x x K exhibit the expected K invariance for sums w i t h a l l ly = Y + | < | ( # + 1). T h e expected finite size effects are observed for sums probing ly = Y + \ > \(K <I >\K X and Al \ x K for the + 1). Figures 5.12 and 5.13 respectively display = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100 contours. K We specifically observe that < l >\K varies linearly in ly according to x < lx >\K = (0.922 ±0.002)/Jy + (0.10 ± 0 . 0 5 ) . As was the case for the subclass A , < lx >\ K provided l Y < \{K squares fit to the l Y (5.17) values are identical between data sets + 1). E q u a t i o n (4.22) is the result of an unweighted linear least< \(K + 1) data, and holds true for each of the K = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100 contours . T h e coefficient of determination for the # = 1 0 0 5 data set of v = 99 degrees of freedom is R 2 Additionally, least-squares fits to 2 = 0.9996, indicating a firm linear correlation. and 3 n d r d 6 order polynomials again reveal that fits to quadratic and higher order are unsuitable. The contours also reveal that < l x >\K decays i n regions where y/3l Y > K + |— the region where K limits the number of classically allowed tetrahedra i n the sum. A s expected, there is also a suppression of the finite size effects from Lorentzian geometries— < lx >\K exhibits very little K dependence when ^(K + |) > l Y > \{K + 1). A l l data sets observe this basic function. The given uncertainties, however, are for the K = 100 data set. Obviously, the data sets with smaller K have larger fit parameter uncertainties since there are fewer data points to fit the function to. T h e probability that a fit with R = 0.9996 and v = 99 degrees of freedom results from an uncorrelated parent population is of the order I O . 5 6 2 - 1 7 5 Chapter 5. Subclass E Two-parameter Anisotropic Calculations Chapter 5. Subclass E Two-parameter Anisotropic Calculations Chapter 5. Subclass E Two-parameter Anisotropic Calculations 110 It also appears that the m a x i m u m value < lx >max\i< occurs at the smallest l satisfy- Y ing \/3Zy > K+\ (the point of transition between Lorentzian tetrahedra and inadmissible geometries). Analysis of these m a x i m u m values < l x >max\K as a function of K for the K = 5,10,15, 2 0 , 3 0 , 4 0 , 5 0 , 6 0 , 8 0 , and 100 data sets reveals < lx >max\K = (0.42 ± 0.02)# + —the result of a linear least-squares fit. R (2 ± 1) (5.18) = 0.984 for this fit of v = 8 degrees of 2 freedom, indicating a strong linear correlation. 7 A g a i n , least-squares fitting to higher orders i n K demonstrates that non-linear descriptions are inappropriate. We now focus on AIX\Kl Calculations reveal that AI \K X < \{K + 1). Furthermore, AIX\K Y where l Y < \(K is linear function in l Y for is cutoff independent provided we consider regions + 1). T h e result of an unweighted linear fit to l y < \(K + 1) for the K — 100 data set reveals AI \K X with R 2 = 0.9996. = (0.4780 ± 0.009)/y + (0.08 ± 0.03), (5.19) 8 Furthermore, evaluating A l x l (5.20) K < lx >\K for the K = 100 contour shows the relative uncertainty converges towards a value in the range 0-515 < *X\* K < 0.520 i n the cutoff independent region of our system (see Figure 5.14). A g a i n , this is larger than the subclass A result of ~ 0.285. In summary, Figure 5.15 displays the surfaces both Zy and K for 0.5 < l Y 9 of < l x > and Alx as functions of < 20.5, 0 < K < 20. T h e probability of this fit resulting from an uncorrelated data set is of the order 10 . The likelihood of an uncorrelated data set with v = 99 degrees of freedom to yield this R of the order I O " 1 7 1 . 9 7 2 8 9 value is A s with our analysis of the wavefunctions, our theory is strictly defined only for the regions ly (# + !)• < Chapter 5. Subclass E Two-parameter Anisotropic Calculations 0.55 0.545 Al X 0.54 K <lx> K 0.535 0.53 0.525 0.52 Figure 5.14: <'x>|x as a function of ly for #=100. Chapter 5. Subclass E Two-parameter Anisotropic Calculations Chapter 5. Subclass E Two-parameter Anisotropic Calculations Subclass E < l 5.1.4 Y > and Al Along Contours of l Y x 113 and K Contours of Fixed lx For a constrained value of l the expectation value of ly is given by x , , Y Eyio = + (y l ) \ ^ M [ X , y , y , y , y , y } \ B 2 ( ^ \^M [X,y,y,y,y,y}\ U x ' 2 =0 B where the sum in Y is over b o t h integer and half-integer values. Calculations for < ly >\i and Aly\i x x reveal that they are non-linear, strongly in- dependent functions. T h i s dependence persists for all values of K. < l Y > \ l x and Al \i Y dependence of < l x Y for the l = 10.5 data set. It is again easy to understand the cutoff x >\i x and Al \i . geometries up to Y = K > X. Figure 5.16 displays Y x Since K > X, the sum includes a l l admissible T h e 6j-symbols for such geometries are, i n general, non- vanishing. O u r sum over histories w i l l therefore experience regulation due to the size of K. F r o m Figure 5.16 it also appears that < ly >\i x K 3> l . x and Al \i Y x become linear in K for Then, taking the ratio -4^f< W (5-22) >\lx for the lx = 10.5 contour shows that the relative uncertainty seems to converge to a value in the range 0.2038 < < 0.2039 (see Figure 5.17). T h i s result is smaller than ~ 0.25—the value for the subclass A system. Contours of Fixed K Evaluating < l Y > and Aly along contours of fixed K again reveals finite size effects and lx dependence in the results. Figures 5.18 and 5.19 respectively show the results for K = 10, 20, 30, 40, 50, 60, 80, and 100 data sets. Chapter 5. Subclass E Two-parameter Anisotropic Calculations 114 Chapter 5. Subclass E Two-parameter Anisotropic 100 Calculations 200 115 300 K Figure 5.17: Relative uncertainty ^ ^ j ' * as a function of K. 400 Chapter 5. Subclass E Two-parameter Anisotropic Calculations 116 Chapter 5. Subclass E Two-parameter Anisotropic Calculations 117 3.525 * 3.5 ^ 3.475 < 3.45 3.425 3.4 0 10 15 20 •X 6.9 6.85 6.8 6.75 6.7 6.65 10 15 20 25 30 0 10 IX 20 30 40 •X 10 20 30 40 50 60 IX 'X 'X Figure 5.19: Al \ Y K as a function of l X Chapter 5. Subclass E Two-parameter Anisotropic Calculations 118 Furthermore, M\ Y K <l l < *y > >\K ( 5 - 2 3 ) for the K — 100 contour is shown i n Figure 5.20. T h e relative uncertainty increases through its m a x i m u m value of ~ 0.2117 and then decays w i t h increasing l . x occurs at the m a x i m u m value of Aly\ . K T h e peak In comparison, the subclass A m a x i m u m relative uncertainty is ~ 0.4523. In summary, Figure 5.21 displays the surfaces of < ly > and Aly as functions of both lx and K for 0.5 < l x 5.2 < 20.5, 0 < K < 20. Comparison of Subclass A and E Results Now let us summarize our findings for the subclass E tessellation. A s expected, we observed: 1) finite size effects for expectation values < l x > and < ly >, and uncertainties Al x and Aly: 2) asymptotic approach to constant values w i t h increasing K for relative uncertainties and and ratios ^ 3) cutoff invariance for < l x Y >2 ^ > and \ T Y , < lx > AI \ , x 1y \K a n d ^^1^-, when the condition (2ly — lx) < \ limits the total number of system configurations; and, 4) finite size effects for < ly > \ , Al \ t Y t , < ly > \ K and Al \ . Y K T h a t is, using configuration l i m i t i n g arguments, we were able to correctly predict the occurrence of finite size effects and cutoff independence. However, we also found that it was the subclass E tetrahedron which tended to be the more anisotropic of the two (for the unconstrained space of histories calculations), and that it generally displayed the larger relative uncertainty for lx- Chapter 5. Subclass E Two-parameter Anisotropic Calculations 119 0.27 7 Figure 5.20: Relative uncertainty as a function of l . x Chapter 5. Subclass E Two-parameter Anisotropic Calculations 20 0.5 Figure 5.21: < l > and Aly as functions of lx and K. Y Chapter 5. Subclass E Two-parameter Anisotropic Calculations 121 However, there is more can we say about the subclass E system. In particular, we can compare the slopes of its linear least-squares fits to those of the subclass A system. Introducing the notation ( ) i pe,A and ( ) i e,E s 0 to respectively designate the linear fit s op slopes of the subclass A and E systems, we write the results as (< lx >)slope,A (< lx >)slope,E slope,A {Alx)slope,E (< ly >)slope,A (< ly >) lope,E = ( L 2 9 2 ± Q 0 0 6 ) > ( 5 2 4 ) = (0.793 ± 0 . 0 0 4 ) , (5.25) = (0.9463 ± 0 . 0 0 0 7 ) , (5.26) s (Aly) sipped = (1.07 ± 0 . 0 2 ) , (Aly) (5.27) slope,E (< lx > \ )slope,A _ K (1.475 ± 0 . 0 0 4 ) , (5.28) = (1.88 ± 0 . 0 9 ) (5.29) (< lx > \x)slope,E lx ^max \x)slope,A lx ^max \K)slope,E and {Alx\ )slope,A K \ ^ X\K) -(0.81 ±0.02). (5.30) slope,E T h e result of equation (5.24) shows us that < lx > varies more rapidly w i t h K for the subclass A tetrahedron than for the subclass E tetrahedron. Therefore, while both tetrahedra are K dependent, the subclass A tetrahedron shows a greater variation w i t h respect to K. Conversely, equation (5.25) shows that the subclass A tetrahedron uncertainty Al x varies more slowly w i t h K than does Al for the subclass E tetrahedron. In x contrast, equations (5.26) and (5.27) respectively reveal that it is the subclass E < ly > which varies more rapidly with K, and the subclass E Aly which varies less rapidly i n K. T h e fixed K contour results < l x > \ , < lx K >max \ K and Al \ x K from respective equations (5.28), (5.29) and (5.30) also show that it is the subclass A expectation value Chapter 5. Subclass E Two-parameter Anisotropic Calculations 122 of lx, but the subclass E uncertainty i n lx, which show the greater variation w i t h respect to changes i n K. T h i s brings our two-parameter anisotropic study and the thesis to a close. Chapter 6 Conclusion We began this study by citing the challenges of quantum cosmology and discussing how the consistent histories approach addresses the problem of making measurements on closed quantum mechanical systems without external observers. It was discussed how Ponzano-Regge theory (a fully specified theory of 3-dimensional gravity) could help lead us towards the full solution of 4-dimensional quantum cosmology. We then introduced the terminology relevant to the study of Ponzano-Regge theory. We provided a geometric (tetrahedral) representation of the 6j-symbol and for- mulated the Ponzano-Regge wavefunction fy Mg (2-dimensional) boundary for 3-dimensional manifolds M B with dM . B We proceeded by investigating the simplest cases i n search of quantities independent of the cutoff inherent i n the theory. O u r first case study was for a 2-spherical boundary. We modelled this boundary as a tessellation into a single isotropic tetrahedron. We found that all tetrahedral edge length expectation values and uncertainties were cutoff dependent due to the lack of intrinsic scale in the formulation of the problem. (In fact, the expectation values and uncertainties were linearly dependent on K.) F i n i t e size effects were thus observed. Additionally, we saw the relative uncertainty of the tetrahedral edge lengths asymptote towards a constant value w i t h increasing cutoff. The next case studies involved the investigation of two-parameter anisotropic boundaries. We again modelled a topologically 2-spherical boundaries as single tetrahedra. 123 Chapter 6. Conclusion 124 However, since we were assuming an anisotropic boundary, we parameterized the tetrahedra w i t h two edge lengths. We studied two of the five subclasses of the two-parameter anisotropic class of tetrahedra. Firstly, we again observed finite size effects when calculations failed to limit space of histories accessible to the system. The tetrahedra displayed characteristics similar to those of the completely isotropic study under such circumstances. In particular, we observed linear dependence between the cutoff and the edge length expectation values and uncertainties. We also found the tetrahedra tended to be deformed away from isotropy—perhaps this is an indication of i n i t i a l anisotropy i n the primordial universe. We then discovered that is is possible to obtain cutoff invariant expectation values and uncertainties providing we restrict the space of histories by fixing an appropriate set of tetrahedral edge lengths. The evaluation of such conditional probabilities also displayed anisotropy i n the tetrahedral geometry. Furthermore, the two anisotropic subclasses displayed qualitatively similar traits—both were peaked or flattened away from isotropy under the same conditions, and their expectation values, uncertainties and relative uncertainties displayed similar properties and dependencies. Such results are an indication ' that Ponzano-Regge theory is tractable and can be sensibly implemented in the consistent histories approach to quantum cosmology. The study we presented was a starting point to a much larger question: "What is the evolution of the universe?" Since the composition of manifolds is inherent in the formulation of Ponzano-Regge wavefunctions, we should be able to study the relationship between the volumes and geometries of different manifold boundaries. We can thereby observe the evolution of the spatial aspects of a 3-dimensional universe between different boundary slicings or times. Thus, the theory naturally lends itself to the study of the universe's dynamics. T h i s w i l l be the subject of future investigations. In review, I believe we can agree that our original goal was achieved—we have studied Chapter 6. Conclusion 125 a discrete theory of 3-dimensional gravity and have gained insight on the types of difficulties it presents and the sorts of solutions it offers to the field of quantum cosmology. Bibliography [1] L . C l a y p o o l , Primus: A n t i p o p , © S t u r g e o n ( B M I ) (Los Angeles, 1999). [2] J . B . Hartle, Quantum cosmology: Problems for the 21st century, [gr-qc/9701022]. [3] J. B . Hartle, Quasiclassical domains in a quantum universe, [gr-qc/9404017]. [4] J . B . Hartle, Space-time quantum mechanics and the quantum mechanics of spacetime, [gr-qc/9304006]. [5] M . G e l l - M a n n and J . B . Hartle, Phys. Rev. D47 (1993) 3345; Classical equations for quantum systems, [gr-qc/9210010]. [6] R Griffiths, J . Stat. Phys. 36 (1984) 219. [7] R . Ormnes, The Interpretation of Quantum Mechanics, (Princeton University Press, Princeton, 1994). [8] J . B . Hartle and D . Marolf, Phys. Rev. D56, 6247 (1997); Comparing formula- tions of generalized quantum mechanics for reparametrization-invariant systems, [gr- qc/9703021]. [9] F . Dowker and A . Kent, Phys. Rev. Lett. 75, 3038 (1995); Properties of consistent histories, [gr-qc/9409037]. [10] J . B . Hartle, Phys. Rev. D49, 6543 (1994); Unitarity and causality in generalized quantum mechanics for nonchronal space-times, [gr-qc/9309012]. [11] J . Halliwell and A . Zoupas, Phys. Rev. D52, 7294 (1995); Quantum state diffu- sion, density matrix diagonalization and decoherent histories: ph/9503008]. A Model, [quant- [12] G . Ponzano and T . Regge, Semiclassical Limit of Racah Coefficients, Spectroscopic and G r o u p Theoretical Methods in Physics, edited by F . B l o c k (North-Holland, A m s t e r d a m , 1968). [13] T . Regge, Nuovo Cimento 19, 558 (1961). [14] V . Turaev and O. V i r o , Topology 31, 865 (1992). 126 Bibliography 127 [15] H . Ooguri, N u c l . Phys. B382, 276 (1992); Partition Changing Amplitudes Functions and Topology- in the 3D Lattice Gravity of Ponzano and Regge, [hep- th/9112072]. [16] J . Iwasaki, J . M a t h . Phys. 36, 6288 (1995); A Definition of the Ponzano-Regge Quantum Gravity Model in Terms of Surfaces, [gr-qc/9505043]. [17] J . W . Barrett and L . Crane, Class. Quant. G r a v . 14 2113 (1997); An Algebraic Interpretation of the Wheeler-deWitt Equation, [gr-qc/9609030]. [18] G . Racah, Physical Review 62, 438 (1942). [19] J . W . Barrett and T . J . Foxon, Class. Quant. G r a v . 11 543 (1994); Semi-classical Limits of Simplicial Quantum Gravity, [gr-qc/931006] [20] S. Davids, Semiclassical Limits of Extended Racah Coefficients, [gr-qc/9807061] [21] A . Cayley, Cambridge M a t h . J . 2, 267 (1841). Appendix Below we find the C programming language code for the function 6jfunction.c which evaluates the 6j-symbol of equation (2.10). 6jfunction.c was developed using the Gnu Multiple Precision Arithmetic Library version 2.O.2. 1 6jf unction, c reproduces the results of the Mathematica (version 4.0.1.0 and preceding version) computational software package for all tested cases. The test cases included: 2 (1) A l l 6j-symbols with ] x =j 2 =j 3 100, as well as X = 111, 122, 133, =j 4 =j 5 =j 6 = X for X = 0, 1, 2, 98, 99, 177, 188, 199, as well as X = 222, 333, 444, and 555; (2) A l l < lx > presented for the completely isotropic tetrahedron of Chapter 3; and, (3) A l l < l x >\ t presented for the subclass A tetrahedron of Chapter 4. Fur- thermore, it was found that 6jfunction, c produced results upwards of twelve times as fast as the commercial software package. The user of 6jfunction, c should, however, 3 note that no guarantee is placed on the function's reliability to accurately evaluate all 6j-symbols. In fact, it will likely be necessary to increase the minimum precision of the 4 Gnu Multiple Precision variables when calculation involves factorials much larger than those encountered in the test cases. A t the time of publication, documentation for the Gnu Multiple Precision Arithmetic Library was available from the G N U Project web server at world wide web address http://www.gnu.org/. Also at time of publication, the latest version of the M P library was available by anonymous ftp from 'prep.ai.mit.edu'—the file name was '/pub/gnu/gmp-M.N.tar.gz'. C a u t i o n should be exercised when using Mathematica in Ponzano-Regge theory, since the 6j-function of the commercial package is defined differently than in equation (2.10)—it is apparent that Mathematica uses different admissibility conditions and will, in fact, give non-zero results for some vanishing PonzanoRegge 6j-symbols. I n fact, it was not uncommon for 6 j f u n c t i o n . c to yield results twenty times faster than Mathematica. As well, the relative time saving tended to increase with the size of the 6j-symbol arguments. Obviously, the author of 6 j f u n c t i o n . c provides no performance guarantees for the function, and all users should note that they assume all risks when using 6 j f u n c t i o n . c . : 2 3 4 128 Appendix 129 The 6 j f u n c t i o n . h header file: /* 6 j f u n c t i o n . h , the header f i l e f o r 6 j f u n c t i o n . c */ double s i x j f u n c t i o n ( d o u b l e j j l , double j j 2, double j j 3 , double j j 4 , double j j 5 , double j j 6 ) ; /* End of 6 j f u n c t i o n . h . */ The 6jfunction, c code: /* Using GMP ( v e r s i o n 2.0.2) v a r i a b l e s t h i s program c a l c u l a t e s the 6 j — symbol and r e t u r n s a type double. The r e s u l t s are e x a c t l y the same (to the d i s p l a y e d p r e c i s i o n ) as those of Mathematica f o r a l l t e s t e d cases. The t e s t cases i n c l u d e d : (1) A l l N[SixJSymbol[{X,X,X},{X,X,X}]] with X=0, 1,2,...98,99,100, X=lll,122,133,...,177,188,199, as w e l l as X=222,333,444, 555; (2) Expectation value of the X edge l e n g t h of the [{X,X,X},{X,X,X}] tetrahedron f o r K=0,1,2,...200; (3) E x p e c t a t i o n values of the X edge l e n g t h of the [{X,X,X},{Y,Y,Y}] tetrahedron f o r a l l K f o r Y=10, 20, 30, ...,80,90,100. */ /* L i s t l i b r a r y dependencies and d e f i n e g l o b a l constants. */ #include <stdio.h> #include <math.h> #include "gmp.h" #include " 6 j f u n c t i o n . h " #define NMAX 14 #define DMAX 18 double s i x j f u n c t i o n ( d o u b l e j j l , double jj5,double double j j 2 , double j j 3 , double j j 4 , jj6) { /* L i s t f u n c t i o n prototypes. */ double sixjsymbol(double, /* Declare double double, double, double, double, double); s i x j v a l u e and i n i t i a l i z e t o 0. */ sixjvalue=0; /•Tests t o see i f the j - v a l u e s are "admissible". I f " i n a d m i s s i b l e " , then Appendix 130 r e t u r n s a v a l u e of z e r o . I f " a d m i s s i b l e " , then c a l c u l a t e s the 6 j symbol v a l u e and r e t u r n s the f l o a t i n g p o i n t v a l u e . * / i f ( ( ( ( j j l + j j 2 + j j 3 ) - f l o o r ( j j l + j j 2 + j j 3 ) ) < Q . 0 0 1 II ( ( j j 1 + j j 2 + j j 3 ) f l o o r ( j j l + j j 2 + j j 3 ) ) > 0 . 9 9 9 ) && ( ( ( j j 3 + j j 4 + j j 5 ) - f l o o r ( j j 3 + j j 4 + j j 5 ) ) < 0.001 | | ( ( j j 3 + j j 4 + j j 5 ) - f l o o r ( j j 3 + j j 4 + j j 5 ) ) > 0 . 9 9 9 ) && ( ( ( j j 5 + j j 6 + jjl)-floor(jj5+jj6+jjl))<0.001 I I ((jj5+jj6+jjl)-floor(jj5+jJ6+ j j l ) ) > 0 . 9 9 9 ) && ( ( ( J j 2 + j j 4 + j j 6 ) - f l o o r ( j j 2 + j j 4 + j j 6 ) ) < 0 . 0 0 1 II C(jj2+ J j 4 + j j 6 ) - f l o o r ( j j 2 + j j 4 + j j 6 ) ) > 0 . 9 9 9 ) && ( ( ( j j 1 + j j 2 + j j 4 + j j 5 ) floor(jjl+jj2+jj4+jj5))<0.001 I I ((jjl+jj2+jj4+jj5)-floor(jjl+jJ2+ j j 4 + j j 5 ) ) > 0 . 9 9 9 ) && ( ( ( j j 2 + j j 3 + j j 5 + j j 6 ) - f l o o r ( j j 2 + j j 3 + j j 5 + j j 6 ) ) < 0.001 II ( ( J j 2 + j j 3 + j j 5 + j j 6 ) - f l o o r ( j j 2 + j j 3 + j j 5 + j j 6 ) ) > 0 . 9 9 9 ) && (((Jjl+jj3+jj4+jj6)-floor(jjl+jj3+jj4+jj6))<0.001 I I ( ( j j 1+j j3+j J4+ Jj6)-floor(jjl+jj3+jj4+jj6))>0.999)) { sixjvalue=sixjsymbol(jjl,jj2,jj3,jj4,jj5,jj6); return (sixjvalue); } else { sixjvalue=0; return (sixjvalue); } } / * L i s t i n g of f u n c t i o n s i x j s y m b o l whose prototype i s g i v e n above. * / double s i x j s y m b o l ( d o u b l e J l , double J 2 , double J 3 , double J 4 , double J 5 , double J6) / * Returns value f o r the 6j-symbol whose argument i s the ordered set { { J l , J 2 , J 3 } , { J 4 , J 5 , J 6 } } . */ { /* L i s t function prototypes. */ l o n g longcompare(const v o i d * p o i n t l , const v o i d * p o i n t 2 ) ; / * D e c l a r e and i n i t i a l i z e v a r i a b l e s . * / double zminl=0; double zmin2=0; l o n g zmin3=0; l o n g zmin3factor=0; double sixjsum=0; double DoubleNum[NMAX]; 131 Appendix double DoubleDen[DMAX] ; i n t Ncount=0; i n t Dcount=0; long RoundNum[NMAX]; long RoundDen[DMAX] ; long OrderedRoundNum[NMAX]; long OrderedRoundDen[DMAX]; unsigned long UnsignedLongl=0; i n t Fcount=0; long Z1=0; long Z2=0; i n t FFcount=0; /* Declare a l l GMP v a r i a b l e s . */ mpf_t mpzmin3factor; mpf_t s i x j s u m f a c t o r l ; mpf_t sixjsumfactor2; mpf_t FinalArray[DMAX]; mpf_t Intermediate1; mpf_t Unity; mpz_t mpArray[12]; mpz_t mpArraylndexCount; mpz_t mpArrayValue; mpz_t mpFinalArrayValue; 'mpf_t mpFloatFinalArrayValue; mpf_t mpInverseFinalArrayValue; /* I n i t i a l i z e a l l GMP v a r i a b l e s . */ mpf_init2 (mpzmin3factor, 256); mpf_init2 ( s i x j s u m f a c t o r l , 512); mpf_init2 (sixjsumfactor2, 512); f o r (FFcount=0; FFcount<DMAX; FFcount++) { mpf_init2 ( F i n a l A r r a y [ F F c o u n t ] , 512); } mpf_init2 (Intermediate1, 512); mpf_init2 (Unity, 256); f o r (FFcount=0; FFcount<12; FFcount++) { mpz_init (mpArray[FFcount]); } mpz_init (mpArraylndexCount); 132 Appendix mpz_init (mpArrayValue); mpz_init mpf _ i n i t 2 (mpFinalArrayValue); (mpFloatFinalArrayValue., mpf_init2 (mpInverseFinalArrayValue, 512) ; 512); /* Set GMP v a r i a b l e s Unity and s i x j s u m f a c t o r . */ mpf_set_si (Unity, 1); mpf_set_si ( s i x j s u m f a c t o r 2 , 0 ) ; /•Makes sure a l l the f a c t o r i a l s while(zminl-Jl-J2-J3<0 are d e f i n e d . * / I I zminl-J3-J4-J5<0 || zminl-J5-J6-Jl<0 || z m i n l - J2-J4-J6<0) { zmin2=++zminl; } /•Executes c o n d i t i o n a l loop which evaluates the sum over i n t e g e r z i n the s i x - j symbol as w e l l as the d e l t a ( n l , n 2 , n 3 ) ' s . * / if(Jl+J2+J4+J5-zmin2>=0 && J2+J3+J5+J6-zmin2>=0 && Jl+J3+J4+J6-zmin2>=0) { while(Jl+J2+J4+J5-zmin2>=0 && J2+J3+J5+J6-zmin2>=0 && J1+J3+J4+J6zmin2>=0) { zmin3=zmin2++; /* F i r s t d e f i n e the elements of DoubleNum and DoubleDen. */ DoubleNum[0]=Jl+J2-J3; DoubleNum[1]=J1+J3-J2; DoubleNum[2]=J2+J3-J1; DoubleNum[3]=J3+J4-J5; DoubleNum[4]=J3+J5-J4; DoubleNum[5]=J4+J5-J3; DoubleNum[6]=J5+J6-J1; DoubleNum[7]=J5+J1-J6; DoubleNum[8]=J6+J1-J5; DoubleNum[9]=J2+J4-J6; DoubleNum[10]=J2+J6-J4; DoubleNum[11]=J4+J6-J2; DoubleNum[12]=zmin3+l; DoubleNum[13]=zmin3+l; DoubleDen[0]=J1+J2+J3+1; DoubleDen[1]=J3+J4+J5+1; DoubleDen[2]=J5+J6+Jl+1; Appendix 133 DoubleDen[3]=J2+J4+J6+1; DoubleDen[4]=J1+J2+J4+J5-zmin3; DoubleDen[5]=J1+J2+J4+J5-zmin3; DoubleDen[6]=J2+J3+J5+J6-zmin3; DoubleDen[7]=J2+J3+J5+J6-zmin3; DoubleDen[8]=J1+J3+J4+J6-zrain3; DoubleDen[9]=J1+J3+J4+J6-zmin3; DoubleDen[10]=zmin3-Jl-J2-J3; DoubleDen[11]=zmin3-Jl-J2-J3; DoubleDen[12]=zmin3-J3-J4-J5; DoubleDen[13]=zmin3-J3-J4-J5; DoubleDen[14]=zmin3-J5-J6-Jl; DoubleDen[15]=zmin3-J5-J6-Jl; DoubleDen[16]=zmin3-J2-J4-J6; DoubleDen[17]=zmin3-J2-J4-J6; /* Takes the array DoubleNum and rounds i t s elements t o the nearest i n t e g e r s , then puts the elements i n descending order. */ for(Ncount=0; Ncount<NMAX; Ncount++) { /* Round o f f the array of elements. */ RoundNum[Ncount]=floor(DoubleNum[Ncount] if(DoubleNum[Ncount]-RoundNum[Ncount] ); <0.5) { OrderedRoundNum[Ncount]=RoundNum[Ncount] ; > else { OrderedRoundNum[Ncount]=RoundNum[Ncount]+1; > } /* Sort the array RoundNum i n descending order. */ qsort(OrderedRoundNum, NMAX, sizeof(OrderedRoundNum[0]), longcompare); /* Takes the array DoubleDen and rounds i t s elements t o the nearest i n t e g e r s , then puts the elements i n descending order. */ for(Dcount=0; Dcount<DMAX; Dcount++) { /* Round o f f the a r r a y of elements. */ RoundDen[Dcount]=floor(DoubleDen[Dcount]); 134 Appendix if(DoubleDen[Dcount]-RoundDen[Dcount]<0.5) { OrderedRoundDen[Dcount]=RoundDen[Dcount]; > else { OrderedRoundDen[Dcount]=RoundDen[Dcount]+1; > > /* Sort the a r r a y RoundDen i n descending order. */ qsort(OrderedRoundDen, DMAX, sizeof(OrderedRoundDen[0]), longcompare); /* Computes f i r s t NMAX elements of F i n a l A r r a y : Takes r a t i o of f a c t o r i a l s of f i r s t NMAX elements of OrderedRoundNum and OrderedRoundDen. */ for(Fcount=0; Fcount<NMAX; { Fcount++) if(OrderedRoundNum[Fcount]==OrderedRoundDen[Fcount]) { mpf_set (FinalArray[Fcount], Unity); > else if(OrderedRoundNum[Fcount]>OrderedRoundDen[Fcount]) { Z1=0; mpf_set (Intermediate1, U n i t y ) ; while(OrderedRoundNum[Fcount] -Zl> OrderedRoundDen[Fcount]) { Z2=Z1++; UnsignedLongl=(OrderedRoundNum[Fcount]-Z2); mpf_mul_ui (Intermediate1, Intermediate 1, UnsignedLongl); } m p f _ s e t ( F i n a l A r r a y [Fcount], Intermediate1); } else { Z1=0; mpf_set (Intermediate1, U n i t y ) ; while(OrderedRoundDen[Fcount]-Zl> OrderedRoundNum[Fcount]) { Appendix 135 Z2=Z1++; UnsignedLongl=OrderedRoundDen[Fcount] -Z2; mpf_mul_ui (Intermediate1, I n t e r m e d i a t e l , UnsignedLongl); } mpf_div ( I n t e r m e d i a t e l , Unity, > > mpf_set(FinalArray[Fcount], Intermediatel); Intermediatel); /* Computes the remaining DMAX-NMAX elements of F i n a l A r r a y : Takes inverse of f a c t o r i a l of the f i n a l DMAX-NMAX elements of OrderedRoundDen. */ for(Fcount=NMAX; Fcount<DMAX; Fcount++) { /* Loop which acts as a f u n c t i o n f o r the f a c t o r i a l : Returns the value (OrderedRoundDen[Fcount])! as a m u l t i p l e p r e c i s i o n integer.*/ /* Declare and i n i t i a l i z e v a r i a b l e s . */ long BiggestArrayIndex=0; long OrderedRoundDenValue=0; /* Set GMP v a r i a b l e s mpArray. */ mpz_ set. . s i (mpArray[0], 1); mpz_ set. . s i (mpArray[1] , l ) ; mpz_ set. . s i (mpArray[2], 2); mpz_ set. . s i mpz_ set. . s i mpz_ set. . s i mpz_ set. . s i (mpArray[3], 6); (mpArray[4] , 24); (mpArray[5] , 120); (mpArray[6] , 720) ; mpz_ set. . s i (mpArray[7], 5040); mpz_ .set. . s i (mpArray[8], 40320); mpz_ .set. . s i (mpArray[9], 362880); mpz_ .set. . s i (mpArray[10] , 3628800); mpz_ .set. . s i (mpArray[11] , 39916800) 0rderedRoundDenValue=0rderedRoundDen[Fcount]; /* I f OrderedRoundDenValue i s 0,1,...,10,11 then take f a c t o r i a l from mpArray v a r i a b l e s . */ if (OrderedRoundDenValue { ==0) mpz_set (mpFinalArrayValue, mpArray[0] ); > e l s e i f (OrderedRoundDenValue == 1) { mpz_set (mpFinalArrayValue, m p A r r a y [ l ] ) ; } e l s e i f (OrderedRoundDenValue == 2) { mpz_set (mpFinalArrayValue, mpArray[2]); > e l s e i f (OrderedRoundDenValue == 3) { mpz_set (mpFinalArrayValue, mpArray[3]); > e l s e i f (OrderedRoundDenValue == 4) { mpz_set (mpFinalArrayValue, mpArray[4]); } e l s e i f (OrderedRoundDenValue == 5) { mpz_set (mpFinalArrayValue, mpArray[5]); > e l s e i f (OrderedRoundDenValue == 6) { mpz_set (mpFinalArrayValue, mpArray[6]); } e l s e i f (OrderedRoundDenValue == 7) { mpz_set (mpFinalArrayValue, mpArray[7]); } e l s e i f (OrderedRoundDenValue == 8) { mpz_set (mpFinalArrayValue, mpArray[8]); } e l s e i f (OrderedRoundDenValue == 9) { mpz_set (mpFinalArrayValue, mpArray[9]); > e l s e i f (OrderedRoundDenValue == 10) { mpz_set (mpFinalArrayValue, mpArray[10]) } e l s e i f (OrderedRoundDenValue == 11) 137 Appendix { mpz_set (mpFinalArrayValue, mpArray[11]); > /* I f OrderedRoundDenValue > 11 then c a l c u l a t e the f a c t o r i a l with a loop. */ else { BiggestArrayIndex=ll; mpz_set (mpArrayValue, mpArray[11]); while (BiggestArraylndex < OrderedRoundDenValue) { ++BiggestArrayIndex; mpz_set_si (mpArraylndexCount, BiggestArraylndex); mpz_mul (mpArrayValue, mpArrayValue, mpArraylndexCount); } mpz_set (mpFinalArrayValue, mpArrayValue); > mpf_set_z (mpFloatFinalArrayValue, mpFinalArrayValue); mpf_div (mpInverseFinalArrayValue, U n i t y , mpFloatFinalArrayValue); mpf_set ( F i n a l A r r a y [ F c o u n t ] , mpInverseFinalArrayValue); > /* Computes sixjsum from the elements of F i n a l A r r a y . */ mpf_set ( s i x j s u m f a c t o r l , U n i t y ) ; for(Fcount=0; Fcount<DMAX; Fcount++) { mpf_mul ( s i x j s u m f a c t o r l , FinalArray[Fcount]); sixjsumfactorl, } mpf_sqrt ( s i x j s u m f a c t o r l , s i x j s u m f a c t o r l ) ; zmin3factor=pow(-l, zmin3); mpf_set_si (mpzmin3factor, zmin3factor); mpf_mul ( s i x j s u m f a c t o r l , mpzmin3factor, s i x j s u m f a c t o r l ) ; mpf_add ( s i x j s u m f a c t o r 2 , s i x j s u m f a c t o r 2 , s i x j s u m f a c t o r l ) ; sixjsum=mpf_get_d ( s i x j s u m f a c t o r 2 ) ; > } else { Appendix 138 sixjsum=0; } / * C l e a r a l l GMP v a r i a b l e s . mpf_clear (mpzmin3factor); mpf_clear (sixjsumfactorl); mpf_clear (sixjsumfactor2); for (Fcount=0; */ Fcount<DMAX; Fcount++) { mpf_clear ( F i n a l A r r a y [ F c o u n t ] ) ; } mpf_clear (Intermediate1); mpf_clear ( U n i t y ) ; mpz_clear (mpArrayIndexCount); mpz_clear (mpArrayValue); mpz_clear (mpFinalArrayValue); mpf_clear ( m p F l o a t F i n a l A r r a y V a l u e ) ; mpf_clear (mpInverseFinalArrayValue); f o r (FFcount=0; FFcount<12; FFcount++) { mpz_clear (mpArray[FFcount]); > r e t u r n sixjsum; / * L i s t i n g of f u n c t i o n longcompare (compares s i z e s of two type long v a r i a b l e s , and r e p o r t s the r e s u l t to qsort) whose prototype i s l i s t e d function sixjsymbol. * / long longcompare(const v o i d * p o i n t l , const v o i d *point2) { r e t u r n ( - *(long * ) p o i n t l + * ( i n t } *)point2); in
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Conditional probabilities in the quantum cosmology of Ponzano-Regge theory Petryk, Roman J.W. 1999
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Title | Conditional probabilities in the quantum cosmology of Ponzano-Regge theory |
Creator |
Petryk, Roman J.W. |
Date Issued | 1999 |
Description | We examine the discrete Ponzano-Regge formulation of (2+1)-dimensional gravity in the context of a consistent histories approach to quantum cosmology. We consider 2- dimensional boundaries of a 3-dimensional spacetime. The 2-dimensional boundaries are tessellated as the surface of a single tetrahedron. Two classes of the tetrahedral tessellation are considered—the completely isotropic tetrahedron and the two-parameter anisotropic tetrahedron. Using Ponzano-Regge wavefunctions, we calculate expectation values and uncertainties for the edge lengths of these tetrahedra. In doing so, we observe finite size effects in the expectation values and uncertainties when the calculations fail to constrain the space of histories accessible to the system. There is, however, an indication that the geometries of the tetrahedra (as quantified by the ratios of their edge lengths) freeze out with increasing cutoff. Conversely, cutoff invariance is observed in our calculations provided the space of histories is constrained by an appropriate fixing of the tetrahedral edge lengths. It is thus suggested that physically meaningful results regarding the early state of our universe can be obtained providing we formulate the problem in a careful manner. A few of the difficulties inherent in quantum cosmology are thereby addressed in this study of an exactly calculable theory. |
Extent | 5327655 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-07-07 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0089411 |
URI | http://hdl.handle.net/2429/10324 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2000-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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