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Conditional probabilities in the quantum cosmology of Ponzano-Regge theory Petryk, Roman J.W. 1999

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C O N D I T I O N A L PROBABILITIES 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 FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF SCIENCE in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS AND ASTRONOMY 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 in partial fulfillment of the requirements for an advanced degree at the University of Br i t i sh Columbia , I agree that the Libra ry 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 Astronomy The University of Br i t i sh Columbia 6224 Agricul tural Road Vancouver, B . C . , Canada V 6 T 1Z1 Date: Abstract We examine the discrete Ponzano-Regge formulation of (2+l)-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 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 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 wi th 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. i i 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 7 2.1 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 . 12 2.2 The Ponzano-Regge Partition Function and Wavefunction 14 2.2.1 Tessellating Manifolds MB and Their Boundaries dMB 14 2.2.2 The Ponzano-Regge Partition Function 16 2.2.3 Ponzano-Regge Wavefunction (for Manifolds with Boundary) . . . 17 iii 3 The Isotropic Class of Tetrahedron 20 3.1 The Isotropic Wavefunction 20 3.2 Isotropic Expectat ion Values < lx > and Uncertainties Alx 23 3.3 Semiclassical Isotropic Analysis 31 3.3.1 Semiclassical Isotropic Expectation Values < Ix >sc and Uncer-tainties AlxSc 31 4 Subclass A Two-parameter Anisotropic Calculations 37 4.1 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 < lY > and Uncertainties AlY • • 47 4.1.4 Subclass A < lx > and Alx along contours of lY and K 55 4.1.5 Curvature Expectation Values < 0 > 67 4.1.6 Subclass A < lY > and AlY A long Contours of lx and K 80 5 Subclass E Two-parameter Anisotropic Calculations 90 5.1 The Subclass E Wavefunction 91 5.1.1 Subclass E Expectation Values < lx > and Uncertainties Alx • • 94 5.1.2 Subclass E Expectation Values < lY > and Uncertainties AlY . . 95 5.1.3 Subclass E < lx > and Alx along contours of lY and K 99 5.1.4 Subclass E < lY > and AlY A long Contours of lx and K 113 5.2 Comparison of Subclass A and E Results 118 6 Conclusion 123 Bibliography 126 iv A p p e n d i x 128 v List o f Tables Two-parameter anisotropic subclasses vi Lis t of Figures 2.1 A representation of the 6j-symbol—right- and left-handed labelling of the tetrahedron t • 10 2.2 The tetrahedron and semiclassical 6j-symbol 13 2.3 Tessellating the boundary of MB 15 2.4 Composit ion of manifolds MB and NB wi th common boundary dMB = dNB. 19 3.1 Tessellation of the single completely isotropic boundary 22 3.2 Isotropic * and | * | 2 as functions of K and X 24 3.3 Isotropic ^ and | ^ | 2 as functions of X (A-dependence suppressed). . . . 25 3.4 Isotropic < l\- > and Alx as functions of K 28 3.5 < lx >sc and AlXsc as functions of K 33 3.6 Comparing exact and semiclassical isotropic < lx > and Alx as functions otK : 35 4.1 Tessellation of the single ellipsoidal anisotropic boundary 39 4.2 Subclass A * and | * | 2 as functions of X and Y for K=20 42 4.3 Subclass A * and | * | 2 as functions of K and X for y = 2 0 43 4.4 Subclass A \& and |vl>|2 as functions of K and Y for A r =20 44 4.5 Subclass A ^ as a function of X and Y for #=20. 45 4.6 Subclass A < lx > and Alx as a functions of K 48 4.7 Subclass A as a function of A" 49 <lx> 4.8 Subclass A <lY > and Aly as functions of i f 51 vii 4.9 Subclass A - g * - as a function of K 52 4.10 Subclass A as a function of K 53 4.11 Subclass A %*- as a function of K 54 4.12 < lx > \i as a function of K for contours of fixed ly. . . 58 4.13 Alx\iY as a function of K for contours of fixed ly 59 4.14 as a function of K for Zy=100.5 60 4.15 < lx >\K as a function of Zy 63 4.16 Alx\K as a function of ly 64 4.17 Alx\K as a function of ly 66 4.18 - ^ ^ f - as a function of ly for AT=100 68 4.19 < /x > and Alx as functions of ly and 69 4.20 Angles at the vertices of the tetrahedron 70 4.21 < 9Yy > \ K as a function of ly 72 4.22 < QXy > \ K as a function of lY 73 4.23 < Oyy > and < 9Xy > as functions of lY and K 74 4.24 A9YY\K as a function of ly 76 4.25 A6Xy\K as a function of ly 77 4.26 < Qyy > \ K and < QXy > \ K a s functions of ly for K=100 78 4.27 A 0 y y | ^ and AQXy\K as functions of lY for 7^=100 79 4.28 < Qyy >max\K a n < ^ < >rnin\K a S functions of ly for K=100 81 4.29 < QXY >max\K a n ^ < 6 A T > m m l i c as functions of lY for 7C=100 82 4.30 <lY >\, and A / y l , as functions of K 84 4.31 Relative uncertainty < ^ ^ ' A " as a function of 7T 85 4.32 < ly > \ K and AlY\K as functions of lx 86 4.33 Relative uncertainty <l*>\ a s a m n c t i ° n °f lx 88 4.34 < ly > \ K and A / y | A , as functions of lx and K 89 v i i i 5.1 Tessellation of the single anisotropic ovoidal boundary 92 5.2 Subclass E and A two-parameter anisotropic 6j-symbols as functions of X and Y 93 5.3 Subclass E < lx > and Alx as a functions of i f 96 5.4 Subclass E as a function oi K 97 5.5 Subclass E < lY > and AlY as functions of K 98 5.6 Subclass E as a function oi K 100 < J y > 5.7 Subclass E as a function of i f 101 < « y > 5.8 Subclass E as a function of i f 102 5.9 < lx > \i as a function of i f for contours of fixed lY 104 5.10 Alx\iY a s a function of i f for contours of fixed lY 105 5.11 as a function of i f for Zy=100.5 106 <lX>\ly 5.12 < lx >\K as a function of lY 108 5.13 Alx\K as a function of lY 109 5.14 as a function of lY for i f=100 I l l 5.15 < lx > and Alx as functions of lY and i f 112 5.16 < lY > I, and AlY\, as functions of i f 114 1 'lx Hx 5.17 Relative uncertainty < / y ^ ' | X as a function of i f 115 5.18 < lY > \ K and as a function of lx 116 5.19 AlY\K as a function oi lx 117 5.20 Relative uncertaintv j^fyJf as a function of lx 119 " < l Y > \ K 5.21 < lY > \ K and AlY\K as functions of lx and K 120 ix Acknowledgements I send out my deepest thanks to everyone without whose aid, patience, and caring I could not have achieved al l that I did. In particular, I thank my thesis advisor Kr i s t i n Schleich for al l her assistance, helpful cri t icism, and insightful commentary. Wi thout her editorial critique of its many drafts, this thesis would have been far less than what it has become. Our many hours of conver-sation were both 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 Don W i t t and Matthew Choptuik for their assistance and advice on many issues, particularly computer hardware and programming. As well, I owe much gratitude to Gordon 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 Astronomy of the University of Br i t i sh Columbia . It was very pleasant interacting and working wi th all of them. A n d not least, I owe much to my loving wife and family. I thank my wife Andrea 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 Mary (and Mihaj lo) , father B i l l , brother Michael (also known as C . G . ) , and grandmother Mar tha for al 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 in 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 in physics, we would conclude that the present state 1The Schrodinger wave equation and classical Einstein equation are two such dynamical physical laws. 2We 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 Chapter 1. Introduction 2 of the universe should be explicated in terms of both its in i t ia 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 ini t ia l 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 ini t ia l state of the universe. This 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. This is one of the central problems of quantum cosmology. It has been suggested (cf. [2]) that the resolution of this quantum to classical cosmo-logical problem wi l l likely involve answers to three questions: 1) Wha t is the true theory of gravitational dynamics? 2) Wha t are the ini t ial conditions for the wavefunction of the universe? 3) How does one extract classical or quasi-classical behaviour from a quantum me-chanical system without external observers—in fact, is it even meaningful 3 to pose 3 This problem is in some sense tantamount to the philosophical question of whether one's own mind is capable of understanding itself. Chapter 1. Introduction 3 this question? A particularly insightful discussion in response to the third question is to be found in [3]. Furthermore, Hartle [4], Ge l l -Mann [5], Griffiths [6] and Ormnes [7] have proposed an interpretation of quantum mechanics applicable to closed systems without external observers. This interpretation is based in the consistent histories approach to quantum mechanics. The consistent histories (or decoherent histories) approach can be understood as fol-lows. 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 ini t ia l quantum state of the system. Each such sequence or history is associated with the probability of the resulting state. 4 If the overlap in probabilities of the set of final states is small, they are said to be decoherent or consistent. However, not all sets of histories wi l l decohere—whether they do or don't wi l l depend on the ini t ia l 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 in a classical manner. 3 The consistent histories approach thereby provides a solution to the third question of our list (i.e., how classical behaviour arises wi thin 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 ini t ia l 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 4This probability corresponds to the modulus squared of the resulting state amplitude. 5Remember 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. 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 in the quantum to classical cosmological problem in 3-dimensional gravity. Ultimately, our goal is to formulate 4-dimensional gravity. However, many issues in defining the quantities of the theory are not resolved. We are not even sure of which theory of gravitational dynamics to use. The Ponzano-Regge theory of gravity is 3-dimensional not 4-dimensional. However, unlike 4-dimensional gravity, Ponzano-Regge is a completely specified theory. The issues of the formulation of in i t ia 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. This thesis wi l l study these issues. Firstly, however, we note that Ponzano-Regge gravity is formulated in terms of the l imi t of a cutoff theory—this distinguishes it from the Turaev-Viro formulation of 3-manifold invariants [14]. Clearly, then, one must try to understand the nature of this l imi t in the computation of quantum amplitudes. We wi 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. Introduction 5 1.2 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 Wheeler-de 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 The Structure of the W h a t Remains 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), 6Their study also revealed a tessellation independent property of the system. The Ponzano-Regge theory is, however, not rigorously a topological field theory. Chapter 1. Introduction 6 Z[M], 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 Chap-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 in two measured quantities: the expectation value of tetrahedral edge lengths < lx > and the uncertainty of the tetrahedral edge lengths AlX- Chapter 2 we also evaluate the expectation values and uncertainties in the semiclassical l imit 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 Alx 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 con-tains 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 in as lucid yet succinct a manner as possible. To this end, equations have been presented using the most appro-priate notation. Al though the notational conventions of no particular work referenced in the study wi l l be strictly adhered to, the reader wi 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. And so we begin... 2.1 The 6j-symbol 2.1.1 Defining and Representing the 6j-symbol Let ji, J2) h i Hi J5i J6 be non-negative integers or half integers. An unordered 3-tuple of this set of jf-values is written as The unordered 3-tuple (2.1) is then said to be admissible if the triangular inequalities (ja,jb,jc) where a,b,c = 1,2, ...,6 and a^b^c. (2.1) life - J c l < 3 a < jb+jc (2.2) are met and the sum over j-values 3a + 3b + 3c (2.3) 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 { h , k , k ) (2.5) 7 Chapter 2. Mathematical and Physical Preliminaries 8 are admissible. We are automatically guaranteed integer values for the sums « 5 = fa + fa + fa + fa, n6 = 32 + fa + 35 + fa, ™7 = fa + fa + fa + fa (2.6) if the four 3-tuples of equation (2.5) are admissible. Admiss ib i l i ty furthermore guarantees that ng<nh, 0 = 1,2,3,4, h = 5,6,7, (2.7) where ng are given by ni = fa + fa + fa n2 = fa + fa + fa, n 3 = fa + fa + fa, n4 = fa + fa + fa, (2.8) and n-h are given by equation (2.6). We can now associate a 6j-symbol f fa fa fa ) (2-9) I fa 35 fa J 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 explicit ly given (cf. [12], [14], [18]) as fa fa fa ) v_ \=AYJB{z) (2.10) 34 fa fa ) where A = [A(fa,fa,fa)A(fa,fa,fa)A{fa,fa,fa)A(j2,fa,fa)}K (2.11) B{z) = ( - i r(z+ l)![(z-n 1)!(^-n 2)!^-n 3)!(z - n 4 ) ! ( n 5-z)!(n 6-^!(n 7- 2)!]- 1, (2.12) and the sum is over al l non-negative integer values of z resulting in non-negative factorial arguments. A(ji,jj,jk) is here denned according to A/- • • \ (fa + 3j ~ Jk)KJi + fa ~ jjV-tij + 3k ~ , . A ( * ' ' " ^ = ^ T ^ T i ) ! ( 2 - 1 3 ) 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 in discussions that follow. There is a natural geometric representation for the 6j-symbol—it is the standard 3-dimensional 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 j2 and j 3 . Next find the edge which shares no vertices with ji and label it j\. Then find the edges which share no vertices with j2 and jz and respectively label them j5 and j6. One could at this point ask which edge to label j2 and which edge to label j 3 once we have chosen j\—there are 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. (2.15) 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 Although all diagrams wi l l show the edges of tetrahedra labelled in terms of the j-values, it is important to note that the lengths of these edges associated with the assigned j-values are given by 2 k=3i + \ , i = l , 2 , . . . , 6 . (2.16) It is useful to study the implications of the triangular inequalities in relation to t and its edge lengths. Examining the geometry of t, we see that equation (2.2) simply guarantees that the edges li, l2, Is form a closed triangle of non-zero surface area (i.e., if \ji — j2\ < 33 5- ji + 3 2 then |/i — l2\ < l3 < l\ 4-12). Similarly, the triangular inequalities of 3-tuples (J3, ji, 3 s ) , ( J 5 , 3 6 , 3 i ) , and (j2,J4,je) respectively guarantee the remaining three faces form triangles of non-zero surface area. That is, admissibility guarantees that the edges ji, 3 2 , •••) je form a closed tetrahedron of non-zero volume V. The triangular'inequalities do not, however, guarantee that the associated tetrahedron has real positive volume (see Subsection 2.1.2 and Chapter 4). Without additional restriction, it is possible to construct hyperflat (V2 < 0) tetrahedra. This occurs when the sum of angles between the three edges forming a vertex is greater than 2-7T. Clearly, the V2 > 0 tetrahedra can be embedded in a 3-dimensional Euclidean space, while the V2 < 0 tetrahedra cannot. There is, however, an interpretation by which the V2 < 0 tetrahedra can be thought of as being embeddable in a 3-dimensional Lorentzian space (cf. [19], [20]). We wi l l therefore respectively refer to the V2 > 0 and V2 < 0 tetrahedra as Euclidean and Lorentzian. Occasionally, we wi l l in this sense refer to the Euclidean or Lorentzian regimes of the tetrahedra. 2The 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. Chapter 2. Mathematical and Physical Preliminaries 12 2.1.2 Quantum Gravity and the Semiclassical Limit of the 6j-symbols 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 in the semiclassical form of the 6j-symbol. Consider the case where a particular tetrahedron in the given triangulation has edge lengths given by equations (2.1.1), as displayed in Figure 2.2. For sufficiently large ji, j2, j 6 , Ponzano and Regge [12] demonstrate the associated 6j-symbol is approximated by 3\ 32 33 34 3b J6 , s c 7 = - (go, + \)0, :7T (2.17) where 0$ is the interior angle between the outward normals of the two tetrahedral faces sharing the lfh edge, and where the square of the tetrahedral volume V is given by (cf. [12], [21]) V2 0 / 4 2 l52 l i 1 h 2 0 Z 3 2 l22 1 ii ii 0 li 1 l i l i l i 0 1 7 2 b2  l l 1 1 (2.18) 1 1 1 1 0 In order to obtain a meaningful (non-imaginary) result from equation (2.17) the tetra-hedral 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 V2. We now note that the Regge action of the tetrahedron in Figure 2.2 is given by [13] S Regge = '^J-i&i — ^2Ui + ^)^i-i=l i=l  1 (2.19) Chapter 2. Mathematical and Physical Preliminaries 13 Figure 2.2: The tetrahedron and semiclassical 6j-symbol. Chapter 2. Mathematical and Physical Preliminaries 14 —that is, the gravitational contribution at each edge of the tetrahedron is liQ{ = (ji + \)0i. For a complex of tetrahedra with n internal edges SRegge would be written SReg9e = J2Ui + lWi. (2.20) i=x 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 ia the W K B method) for the semiclassical approximation of the 6 j - symbol in the Lorentzian regime (V2 < 0). 2.2 The Ponzano-Regge Partit ion Function and Wavefunction 2.2.1 Tessellating Manifolds MB and Their Boundaries dMB. Before we begin discussion of Ponzano-Regge part i t ion functions and wavefunctions, it is crucial to understand that these theories are formulated in 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. This tessellation is not unique—different T can map the manifold to a single tetrahedron or to many tetrahedra. As well, the connectivity of tetrahedral vertices is not in general unique. That 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 wi l l also depend on the particular choice of T. Since we wi l l be primarily concerned with Ponzano-Regge wavefunctions, consider the tessellation of the 2-manifold boundary dMB as shown in Figure 2.3. The figure displays two distinct mappings of the closed manifold boundary 8MB. Ti is a mapping which leads to the crudest tessellation possible—8M B is mapped to a single tetrahedron t wi th edges labelled ji, j2, je- The map T2 Chapter 2. Mathematical and Physical Preliminaries 15 results in a much finer (and more precise) tetrahedral modeling of 8MB- Wh ich mapping T we choose wi l l depend on how closely we want to approximate the manifold wi th a tetrahedral mesh 3 . Figure 2.3: Tessellating the boundary of MB. 3It 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. Chapter 2. Mathematical and Physical Preliminaries 16 2.2.2 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 2 and S 3 , respectively. Also, let sm be the number of objects contained in set Sm. (For example, SQ is the number of vertices in 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 ith 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 bound-ary) is (cf. [12]) Z[M] = lim E A - ° f t ( - l ) ^ ( 2 j i + l) f[[tn] (2.21) <h i = l n=l where [tn] is given by [tn] = (_l)-0"l+j2+i3+i4+j5+j6) I Jl H ^ \ (2.22) U 4 J5 k where \ (2-23) J4 Jb 36 is the 6j-symbol for the nth 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 6-tuples instead of restricting ourselves to only admissible 6-tuples in Z[M]. Also notice Chapter 2. Mathematical and Physical Preliminaries 17 that since l im^-^ooA is 0(K3), Z[M] w i l l decay rapidly in K unless we eliminate A by normalizing the parti t ion function. Regge [13] demonstrated that the sum of contributions to SRegge from all tetrahedra in a tessellation approaches a value proportional to the action of Einstein gravity, X ( M ) , provided the number of edges and vertices in the tessellation becomes very large. That is, where R is the Riemann curvature scalar of M and dV is the volume element on M. Comparing (2.21) to (2.25) reveals that the parti t ion 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 simpli-cial geometry. The Ponzano-Regge parti t ion function thereby provides a precise (exact) formulation of 3-d gravity. 2.2.3 Ponzano-Regge Wavefunction (for Manifolds with Boundary) We have just seen that the Ponzano-Regge parti t ion function provides a means of per-forming calculations on a 3-dimensional manifold, so one may ask "Why should we bother to consider Ponzano-Regge wavefunctions?". The reasons are obvious. Suppose we be-gin with a (2+l)-dimensional spacetime manifold. If we perform a slicing in the time dimension on this manifold we wi l l end up wi th a foliation of spacelike surfaces. These spacelike surfaces wi l l be 2-manifold boundaries of the original (2+l)-dimensional space-time 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 t ime—a very useful result! (2.25) Chapter 2. Mathematical and Physical Preliminaries 18 For this purpose, we now consider a 3-manifold MB wi th a closed 2-manifold boundary 8MB- A s before, we tessellate the manifold and respectively denote the sets of vertices, edges, faces and tetrahedra o/f the boundary of T[MB] as So, Si, S2 and S 3 . We again let sm be the number of objects in the set Sm. B y analogy, T[8MB] is specifically the tetrahedral tessellation of the manifold boundary 8MB wi th fixed connectivity and number of tetrahedra. We denote the sets of vertices, edges, faces and tetrahedra in T[8MB] as B0, Bi, B2 and B3, respectively. We define bm to be the number of objects in set Bm (i.e., bo is the number of vertices in the manifold boundary, and 61 is the number of edges in the boundary). Now, we again let K (the cutoff) be a non-negative integer or half-integer. Furthermore, we let (j) D e a n admissible assignment of a non-negative integer or half-integer < K (i = 1, 2, bi) to the iih edge in Bx, and ji < K (i — h+l, bi+2, Si) to the ith edge in Si. (Notice that we begin our labelling of edges at the boundary with i = 1 and work our way to the interior once al l boundary edges are labelled.) We now state the proposed definition for the wavefunction of a manifold with bound-ary: Definition 2. The Ponzano-Regge wavefunction for a manifold MB with boundary 8MB is given by * [ M B , { j l } ] = ^ m o 5 : A - ^ ) n ( - i ) j i ( 2 j l + i ) ^ n ( - i ) 2 j ! ( 2 j , + i ) n (f> i=l i=bi+l n= l (2.26) wnere A and [tn] are defined as in (2.21) trough (2.24). This definition of ^ [ M B , {Ji}] is chosen in order to satisfy the composition law Z[M#N] = l im Yl *K[Mb, {Ji}]*K[NB, {Ji}], (2.27) °°{{Ji}\Ji<K} where W K [ M B , {Ji}] and ^K[NB, {Ji}] are respectively the wavefunctions on manifolds Chapter 2. Mathematical and Physical Preliminaries 19 MB and NB wi th common boundary dMB = dNB and identical boundary tessellation T[dMB] = T[dNB] before the l imit K —> oo is taken. The action of such a composition is represented in Figure 2.4. Figure 2.4: Composit ion of manifolds MB and NB wi th common boundary dMB = dNB. The wavefunction ^[MB, {J,}] is the starting point of our study. We wi l l want to begin by addressing the issue of how to find good physical characterizations of our space-time. In order to perform meaningful calculations on our (2+l)-dimensional spacetime manifold, we wi l l need to search for quantities which are invariant in K. If the quantities show dependence on K (otherwise known as finite size effects) they wi l l be physically meaningless due to the infinite l imit in 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 wi th the Ponzano-Regge wave-function is to evaluate expectation values and uncertainties for a 3-manifold MB wi th a single, completely isotropic, spacelike, 2-manifold boundary dMB (a 2-sphere). We wi 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 wi l l thus perform the simplest calculations on this, our simplest of systems, in order to gain general insight on the issues at hand. Perhaps then we wi l l be able to successfully address the problem for a far less t r iv ia l set of systems. 3.1 The Isotropic Wavefunction The single, completely isotropic boundary can be tessellated by the action of T as a single tetrahedron whose edges are all of identical length. That is, the faces of this tetrahedron are al 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 wi th 6 J -va lues—Ji , J 2 , J3, J4, J5 and J 6 —according to the conventions outlined in Section 2.2.3. Since the tetrahedron is to be made completely isotropic we fix J1 = J2 = J3 = JA = J5 = J6 = X. (3.1) 20 Chapter 3. The Isotropic Class of Tetrahedron 21 Then, according to equation (2.26), the non-normalized wavefunction 1 i is * [ M B , {X, X, X, X, X, X}} = l im ( V2 £ (2p + l ) 2 ( 2 X + 1) 3 V>=o,i,..,K j X X X 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 wi th the 6j-symbol is inadmissible unless X is integer for Ji = J2 = J3 — J4 = J5 = Je = X. 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. That is, it makes no difference whether we choose to include or exclude vanishing probability amplitudes |\&| 2 from the pending expectation value calculations. We could now investigate to see what cutoff invariant information is given by wave-function \I> or its corresponding probability amplitude by considering behaviour for many fixed, increasing values of K. From 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 second term increases as X 3 , and the third term is an oscillating function of X. 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. Expl ic i t calculation precisely exhibits these characteristics for the wavefunction for all ranges of K and X. Figure 3.2 displays * for the range 2 0 < K < 20, 0 < X < 20. The probability amplitude | \ l / | 2 exhibits similar behaviour (see Figure 3.2)—the exceptions 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 1 For 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. 2 Strict ly 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 | * | 2 over extended (disallowed) ranges of X. Also note that many values of * and |\t| 2 exceed the range of the plot—these regions appear as holes in the plotted surfaces. Chapter 3. The Isotropic Class of Tetrahedron 22 Je=X J1=X Figure 3.1: Tessellation of the single completely isotropic boundary. Chapter 3. The Isotropic Class of Tetrahedron 23 gives litt le information on the system under scrutiny. This is why the expectation values of lx = X + | and corresponding uncertainties must be investigated. In order to isolate the dependence in X we calculate \I/ and \^f\2 without the renor-malization factor A . The results for 0 < K < 20, 0 < X < 20 are shown 3 in Figure 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 as well. 3.2 Isotropic Expectation Values < lx > and Uncertainties Alx Since al 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 wi l l therefore be given by . Ef = 0 Q + ^MB[X,X,X,X,X,X}\2 X S^K hi, f P ' ^ ' where the explicit form of ^ MB is given by equation (3.2) and the sum in x runs over al l integer values 4 less than or equal to K. Note the term in the denominator—this is the normalization factor for the wavefunction. Since K must be allowed to take on all 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 in the calculation. B y computing this partial expectation value for many increasing fixed-# values, we wi l l be able to refine our fixed-# approximation to the exact Ponzano-Regge wavefunction. 3 A s above. 4Rigorously 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 with change in 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 in the numerator, but also in the normalization factor. As 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 < lx > would oscillate or perhaps increase with K, but this would be very speculative. A back-of-the-envelope calculation can be performed in the following manner. If we assume the 6j-symbol is constant when averaged over a sufficiently large range of X, we find K 7 <lx>~ ~ £ ~ K (3.5) f&dx K l 2 o in the l imit of large K. That is, < lx > w i l l likely exhibit linear dependence in K. B y performing the required computer-assisted calculations, we find that < lx > does in fact exhibit linearity in K. Figure 3.4 displays the results of calculation for 180 < K < 200. The solid line represents the central value of < lx >= (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 (R2) which is found to be R2 = .99995, indicating a firm linear correlation and suggesting that the choice of fitting function was appropriate. 5 Addit ionally, 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 5 The probability that the observed data with v = 199 degrees of freedom and R2 — .99995 could have come from an uncorrelated parent population is of the order 1 0 - 4 3 0 . Chapter 3. The Isotropic Class of Tetrahedron 27 improvement in R2—this 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 in K. Since an increase in 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. The amplitude of these extra configurations wi l l in gen-eral be non-vanishing. The expectation value wi l l therefore increase unless we l imit 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 in flat 3-dimensional space. If we integrate, not over al l space, but up to some maximum radius R we obtain R R J rdV J 4irr3dr „ <r> = °~ir- = -R = !*• JdV J 4-nr2dr o o That is, the expectation value < r > varies linearly in the radius R. If we fail to fix this maximum radius, the scale of the sphere wi l l grow without bound as we integrate out to infinity. Unless we constrain the size of our structure in some other way, its expected size wi l l be determined by the volume of space, so any measurements we make on its geometry wi l l lack invariance in R. Furthermore, simple geometric analysis reveals the maximum 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 lmax and R is of the same order of mag-nitude as the ratio between < lx > and K in equation (3.6). We now understand the relationship between < lx > and K, so let us turn our attention to the calculation of uncertainty Alx. The uncertainty of edge length expectation value is given by the root-mean-square deviation from the mean Alx = yj< (lx- < lx >Y >. (3.10) Just as for < lx >, it is impossible to evaluate equation (3.10) as it is written due to the infinite l imi t in K. We must again evaluate the partial form of the given function for a set of many increasing values of K. From the behaviour of < lx >, 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 Alx oc K. (3.11) Let us elaborate on why this is so. Again 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 8 < i \ > - % - K\ (3.i2) J-gidx ^ 1 2 J o and therefore Alx ~ K (3.13) in the l imi t of large K. That is, Alx should exhibit linear dependence in K. Performing the required computer-assisted computations, we find that the relation-ship between the uncertainty and cutoff is of the expected form (3.11). The result of an Chapter 3. The Isotropic Class of Tetrahedron 30 unweighted linear least-squares fit to the generated Alx values for 0 < K < 200 is Alx = (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 R2 = 0.99995, indicating a firm linear correlation for the 0 < K < 200 data set. Least-squares fits to 2 n d and 3 r d order polynomials can also be performed. The 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 in R2 for the higher order fits. The linear fit is thus found to be appropriate. We again return to our simple physical analogy of a sphere in flat 3-dimensional space. Integrating up to the maximum radius R we obtain R R J Y W J47rrldr < r * > = ° — = °-R = ' f l ? , (3.15) fdV fiirr2dr o o for the normalized expectation value of r2. Then (3.7) and (3.15) yield the result A r = JloR- (3-16) That is, the uncertainty A r varies linearly in R. Again , failure to fix the maximum radius in effect allows a greater volume of integration to determine the scale of measurements on the system. The uncertainty of the maximum edge length tetrahedron contained within the sphere of radius < r > is therefore Almax = ^ A r ~ 0.316i?. (3.17) We also observe that the ratio between Almax and R in our analogous calculation (3.17) is of the same order of magnitude as the ratio between Alx and K in equation (3.14). In this sense, the results for the isotropic tetrahedron fit well wi th interpreting ^ as the distribution of topologically spherical structures in flat space. Chapter 3. The Isotropic Class of Tetrahedron 31 3.3 Semiclassical Isotropic Analysis Now that we have determined the behaviours of < lx > and Alx we can ask how well 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 6 than equation (2.10) with its many factorials—so why not use it? 3.3.1 Semiclassical Isotropic Expectation Values < lx >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). The semiclassical isotropic wavefunction and expectation values are respectively given by / V 2 (XXX VSC[MB, {X, X, X, X, X, X}} = l im K—>oo £ ( 2 p + l ) 2 ( 2 X + 1 ) 3 , [P=O,I,...,K ) (XXX (3.18) and Ex=0 (x + \)\^sc,MB\xixixixixi x] ^2x=0 \^SC,MB iXi XI X I X I X I X where equation (2.17) gives X X X ) 2 i ( c o 5 ( 6 ( 7 r - a r c c o s ( i ) ) ( X + i ) + f)) < lX >sc ~ ^ K ,,T, r 1.2 > I"3"19) (3.20) X X X)sc 0 ^ ( 2 ^ + 1)2 and we choose the principal value of arccos( |) . Since the 6j-symbol should vanish for non-integer x, the sum in x again runs over al l integers up to or equal to K. Investigation of < lx >sc reveals that it exhibits linear dependence in K just as do the exact values < lx > (see Figure 3.5). The result of an unweighted linear least-squares fit 3 This 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) wi thin uncertainties. The coefficient of determination is again found to be R2 = .99995 for the 0 < K < 200 data, and fitting to quadratic and cubic polynomials again demonstrates the unsuitability of non-linear descriptions. The degree to which the semiclassically approximated < lx >sc estimates the exact < lx > is further studied from the difference <lx>~<lx >sc (3.21) at given K. For the range 0 < K < 200 it is generally found that < lx > — < lx >sc becomes smaller with increasing K (see Figure 3.6), indicating that the semiclassical approximation improves with 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 wi th increasing ] \ . The apparent three-fold periodicity of the difference is, however, unanticipated. Considering the difference for every third 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 2cos(c 37s: + c 4 )) , (3.22) where C! = 4.240, c 2 = -0.9483, c 3 = 0.1104, c 4 = 0.6983, c 5 = 106.3, c 6 = 1.639, (3.23) for K = 99,102,105,.. . ,198, Ci = 885.5, c 2 = 0.9539, c 3 = -0.1103, c 4 = 0.3442, c 5 = 210.7, c 6 = 2.451, (3.24) 7Since 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 2 = 0.9528, c 3 = 0.1106, c 4 = 1.717, c 5 = -21.98, c 6 = 0.8400, (3.25) for K - 101,104,107, ...,200. The uti l i ty of equation (3.22) predominantly lies in its ability to provide an order-of-magnitude estimate of < lx > — < lx >sc for 200 < K < 300. Similarly, the study of AlXsc reveals a linear relationship in K (see Figure 3.5). The result of a linear least-squares fit to AlXsc values for 0 < K < 200 is found to be the same as that of equation (3.14) within uncertainties. The coefficient of determination is again found to be R2 = .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 Alxsc and exact Alx is further studied from the difference A / x - AlXsc (3.26) at specified K. For the range 0 < K < 200 it is generally found that the difference be-comes smaller wi th increasing K (see Figure 3.6). A s wi th the difference < lx > — < lx > s this result is not unexpected in light of [12]. Wha t is again unanticipated is the three-fold periodicity of the difference. Considering Alx — Alxsc f ° r every third value of K reveals that, from values K ~ 100 to K = 200, the quantity behaves as a damped si-nusoidally oscillating function. Unweighted least-squares fits reveal that the three-fold periodic pattern for 99 < K < 200 is reasonably approximated 8 by the function Alx-Alxsc= ( v , 0 7 , C l 2 ( l + c 8 c o s ( c 9 K + c 1 0 ) ) , (3.27) [is. + Cn) 8 A s in the case of equation (3.22), (3.27) was fit to Alx — AlxSc 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. Chapter 3. The Isotropic Class of Tetrahedron 35 0.03 0.02 0.01 A X V A X' v -0.01 -0.02 X < -0.0025 -0.005 -0.0075 -0.01 -0.0125 -0.015 -0.0175 50 100 120 140 160 180 200 100 K 150 200 Figure 3.6: Comparing exact and semiclassical isotropic < lx > and Alx as functions of K. Chapter 3. The Isotropic Class of Tetrahedron 36 where c 7 = -0.05588, c 8 = 0.1790, c 9 = 0.1091, cw = 0.9668, c n = -7 .141 , c12 = 0.9474, (3.28) for K = 99,102,105, ...,198, c 7 = -0.3863, c 8 = 0.1803, c 9 = 0.1114, cw = 2.693, cn = 40.69, c12 = 1.263, (3.29) for K = 1 0 0 , 1 0 3 , 1 0 6 , 1 9 9 and c 7 = -7 .468, c 8 = -0.1853, c 9 = 0.1089, c i 0 = 2.050, cn = 110.9, cl2 = 1.724, (3.30) for K = 101,104,107, ...,200. Again , the ut i l i ty of (3.27) is its abili ty to provide an order-of-magnitude estimate of Alx — AlXsc f ° r 200 < K < 300. 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 inter-preted this effect in terms of a natural description of the distribution of spherical objects in flat space—essentially, the infinite l imi t in the cutoff increases the configurations acces-sible 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 wi l l do so by fixing tetrahedral edge lengths, and then proceed wi th the analysis by evaluating conditional probabilities for our system. That is, we wi 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 evalu-ation of wavefunctions, probability densities, expectation values and uncertainties for a 3-manifold MB with a single, anisotropic, 2-manifold boundary 8MB wi th 2-sphere topol-ogy, where the anisotropy is completely parameterized by 2 independent variables. We wi 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 al l five subclasses. We wi l l 37 Chapter 4. Subclass A Two-parameter Anisotropic Calculations investigate subclasses A and E. 1 38 subclass Ji h h h h h A X X X Y Y Y B X X Y X Y Y C X X Y Y Y Y D X Y Y X Y Y E X 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 with two independent edge lengths given by J\ = J2 = J3 = AT, J 4 = J5 = J 6 = Y. (4.1) This tetrahedral representation 2 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[dMB] is chosen such that al l vertices, edges and faces of the single tetrahedron t are in the boundary of the tessellation. 1 Note that there is only one subclass of the isotropic type—the isotropic class itself. 2Since this chapter discusses analysis of the tetrahedron with li = I2 = h = X + | , I4 — Z5 = IQ = Y + | , it is useful to note that in such a case V2 < 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 V2 = - f f f i . Chapter 4. Subclass A Two-parameter Anisotropic Calculations 39 T Je=Y J4=Y Js=Y J2=X J3=X Ji=X Figure 4.1: Tessellation of the single ellipsoidal anisotropic boundary. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 40 Then, according to equation (2.26), the wavefunction 3 is *[MB,{X,X,X,Y,Y,Y}] = I \ 2 (XXX l i m K—>oo £ (2p + i ) 2 \p=0,l...,K J ( 2 X + 1 )2 (2F + 1)2 { }, (4.2) Y Y Y where X = 0, | , 1 , K , and Y = 0, | , 1 , K . Again , as in the isotropic calculations, not all values X lead to an admissible 6j-symbol—admissible X are again integral. In these cases the 6j-symbol wi l l vanish. In contrast, al l integer and half-integer values of Y w i l l result in an admissible 6j-symbol (provided X is integer). As in the previous chapter, we could ask questions regarding the behaviour of \I> as a function of the variables X, Y, and K. From equation (4.2) we see that the first term is again a rapidly decreasing function of K (it is 0(K~6)). The second and third terms 3 3 are 0(X*) and 0(Y*) increasing functions of X and Y respectively. The final term is again a rapidly oscillating function of its arguments. We therefore expect \& and |\I>|2 to decrease rapidly (asymptotically approaching O) as K increases along contours of fixed X and Y. They 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 4 0 < X < 20, 0 < Y < 20. Again notice that the behaviour I 12 I 12 of \& and |\&| are very similar—the predominant difference between them is that \^ >\ is again a strictly positive function with twice the frequency of oscillation of Figures 4,3 and 4.4 respectively display ^/ and |\I/| 2 as functions of X and K (fixed Y), and Y and K (fixed X)5. Notice that ^ and |\I>|2 vanish for X < 2Y. This occurs because the 3 For clarity, the boundary J-values are again explicitly listed as arguments of the wavefunction. For brevity, this explicitness will be dropped as we continue the study. 4 Aga in note that many values of * and | $ | exceed the displayed range of the plot—these regions appear as holes in the plotted surfaces. 5 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. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 41 inadmissibili ty of the 6-tuple gives gives rise to a vanishing 6j-symbol. Also note that Figures 4.3 and 4.4 reveal that ^ and | ^ | decrease more rapidly along contours of fixed Y than along contours of fixed X. The behaviour displayed by al l three sets of plots extends to al l larger ranges of variables X, Y and K. We now observe that there is a behaviour in the subclass A wavefunction and prob-ability densities which was not present in the isotropic functions. We observe this new feature in 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) in Y for given K. This occurs in the Lorentzian regime of the tetrahedron 6 where the ratio y is large enough to cause inad-missibility in the ordered set of 6j-values. This vanishing ratio turns out to be 2. That 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 6j-symbol itself vanishes. This , in turn, results in a vanishing wavefunction and probability density. However, no such relationship holds for sufficiently large ^ . This 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 maximum 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 maximum 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 wi l l see that the vanishing of the wavefunction and proba-bil i ty amplitude for y give rise to cutoff invariants. 6I.e., where V3(Y + \) < X + \ leads to V2 < 0. 7 This 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. Chapter 4. Subclass A Two-parameter Anisotropic Calculations Chapter 4. Subclass A Two-parameter Anisotropic Calculations 43 20 0 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 Calculations 46 4.1.2 Subclass A Expectation Values < lx > and Uncertainties Alx Just as wi th 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 < l x > = S£y=o (x + %)\*MB[x,x,x,y,y,y]\2 ^ ^ 52x,y=o \^MB[x,x,x,y,y,y}\2 where the sums over variables x and y are carried out for al l 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 in K <lx>otK. (4.4) Similarly, we predict the uncertainty (root-mean-squared deviation from the mean) Alx to behave in K as did the isotropic uncertainty—Alx should be linearly dependent on K Alx oc K. (4.5) Performing the necessary calculations we find that (4.4) and (4.5) hold true. In particular, performing the appropriate least-squares fits to the 0 < K < 50 data set (see Figure 4.6) reveals that < lx >= (0.688 ± 0.001) A" + (0.50 ± 0.04) (4.6) 8 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. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 47 with corresponding coefficient of determination 9 R2 = 0.99961, and Alx = (0.2312 ± 0.0005)# + (0.11 ± 0.01) (4.7) wi th coefficient 1 0 R2 = 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 AlY. Study also reveals that the rate of change of < lx > wi th 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. We also observe that the rate of change of Alx is less than that of the isotropic calculation. This is again due to the greater contribution from terms at given K. As well, taking the ratio (4.8) < lx > for 0 < K < 50 shows that the relative uncertainty approaches a value in the range 0-33 < < 0.34 (see Figure 4.7). 4.1.3 Subclass A Expectation Values < ly > and Uncertainties AlY Similarly, the normalized expectation value of lY is given by < l y > = E£ y=o (V + l)\^MB[x,x,x,y,y,y}\2 ^ Ex,y=o \^MB[x,x,x,y,y,y}\2 As in preceding discussions, one would predict < lY > and AlY to behave according to < lY > oc K (4.10) and AlY oc K, (4.11) 9 The 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 48 Figure 4.6: Subclass A < lx > and Alx as a functions of K. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 0.36 • • A/x 0 . 3 5 • • • • • </,> 0.34 • 10 20 30 40 50 ' ' K Figure 4.7: Subclass A as a function of K. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 50 since the sum in 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. Least-squares fits to the 0 < K < 50 data and reveals that < lY >= (0.7514 ± 0.0005)if + (0.53 ± 0.01) (4.12) with corresponding coefficient of determination 1 1 R2 = 0.99996, and AlY = (0.1746 ± 0.0002)# + (0.119 ± 0.007) (4.13) with coefficient 1 2 R2 = 0.99981 (see Figure 4.8). 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 conver-gence towards ~0.916 wi th 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). 1 1 Probability that R2 = 0.99996 is the result of a v — 99 uncorrelated data set is of the order 10 2 1 9 . 12The corresponding probability for an uncorrelated data set is of the order IO - 1 8 6 . er 4. Subclass A Two-parameter Anisotropic Calculations Figure 4.8: Subclass A < lY > and AlY as functions of K. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 0.26 • •• AIY 0.255 0.25 • • • • • • <IY> 0.245 0.24 0.235 • • • • • • • • • . • • . • • • • • • .*• . ••*..• •••••• • • • • • • *»• • 10 20 30 K 40 50 Figure 4.9: Subclass A -=r*- as a function of K. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 0.96 0.94 <lx> • • • • • • • • » • • • • 0.92 <IY> 0.88 •* W 20 30 40 50 • • • K Figure 4.10: Subclass A as a function of K. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 1.4 1.35 M x A. 1 m • • • t > > Aly ^ 1.25 • . • • • • t • K • Figure 4.11: Subclass A -rr^ as a function of K. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 55 4.1.4 Subclass A < lx > and Alx along contours of ly and K Although < lx >, < ly >, Alx and AlY are K dependent, we have just seen that J ^ c , j ^ - and ^ j ^ - seem to asymptotically approach constant values wi th increasing K. In some sense the geometry of the tessellated boundary seems to freeze out in K. There are other instances where we can observe invariants in K. These situations arise when we fix (constrain) certain parameters of the theory and investigate contours of the unconstrained variables. The first of these wi l l be in < lx > and AlX-There are two types of contour to consider: (1) fixed ly', and (2) fixed K. We wi 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 wi 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. Recall that tetrahedra with V2 > 0 are referred to as Euclidean, while those with V2 < 0 are called Lorentzian. The distinction merely refers to the type of 3-dimensional space into which we can embed the tetrahedra. We wi l l continue to use Euclidean and Lorentzian in this sense. We could alternatively refer to the V2 > 0 as the classically allowed regime and V2 < 0 as the classically forbidden regime. This is because we associate a classically allowed wavefunction with a boundary exhibiting characteristics of the Euclidean tetrahedron, and a classically forbidden wavefunction with boundaries exhibiting the characteristics of the Lorentzian tetrahedron. The analogy is the wavefunction in 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 and 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 wi l l exhibit finite size effects. We wi l l now find quantities which are invariant in the cutoff K. That is, they wi l l show no finite size effects. The calculation of these quantities, wi l l however involve contributions from both classically allowed and classically forbidden tetrahedra. We wi 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 lY For a given value of lY — (Y + | ) the expectation value of lx is given by E f = 0 {x + ±)\qMB[x,x,x,Y,Y,Y} f Y:?=0\*MB[X,X,X,Y,Y,Y]\* < l X > \ l Y = sr^K UT, r ^ v v l , 2 • I 4 ' 1 7 ) That is, there is no summation over parameter Y for this case of constrained-^ tetrahe-dra. In section 4.1.1 we saw that ^ vanishes for all y > 2. We would thus expect all probability densities and (x + |)-values to contribute to < lx > \ i y unt i l y exceeds this ratio. That is, for al l K probing the region y > 2 we expect < lx > \i to be a constant. However, unti l this y > 2 ratio condition is met, < lx > \iY w i l l be dependent on (i.e., controlled by) K. That is, in this classically allowed regime, al l 6j-values wi l l form admissible 6-tuples, so the corresponding 6j-symbols w i l l be non-vanishing. There wi l l therefore be non-zero contributions to the sum from al l x < K in this region. In particular, we may guess that <lx > \t w i l l be a non-decreasing function of K. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 57 We similarly expect the uncertainty Alx\W to exhibit strong dependence on K unt i l we probe the region y > 2. The 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 > \iY approaches a constant value for sufficiently large values of K. Also as expected, calculations reveal that the functional behavior of Alx\iY is qualita-tively similar to that of < lx >\iY (see Figure 4.13 for the lY= 20.5, 40.5, 60.5, 80.5, and 100.5 contours). Aga in , the function exhibits cutoff invariance when K is large enough to allow y > 2. There is, however, an interesting feature that we did not predict—lx\i Y and Alx\ly show dramatically litt le variation in the Lorentzian regime. That 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 litt le to the uncertainties and expectation values. Furthermore, taking the ratio A l x l l Y (4.18) <lx> h for the ly = 100.5 contour (see Figure 4.14) reveals that the relative uncertainty ap-&lxhY < <lx>\iY -proaches a value in the range 0.333 < ^ , l x ^ < 0.335. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 58 Figure 4.12: < lx > \i as a function of K for contours of fixed ly-Chapter 4. Subclass A Two-parameter Anisotropic Calculations Figure 4.13: Alx\iY a s a 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 < lx > \K and uncertainties Alx\K as a function oily, we begin to directly probe the geometry of our tessellated boundary. That is, we expect to start observing whether or not the associated tetrahedron is isotropic. We wi l l therefore see whether or not < lx >\K observes <IX>\K = W, (4.19) or simply <lx >\K only. (4.20) We have already seen that < lx > \ l y is regulated by K unt i l y < 2, and is constant afterwards. We have also seen that there is very litt le contribution from geometries probing the Lorentzian regime. However, when we fix K and sum over both X and Y to evaluate < lx >\K, all admissible 6j-values (both Euclidean and Lorentzian tetrahedra) w i l l contribute to the result provided the terms wi th y > 2 enter the sum. That is, provided, ly = Y + | < 7}(K + 1), all tetrahedra wi l l contribute. However, if we allow ly = Y + | > ^(K + 1) our sums wi l l involve al l Euclidean tetrahedra but not al 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. That is, for sufficiently large ^> a ^ tetrahedra wi l l be Euclidean, so \1> wi l l not vanish—the expectation values (space of histories) wi l l therefore be l imited not by lY (the dynamics), but by the cutoff K. We similarly expect ly to regulate the behaviour of Alx\K in the classically forbidden regions with lY < ^(K + 1), and expect to see cutoff dependence for lY > ^(K + 1). It is again difficult to predict the exact behaviour of < lx >\K and Alx\K as functions of ly, but we do know that | ^ | is expected to contribute very li t t le to < lx >\K in the Chapter 4. Subclass A Two-parameter Anisotropic Calculations 62 Lorentzian region where \fZlY > lx is allowed by the condition V3lY>K + ^. (4.21) If contributions in 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 exhibit the predicted features. Figures 4.15 and 4.16 respectively display < lx >\K and Alx\K for the K = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100 contours. In detail, we observe the following. < lx >\K varies linearly in ly according to < lx >\K = (1.360 ± 0.001)/ y + (0.05 ± 0.03), (4.22) when ly = Y + ^ < ^(K + 1). In fact, the < lx >\K values are identical between data sets provided ly < \{K + 1). Equation (4.22) is the result of an unweighted linear least-squares fit to the lY < \(K + l) data, and holds true for each of the K = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100 contours 1 3 . The coefficient of determination for the 7^=100 data set of v = 99 degrees of freedom is R2 = 0.99994, indicating a firm linear correlation. 1 4 Addit ionally, least-squares fits to 2 n d and 3 r d order polynomials again reveal that fits to quadratic and higher order are unsuitable. Furthermore, the contours reveal that < lx >\K oscillates in regions where \fZlY > K + ^—the region where K l imits the number of classically allowed tetrahedra in 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). 13All 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. 1 4 T h e probability that a fit with R2 = 0.99994 and = 99 degrees of freedom results from an uncorrelated parent population is of the order 10~ 2 1 1 . 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 Calculations 64 AA * I K K=20 K= 15 K= 10 K=5 /y K=30 AA * I K K=80 K=60 K=50 K=40 K= 100 Figure 4.16: A / x | ^ as a function of Zy. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 65 It also appears that the maximum value < lx >max\K occurs at the smallest ly sat-isfying (4.21). When we analyze these maximum 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) (4.23) — the result of a linear least-squares fit. R2 — 0.99998 for this fit of v = 8 degrees of freedom, indicating a strong linear correlat ion. 1 5 Again , least-squares fitting to higher orders in K demonstrates that non-linear descriptions are inappropriate. We thus observe that the ratio of < lx > \K to ly is Tf-invariant when lY < ^(K + 1) and that finite size effects become apparent when the cutoff begins to exclude non-vanishing 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 in regions where ly < ^(K + 1) we may get cutoff invariant (i.e., physically meaningful) results, but we wi l l likely not if we ask questions elsewhere. We now turn our attention to AIX\K-Calculations reveal that AIX\K resembles the superposition of a linear and a small-amplitude oscillating function in ly for ly < \(K + 1). Furthermore, AIX\K is cutoff independent provided we consider regions where ly < \{K + 1). Figure 4.17 isolates and displays the K = 100 contour. The result of unweighted linear least-squares fitting to lY < \(K + 1) for the K = 100 data set reveals that AIX\K = (0.388 ± 0.004)Z y + (0.0 ± 0.1), (4.24) with R2 = 0.99124. 1 6 1 5 T h e probability of this fit resulting from an uncorrelated data set is of the order of 1 0 - 2 0 . 1 6 T h e likelihood of an uncorrelated data set with v = 99 degrees of freedom to yield this R2 value is of the order I O - 1 0 3 . Chapter 4. Subclass A Two-parameter Anisotropic Calculations 66 F i gure 4.17: A Z ^ I ^ as a function of7y. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 67 Furthermore, calculating AIX\K (4.25) < lx >\K for the K = 100 contour reveals that the relative uncertainty converges towards a value in the range 0.26 < '4*l*K < 0.31 in the cutoff independent region of our system (see Figure 4.18). In summary, Figure 4.19 displays the surfaces17 of < lx > and Alx as functions of both lY and K for 0.5 <ly < 20.5, 0 < K < 20. 4.1.5 Curvature Expectation Values < 0 > 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 9Xy. 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 > 2, (4.29) Y so we are in the region disallowed by the admissibility conditions. 1 7 A s with our analysis of the wavefunctions, our theory is strictly defined only for the regions lY < Chapter 4. Subclass A Two-parameter Anisotropic Calculations <lx 0.35 • • • 0.325 ...» . . v /v* K=100 0.3 • • • • • • . • • . • • • • • 0.275 0.25 . • • • • . • # • • • • 0.225 20.5 40.5 60.5 80.5 100.5 0.175 ly Figure 4.18: ^ f - as a function of lY for AT=100. Chapter 4. Subclass A Two-parameter Anisotropic Calculations Al X 2 10.r*5 20.5 0 20.5 o Figure 4.19: < lx > and Alx as functions of lY and K Chapter 4. Subclass A Two-parameter Anisotropic Calculations 70 Figure 4.20: Angles at the vertices of the tetrahedron. <eYY>\K = v , ) ' ' - 2 , (4.30) < >\K = r_ _ _ , , ^ , 2 • (4-31) Chapter 4. Subclass A Two-parameter Anisotropic Calculations 71 •K -We wi l l consider expectation values along contours of fixed K because we know that we wi l l be able to extract cutoff independent information from our system provided we restrict our calculations to lY < \(K + 1). The expectation values of 9YY and 9XY for fixed K are therefore given by 2(y+>)» E^0\^MB[X,X,X,Y,Y,Y}\' and E f ^ c o s - 1 ( ^ y ) \yMB[x,x,x,Y,YtY\\ Y%=Q\*MB[x,x,x,Y,Y,Ytf As expected, computations reveal that < 0YY > \ K and < 9XY > \K a r e constant in K provided ly < \{K + 1). Figure 4.21 shows < 6YY > \ K for K = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100, while Figure 4.22 displays the results of < 9Xy > \ K for K = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100. Least-squares fitting to constants determines < QyY >\K = (1.555 ± 0.006), (4.32) and < BXY >\K = ( ° - 7 9 4 ± ° - 0 0 3 ) , (4-33) for lY < \{K -r 1) along the K = 100 contours. Figure 4.23 summarizes the data for K < 20. Since < 9Yy > \ 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 > \ K = < &XY > \ K = f • Calculations and least-squares fitting also revealed that /\9YY\K = (0.500 ± 0.006) (4.34) Chapter 4. Subclass A Two-parameter Anisotropic Calculations < ®YY> K 1. 8 • 1. 6 1 .4 1.2 • . 'mm' -.» • • • • • * 1 0 . 8 : -A A *• * • • K = 5 K = 7 5 K = 2 0 • • • K=30 0 .5 5.5 10.5 15.5 20.5 25.5 W 30 . 5 < QYY> \K 1 . 8 mr * - _ v. • "4 •A * **** " ^ /<=40 /<=50 /<=60 K=80 K=100 'Y Figure 4.21: < 9Y  > \K SLS a function of ly-Chapter 4. Subclass A Two-parameter Anisotropic Calculations K=5 K=10 K=15 K=20 K=30 0 . 7 \ 0 . 5 5 . 5 1 0 . 5 1 5 . 5 2 0 . 5 2 5 . 5 3 0 . 5 0 . 5 2 0 . 5 4 0 . 5 6 0 . 5 8 0 . 5 1 0 0 . 5 Figure 4.22: < 9Xy > \K a s a function of ly. Chapter 4. Subclass A Two-parameter Anisotropic Calculations <eXY> Figure 4.23: < Qyy > and < QXY > as functions of ly and K Chapter 4. Subclass A Two-parameter Anisotropic Calculations 75 A9XY\K = (0.250 ± 0.003) (4.35) for lY < | ( # + 1) along the K = 100 contour. See Figure 4.24 for < 9YY > \ K along the # = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100 contours, and Figure 4.25 for A9XY\K along the K = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100 contours. We have again observed cutoff independent results for the region lY < | ( # + 1). Furthermore, the curvature at the vertex formed by the three length lY edges is given by 6 y y = 2ir - 39YY, (4.36) while the curvature at the vertex formed by two length lx edges and one length lY edge is eXY = 2ir-(26XY + ^). (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 = (1-62 ± 0.02), (4.38) < &XY >\K = (3.649 ± 0.006), (4.39) and A 6 y y | ^ = (1.50 ± 0 . 0 2 ) , (4.40) AQXY\K = (0.500 ± 0.006), (4.41) for lY < T;(K + 1). Figure 4.26 shows the < QYY > \ K and < Q X Y > \ K results for the #=100 contour, and Figure 4.27 shows the A 0 y y | ^ and AQXY\K results for the #=100 contour. B y evaluating < 6 y y > m a x \ K = < QYY > \ K + A Q Y Y \ 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=15 K=20 K=30 o . i 0.5 10 .5 15.5 /, Y 20.5 25 .5 30.5 A 0 YY K 0.6 0 .5 0.4 0.3 0.2 • • • 9 m a .... ., 4 . v>. A=-#7 /fe5Z? /f=£<? /f=<?<? K=100 o . i 0 . 5 20 .5 4 0 . 5 60.5 80 .5 100 .5 'Y Figure 4.24: A9YY\K a s a function of lY. Chapter 4. Subclass A Two-parameter Anisotropic Calculations AG 'XY\K 0 . 2 0.1 0 . 0 5 1* K=5 K=W K=15 K=20 K=30 Y 2 0 . 5 2 5 . 5 AG 'XY\K 0 . 3 0 . 2 5 0 . 2 0 . 1 5 0 . 1 K=40 K=50 K=60 K=80 K=100 0 . 05 60 . 5 1 0 0 . 5 Y Figure 4.25: A9Xy\K as a function of ly-Chapter 4. Subclass A Two-parameter Anisotropic Calculations Chapter 4. Subclass A Two-parameter Anisotropic Calculations AOyy K 2 1 1.5 1 ^ K=100 0.5 0 5 20.5 40 . 5 60 . 5 IY 80.5 100.5 A®XY K 0 . 6 0 . 5 0.4 0 . 3 K=100 0.2 0 .1 0 20.5 5 40 . 5 60 . 5 80.5 100.5 IY Figure 4.27: AQYY\K and A 6 x y | ^ as functions of ly for #=100. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 80 < 0 y y >min\K = <eYY>\K- AQYY\K, (4-43) and < ®XY >max\K = < &XY >\K + &QXY\K, (4-44) < &XY >min\K = < QXY > \ K ~ &QXY\K> (4.45) we determine the maximum and minimum observable curvatures of < QYY > \K ANA" < QXY > \K a t the vertices of the tetrahedron. Figure 4.28 displays the maximum and minimum values of < O y y > 1^- for K=100, while Figure 4.29 displays the maximum and minimum values of < QXY > \K f ° r 7f=100. Interestingly, we find that it is possible to observe negative curvatures for small lY at the vertex formed by the three length lY 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 ly w i l l always be positively curved. In particular, least-squares fitting to the region lY < ^(K + 1) yields (3.12 ± 0 . 0 4 ) , (4.46) (0.12 ± 0.04), (4.47) (4.15 ± 0 . 0 1 ) , (4.48) (3.15 ± 0 . 0 1 ) . (4.49) 4.1.6 Subclass A < lY > and AlY Along Contours of lx and K We wi l l continue the investigation of our A-subclass two-parameter anisotropic tetrahe-dron. Specifically, we wi l l evaluate the expectation values < lY > along contours of fixed lx and K. < 0 y y > max \K < 0 y y >min\K and < QXY >max\n < QXY >TTUTIK Chapter 4. Subclass A Two-parameter Anisotropic Calculations Chapter 4. Subclass A Two-parameter Anisotropic Calculations 82 Figure 4.29: < QXY >max IK- a n d < ®XY >mml/c a s functions of ly for #=100. Chapter 4. Subclass A Two-parameter Anisotropic Calculations 83 Contours of Fixed lx For a constrained value of lx the expectation value of ly is given by ^ , Ef = 0 (y+l)\*MB[X,X,X,y,y,y}\2 < Ly > i — r; : —, . (4 .0U) UX ZU\*"B[X,X,X,V,V,V]\ Calculations for < ly >\ix and AlY\ix reveal that they are strongly dependent non-linear functions in K. This dependence persists for all values of K. Figure 4.30 displays < lY > \ l x and AlY\ix for the lx = 10.5 data set. Al though it is difficult to predict the exact behaviour of < ly > | ; x f rom (4.50) alone, it is easy to understand the cutoff dependence of < ly >\ix and AlY\ix. Notice that the sum involves summing over all tetrahedral geometries with lY > lx. A l l 6j-symbols are non-vanishing for such geome-tries. Therefore, when we increase K we increase the admissible configurations included in our sum. Fini te size effects wi l l therefore be observed for al l such calculations. From Figure 4.30 it also appears that < lY >\ix and AlY\ix become linear in K for K » lx. Then, taking the ratio A I y 1 i » (4.51) < W >\lx for the lx = 10.5 contour shows that the relative uncertainty seems to converge to a value in the range 0.2 < < 0.3 (see Figure 4.31). Contours of Fixed K If we now evaluate < lY > and AlY 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. We again observe cutoff dependent behaviour for < lY > \ K and A l Y K . However, we also see lx dependence in the results. Addit ional ly, because AlY\K tends to decrease with lx while <lY > \ 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 1 - • " * ' * ' * V - ' v - - w - ^ ^ K = 1 0 ° 6O1 • * * • mm * * * * * * * **V V V / A V A ^ K=60 4 0 . A A A A 2 0 ^ ^ W * ^ # = 3 0 • ">Vl*W # = 2 0 • #=yo 2 0 . 5 4 0 . 5 6 0 . 5 8 0 . 5 1 0 0 . 5 Ix AI Y K 2 5 2 0 1 5 v --,->v- **-v**^ i * * » * * ^ * * * ^ * * * * • * . . * * * * • * * * * * * * * * * ~ •** ** **r . * « • • ' • f * J • / . * * * • • • . • A ****, ****, K=100 K=S0 W + * * * * * A . /C=40 - K=60 K=50 K=10 2 0 . 5 4 0 . 5 6 0 . 5 80 .'5 1 0 0 . 5 Figure 4.32: < lY > \K and AlY\K as functions of lX-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 lx and K for 0.5 < lx < 20.5, 0 < K < 20. Chapter 4. Subclass A Two-parameter Anisotropic Calculations My K <ly> K 0 . 4 r 0 . 3 5 -0 . 3 • • • • • • • • • 0 . 2 5 • • • • ••*•.... K=100 0 . 2 . .... • • ... 0 . 1 5 2 0 . 5 4 0 . 5 6 0 . 5 8 0 . 5 1 0 0 . 5 Ix Figure 4.33: Relative uncertainty , Y}? as a function of lX-Chapter 4. Subclass A Two-parameter Anisotropic Calculations 89 2 0 . 5 Figure 4.34: < ly > and Aly as functions of lx and K. Chapter 5 Subclass E Two-parameter Anisotropic Calculations We know any expectation values we measure on the single-tetrahedral tessellation wi l l be cutoff dependent i f conditional probabilities fail to restrict the number of allowed configurations the system can adopt. That is, the lengths and volume of our tessellated boundary wi 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 wi 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 l imit the total number of configurations the tetrahedron could adopt. We believe the behaviour of other two-parameter anisotropic subclasses can be predicted and understood in terms of configurational restrictions as well. We wi 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 5.1 The Subclass E Wavefunction 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, J2 = J 3 = J 4 = J5 = J 6 = Y. (5.1) As mentioned in 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[MB,{X,Y,Y:Y,Y,Y}} = l im K—>oo t V2 £ (2p + i ) 2 \p=0,l,...,K J X Y Y (2X + l)*(2y + l )$ { \, (5.2) Y Y Y where X = 0 , \ , 1 , K , and Y = 0,\,1,..., K. Now let us compare the behaviours of the 6j-symbols (X Y Y (5.3) ( Y Y Y and X X 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. The triangular inequalities presented in 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 92 Figure 5.1: Tessellation of the single anisotropic ovoidal boundary. 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 in observed behaviour wi l l be due to the 6j-symbols alone. Based on the qualitative and quantitative similarities between the two 6j-symbols, 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 < lY >, and uncertainties Alx and A / y ; 2) relative uncertainties a n d and ratios ^ j ^ - and ^ should asymptoti-cally approach constant values with increasing K; 3) cutoff invariance wi l l be observed for < lx > \t , Alx\l , < lx > \ K and Alx\K, when y > 2 (2lY — lx) < \ terms enter the calculations; and, 4) < lY > \ l x , AlY\i , < lY > \ K and Aly\K w i l l exhibit finite size effects. So let us now review the computational results in brief. 5.1.1 Subclass E Expectation Values < lx > and Uncertainties Alx Now, the normalized expectation value of lx is given by 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 < lx > and K. In particular, performing the appropriate least-squares fits to the 0 < K < 50 data set (see Figure 5.3) reveals that < lx >= T^x,y=o (x + ^)\^MB[x,y,y,y,y,y]\2 T:x,y=o \^MB[x,y,y,y,y,y]\2 (5.5) < ix >= (0.533 ± O.OOIJA" + (0.32 ± 0.03) (5.6) Chapter 5. Subclass E Two-parameter Anisotropic Calculations 95 with corresponding coefficient of determination 1 R2 = 0.9995, and Alx = (0.2916 ± 0.0007)K + (0.22 ± 0.02) (5.7) with coefficient2 R2 = 0.99946 (see Figure 5.3). Furthermore, taking the ratio (5.8) < lx > for 0 < K < 50 shows that the relative uncertainty approaches a value in the range 0.546 < < 0.552 (see Figure 5.4). This value is larger than - 0.335—the value obtained for the subclass A tetrahedron. 5.1.2 Subclass E Expectation Values < lY > and Uncertainties AlY Similarly, the normalized expectation value of lY is given by , ^ Ex,y=Q(y + l)\^MB[x,y,y,y,y,y]\2 where x and y are again summed over all integer and half-integer values. Performing the appropriate calculations we find a linearity between < lY > and K. Least-squares fits to the 0 < K < 50 data and reveals that < lY >= (0.794 ± 0 . 0 0 1 ) 7 T + (0.53 ± 0 . 0 4 ) (5.10) with corresponding coefficient of determination 3 R2 = 0.9997, and AlY = (0.1632 ± 0.0003)TsT + (0.07 ± 0 . 0 1 ) (5.11) with coefficient4 R2 = 0.9995 (see Figure 5.5). 1 The probability that this coefficient of determination is the result of a v = 99 uncorrelated data set is of the order I O - 1 6 7 . 2 The corresponding probability for an uncorrelated data set is of the order 10~ 1 6 3 . Probabi l i ty that R2 = 0.9997 is the result of a v — 99 uncorrelated data set is of the order 1 0 - 1 7 5 . 4 The corresponding probability for an uncorrelated data set is of the order I O - 1 6 8 . Chapter 5. Subclass E Two-parameter Anisotropic Calculations Chapter 5. Subclass E Two-parameter Anisotropic Calculations Al x <lx> 0.57 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 50 K Figure 5.5: Subclass E < lY > and AlY as functions of K. 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 in the range 0.203 < < 0.205 (see Figure 5.6). This 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 wi th increasing K (see Figure 5.7). This 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) AlY for the 0.5 < K < 50 data set converges towards a value in the range 1-775 < < 1.825 for increasing K (see Figure 5.8). 5.1.3 Subclass E < lx > and Alx along contours of lY and K We have thus verified that < lx >, < lY >, Alx and AlY are K dependent, and that Tjf"' 7^> l~w a n < ^ seem to asymptotically approach constant values with increasing K. We wi l l now continue searching for cutoff invariant measures on our tessellated bound-ary of the subclass E system by considering two methods of constraint: (1) fixed lY; and (2) fixed K. Let us begin wi th the fixed lY constraint. Chapter 5. Subclass E Two-parameter Anisotropic Calculations 100 0.2 0.18 Aly 0.16 <lv> 0.14 0.12 0.1 Figure 5.6: Subclass E as a function of K. Chapter 5. Subclass E Two-parameter Anisotropic Calculations 101 0.9 <lx> 0.8 0.7 0.6 < \ y > 0 10 20 30 K 40 50 Figure 5.7: Subclass E as a function of K. Chapter 5. Subclass E Two-parameter Anisotropic Calculations 102 3.5 Al x Al 2.5 Y 0 10 20 30 K 40 50 Figure 5.8: Subclass E ^f- as a function of K. Chapter 5. Subclass E Two-parameter Anisotropic Calculations 103 Contours of Fixed lY For a given value of ly = (Y + | ) the expectation value of lx is given by £*0 (x + ±)\*MB[x,Y,Y,Y,Y,Y]f E^ 0 l*M S[x ,y,y,r,y,y]r < l x > 'V = v ^ t f ivr, r_ ^  ,r w ,^1,2 • ( 5 - 1 5 ) That 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 constrainedly tetrahedra. Performing the necessary calculations we clearly observe the following results. < lx > \ l y is a non-decreasing function of K for all values of Y and K (see Figure 5.9 for the lY = 20.5, 40.5, 60.5, 80.5, and 100.5 contours), and it is a constant for ^(K + 1) > ly. Calculations also reveal that the functional behavior of Alx\iY is qualitatively similar to that of < lx >\iY (see Figure 5.10 for the lY= 20.5, 40.5, 60.5, 80.5, and 100.5 contours). Again , the function exhibits cutoff invariance when ^(K + 1) > ly. As in the study of subclass A we note that lx\iY and A.lx\ly show dramatically lit t le variation in the Lorentzian regime. That is, it appears the functions are nearly constant for (K + | ) > \/3lY where values (x + | ) > \/3lY enter the calculations. Lorentzian tetrahedra again contribute remarkably litt le to the uncertainties and expectation values. Furthermore, taking the ratio M x W (5.16) < IX >\ly for the lY = 100.5 contour (see Figure 5.11) shows the relative uncertainty decay to a constant value ^,LX)\Y ~ 0.4835. This is larger than the subclass A result of ~ 0.2785. Contours of Fixed K Again, when we fix K and sum over both x and y to evaluate < lx >\K or Alx\K, our calculations wi l l be regulated by one of two effects. Either K wi l l be large enough to allow all Euclidean and Lorentzian tetrahedra to contribute to the sum (ly w i l l regulate Chapter 5. Subclass E Two-parameter Anisotropic Calculations <lx> 'Y 20 25 30 35 40 45 K <lx> 'Y 65 60 55 50 45 40 35 K= 60.5 60 70 80 90 100110120130 K <lx> Y 40 50 60 70 80 90 K <lx> 80 100 120 140 160 K 100 120 140 160 180 200 220 K Figure 5.9: < lx > \ly as a function of K for contours of fixed lY-Chapter 5. Subclass E Two-parameter Anisotropic Calculations 105 Chapter 5. Subclass E Two-parameter Anisotropic Calculations 107 < lx > \K), or the space of histories w i l l be restricted to exclude some tetrahedral ampli-tudes when y is sufficiently small (K w i l l regulate < lx > \K). Specifically, al l tetrahedra wi l l contribute to the amplitude if lY = Y + | < \{K + 1), but our set of histories w i l l exclude at least one tetrahedral amplitude i f ly = Y + \ > | ( # + 1). Calculations reveal that <lx >\K and Alx\K exhibit the expected K invariance for sums with al l ly = Y + | < | ( # + 1). The expected finite size effects are observed for sums probing ly = Y + \ > \(K + 1). Figures 5.12 and 5.13 respectively display <IX>\K and Alx\K for the K = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100 contours. We specifically observe that < lx >\K varies linearly in ly according to < lx >\K = (0.922 ±0.002)/Jy + (0.10 ± 0 . 0 5 ) . (5.17) As was the case for the subclass A , < lx >\K values are identical between data sets provided lY < \{K + 1). Equation (4.22) is the result of an unweighted linear least-squares fit to the lY < \(K + 1) data, and holds true for each of the K = 5, 10, 15, 20, 30, 40, 50, 60, 80, and 100 contours 5. The coefficient of determination for the #=100 data set of v = 99 degrees of freedom is R2 = 0.9996, indicating a firm linear correlation. 6 Additionally, least-squares fits to 2 n d and 3 r d order polynomials again reveal that fits to quadratic and higher order are unsuitable. The contours also reveal that < lx >\K decays in regions where y/3lY > K + |— the region where K l imits the number of classically allowed tetrahedra in the sum. As expected, there is also a suppression of the finite size effects from Lorentzian geometries— < lx >\K exhibits very litt le K dependence when ^(K + |) > lY > \{K + 1). 5 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. 6 The probability that a fit with R2 = 0.9996 and v = 99 degrees of freedom results from an uncorre-lated parent population is of the order I O - 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 maximum value < lx >max\i< occurs at the smallest lY satisfy-ing \/3Zy > K+\ (the point of transition between Lorentzian tetrahedra and inadmissible geometries). Analysis of these maximum values < lx >max\K as a function of K for the K = 5,10,15, 20,30,40,50,60,80, and 100 data sets reveals < lx >max\K = (0.42 ± 0.02)# + (2 ± 1) (5.18) —the result of a linear least-squares fit. R2 = 0.984 for this fit of v = 8 degrees of freedom, indicating a strong linear correlation. 7 Aga in , least-squares fitting to higher orders in K demonstrates that non-linear descriptions are inappropriate. We now focus on AIX\K- Calculations reveal that AIX\K is linear function in lY for lY < \{K + 1). Furthermore, AIX\K is cutoff independent provided we consider regions where lY < \(K + 1). The result of an unweighted linear fit to ly < \(K + 1) for the K — 100 data set reveals AIX\K = (0.4780 ± 0.009)/y + (0.08 ± 0.03), (5.19) with R2 = 0.9996.8 Furthermore, evaluating A l x l K (5.20) < lx >\K for the K = 100 contour shows the relative uncertainty converges towards a value in the range 0-515 < *X\*K < 0.520 in the cutoff independent region of our system (see Figure 5.14). Again , this is larger than the subclass A result of ~ 0.285. In summary, Figure 5.15 displays the surfaces9 of < lx > and Alx as functions of both Zy and K for 0.5 < lY < 20.5, 0 < K < 20. 7 T h e probability of this fit resulting from an uncorrelated data set is of the order 10 9 . 8 The likelihood of an uncorrelated data set with v = 99 degrees of freedom to yield this R2 value is of the order I O " 1 7 1 . 9 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 Al X K 0.55 0.545 0.54 0.535 <lx> K 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 113 5.1.4 Subclass E < lY > and AlY Along Contours of lx and K Contours of Fixed lx For a constrained value of lx the expectation value of ly is given by , , = Eyio (y + l ) \ ^ M B [ X , y , y , y , y , y } \ 2 ( Y U x ^=0\^MB[X,y,y,y,y,y}\2 ' where the sum in Y is over both integer and half-integer values. Calculations for < ly >\ix and Aly\ix reveal that they are non-linear, strongly in-dependent functions. This dependence persists for all values of K. Figure 5.16 displays < lY > \ l x and AlY\ix for the lx = 10.5 data set. It is again easy to understand the cutoff dependence of < lY >\ix and AlY\ix. Since K > X, the sum includes al l admissible geometries up to Y = K > X. The 6j-symbols for such geometries are, in general, non-vanishing. Our sum over histories wi l l therefore experience regulation due to the size of K. From Figure 5.16 it also appears that < ly >\ix and AlY\ix become linear in K for K 3> lx. Then, taking the ratio -4^f - (5-22) < W >\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). This result is smaller than ~ 0.25—the value for the subclass A system. Contours of Fixed K Evaluating < lY > 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 Calculations 115 100 200 300 400 K Figure 5.17: Relative uncertainty ^ ^ j ' * as a function of K. Chapter 5. Subclass E Two-parameter Anisotropic Calculations 116 Chapter 5. Subclass E Two-parameter Anisotropic Calculations 117 10 15 20 25 30 I 3.525 * 3.5 ^ 3.475 < 3.45 3.425 3.4 6.9 6.85 6.8 6.75 6.7 6.65 0 0 10 10 15 20 •X X 20 •X 30 40 10 20 30 40 50 60 I X 'X 'X Figure 5.19: AlY\K as a function of lX-Chapter 5. Subclass E Two-parameter Anisotropic Calculations 118 Furthermore, MY\K <l >l ( 5 - 2 3 ) < *y \K for the K — 100 contour is shown in Figure 5.20. The relative uncertainty increases through its maximum value of ~ 0.2117 and then decays wi th increasing lx. The peak occurs at the maximum value of Aly\K. In comparison, the subclass A maximum 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 < lx < 20.5, 0 < K < 20. 5.2 Comparison of Subclass A and E Results Now let us summarize our findings for the subclass E tessellation. As expected, we observed: 1) finite size effects for expectation values < lx > and < ly >, and uncertainties Alx and Aly: 2) asymptotic approach to constant values with increasing K for relative uncertainties and and ratios ^ and 3) cutoff invariance for < lx > \ T Y , AIx\1y, < lx > \K a n d ^^1^-, when the condition Y > 2 ^ (2ly — lx) < \ l imits the total number of system configurations; and, 4) finite size effects for < ly > \t , AlY\t , < ly > \ K and AlY\K. That is, using configuration l imi t ing 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 lx. Chapter 5. Subclass E Two-parameter Anisotropic Calculations 20 0.5 Figure 5.21: < lY > and Aly as functions of lx and K. 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 ( )si0pe,A and ( )siope,E to respectively designate the linear fit slopes of the subclass A and E systems, we write the results as (< lx >)slope,A = ( L 2 9 2 ± Q 0 0 6 ) > ( 5 2 4 ) (< lx >)slope,E = (0.793 ± 0 . 0 0 4 ) , (5.25) = (0.9463 ± 0 . 0 0 0 7 ) , (5.26) = (1.07 ± 0 . 0 2 ) , (5.27) (1.475 ± 0 . 0 0 4 ) , (5.28) = (1.88 ± 0 . 0 9 ) (5.29) slope,A {Alx)slope,E (< ly >)slope,A (< ly >)slope,E (Aly) sipped (Aly) slope,E (< lx > \K)slope,A _ (< lx > \x)slope,E lx ^max \x)slope,A and lx ^max \K)slope,E {Alx\K)slope,A - ( 0 . 8 1 ± 0 . 0 2 ) . (5.30) \ ^ X\K) slope,E The result of equation (5.24) shows us that < lx > varies more rapidly with 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 with respect to K. Conversely, equation (5.25) shows that the subclass A tetrahedron uncer-tainty Alx varies more slowly with K than does Alx for the subclass E tetrahedron. In 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 in K. The fixed K contour results < lx > \ K , < lx >max \ K and Alx\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 in lx, which show the greater variation wi th respect to changes in K. This brings our two-parameter anisotropic study and the thesis to a close. Chapter 6 Conclusion We began this study by cit ing 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 the-ory. We provided a geometric (tetrahedral) representation of the 6j-symbol and for-mulated the Ponzano-Regge wavefunction fyMg for 3-dimensional manifolds MB with (2-dimensional) boundary dMB. We proceeded by investigating the simplest cases in search of quantities independent of the cutoff inherent in the theory. Our 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.) Fini te size effects were thus observed. Addit ional ly, we saw the relative uncertainty of the tetrahedral edge lengths asymptote towards a constant value with increasing cutoff. The next case studies involved the investigation of two-parameter anisotropic bound-aries. 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 tetra-hedra with 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 l imit 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 in i t ia l anisotropy in 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 in the tetrahedral geometry. Furthermore, the two anisotropic subclasses dis-played qualitatively similar traits—both were peaked or flattened away from isotropy under the same conditions, and their expectation values, uncertainties and relative un-certainties 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. This wi 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 difficul-ties it presents and the sorts of solutions it offers to the field of quantum cosmology. Bibliography [1] L . Claypool , Primus: Ant ipop , © S t u r g e o n (BMI) (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 space-time, [gr-qc/9304006]. [5] M . Ge l l -Mann 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 . Hal l iwell and A . Zoupas, Phys. Rev. D52, 7294 (1995); Quantum state diffu-sion, density matrix diagonalization and decoherent histories: A Model, [quant-ph/9503008]. [12] G . Ponzano and T. Regge, Semiclassical Limit of Racah Coefficients, Spectroscopic and Group Theoretical Methods in Physics, edited by F . Block (North-Holland, Amsterdam, 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, Nucl . Phys. B382, 276 (1992); Partition Functions and Topology-Changing Amplitudes in the 3D Lattice Gravity of Ponzano and Regge, [hep-th/9112072]. [16] J . Iwasaki, J . Ma th . 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. Grav. 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. Grav. 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 preced-ing version) computational software package2 for all tested cases. The test cases included: (1) A l l 6j-symbols with ] x = j2 = j3 = j4 = j5 = j6 = X for X = 0, 1, 2, 98, 99, 100, as well as X = 111, 122, 133, 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 < lx > \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 fast3 as the commercial software package. The user of 6jfunction, c should, however, note that no guarantee is placed on the function's reliability to accurately evaluate all 6j-symbols.4 In fact, it will likely be necessary to increase the minimum precision of the 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' . 2 Caut ion 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 Ponzano-Regge 6j-symbols. 3 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 Mathe-matica. As well, the relative time saving tended to increase with the size of the 6j-symbol arguments. 4Obviously, the author of 6 j func t ion.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. 128 Appendix 129 The 6 j f u n c t i o n . h header file: /* 6jfunction.h, the header f i l e for 6jfunction.c */ double sixjfunction(double 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 6jfunction.h. */ The 6jfunction, c code: /* Using GMP (version 2.0.2) variables t h i s program calculates the 6j— symbol and returns a type double. The results are exactly the same (to the displayed precision) as those of Mathematica for a l l tested cases. The test cases included: (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 well as X=222,333,444, 555; (2) Expectation value of the X edge length of the [{X,X,X},{X,X,X}] tetrahedron for K=0,1,2,...200; (3) Expectation values of the X edge length of the [{X,X,X},{Y,Y,Y}] tetrahedron for a l l K for Y=10, 20, 30, ...,80,90,100. */ /* L i s t l i b r a r y dependencies and define global constants. */ #include <stdio.h> #include <math.h> #include "gmp.h" #include "6jfunction.h" #define NMAX 14 #define DMAX 18 double sixjfunction(double j j l , double j j 2 , double j j 3 , double j j 4 , double jj5,double jj6) { /* L i s t function prototypes. */ double sixjsymbol(double, double, double, double, double, double); /* Declare sixjvalue and i n i t i a l i z e to 0. */ double sixjvalue=0; /•Tests to see i f the j-values are "admissible". If "inadmissible", then Appendix 130 re turns a value of ze ro . I f "admiss ib le" , then ca l cu l a t e s the 6 j -symbol value and re turns the f l o a t i n g poin 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 j3 ) -f loo r ( j j l+ j j2+ j j3 ) )>0 .999) && ( ( ( j j 3+ j j4+ j j5 ) - f l oo r ( j j 3+ j j4+ j j5 ) )< 0.001 | | ( ( j j3+j j4+j j5 ) - f loor ( j j3+j j4+j j5 ) )>0 .999) && ((( j j5+j j6+ j j l ) - f l o o r ( j j 5 + j j 6 + j j l ) ) < 0 . 0 0 1 I I ( ( j j 5 + j j 6 + j j l ) - f l o o r ( j j 5 + j J 6 + j j l ) ) > 0 . 9 9 9 ) && ( ( ( J j2+j j4+j j6) - f loor ( j j2+j j4+j j6) )<0 .001 II C(jj2+ J j4+j j6 ) - f loor ( j j2+j j4+j j6 ) )>0 .999) && ( ( ( j j1+ j j2+ j j4+ j j5 ) -f loo r ( j j l+ j j2+ j j4+ j j5 ) )<0 .001 I I ( ( j j l + j j 2 + j j 4 + j j 5 ) - f l o o r ( j j l + j J 2 + j j4+jj5))>0.999) && ( ( ( j j2+ j j3+ j j5+ j j6 ) - f loor ( j j2+ j j3+ j j5+ j j6 ) )< 0.001 II ( ( J j2+j j3+j j5+j j6)- f loor( j j2+j j3+j j5+j j6))>0.999) && ( ( ( J j l+ j j3+ j j4+ j j6 ) - f loo r ( j j l+ j j3+ j j4+ j j6 ) )<0 .001 I I ( ( j j 1+j j3+j J4+ J j6 ) - f l oo r ( j j l+ j j 3+ j j4+ j j6 ) )>0 .999 ) ) { s i x j v a l u e = s i x j s y m b o l ( j j l , j j 2 , j j 3 , j j 4 , j j 5 , j j 6 ) ; r e t u rn ( s i x j v a l u e ) ; } e l se { s ix jvalue=0; r e tu rn ( s i x j v a l u e ) ; } } / * L i s t i n g of func t ion s ix jsymbol whose prototype i s g iven above. */ double s ixjsymbol(double J l , double J 2 , double J3 , double J4 , double J5 , double J6) / * Returns value fo 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 func t ion prototypes . * / long longcompare(const v o i d * p o i n t l , const v o i d *poin t2) ; / * Declare 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; long zmin3=0; long zmin3factor=0; double sixjsum=0; double DoubleNum[NMAX]; Appendix 131 double DoubleDen[DMAX] ; int Ncount=0; int Dcount=0; long RoundNum[NMAX]; long RoundDen[DMAX] ; long OrderedRoundNum[NMAX]; long OrderedRoundDen[DMAX]; unsigned long UnsignedLongl=0; in t Fcount=0; long Z1=0; long Z2=0; int FFcount=0; /* Declare a l l GMP variables. */ mpf_t mpzmin3factor; mpf_t sixjsumfactorl; 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 variables. */ mpf_init2 (mpzmin3factor, 256); mpf_init2 (sixjsumfactorl, 512); mpf_init2 (sixjsumfactor2, 512); for (FFcount=0; FFcount<DMAX; FFcount++) { mpf_init2 (FinalArray[FFcount], 512); } mpf_init2 (Intermediate1, 512); mpf_init2 (Unity, 256); for (FFcount=0; FFcount<12; FFcount++) { mpz_init (mpArray[FFcount]); } mpz_init (mpArraylndexCount); Appendix 132 mpz_init (mpArrayValue); mpz_init (mpFinalArrayValue); mpf _ i n i t 2 (mpFloatFinalArrayValue., 512) ; mpf_init2 (mpInverseFinalArrayValue, 512); /* Set GMP variables Unity and sixjsumfactor. */ mpf_set_si (Unity, 1); mpf_set_si (sixjsumfactor2, 0); /•Makes sure a l l the f a c t o r i a l s are defined.*/ while(zminl-Jl-J2-J3<0 I I zminl-J3-J4-J5<0 || zminl-J5-J6-Jl<0 || zminl-J2-J4-J6<0) { zmin2=++zminl; } /•Executes conditional loop which evaluates the sum over integer z i n the s i x - j symbol as well as the delta(nl,n2,n3)'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+J6-zmin2>=0) { zmin3=zmin2++; /* F i r s t define 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 to the nearest integers, then puts the elements i n descending order. */ for(Ncount=0; Ncount<NMAX; Ncount++) { /* Round off 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 to the nearest integers, then puts the elements i n descending order. */ for(Dcount=0; Dcount<DMAX; Dcount++) { /* Round off the array of elements. */ RoundDen[Dcount]=floor(DoubleDen[Dcount]); Appendix 134 if(DoubleDen[Dcount]-RoundDen[Dcount]<0.5) { OrderedRoundDen[Dcount]=RoundDen[Dcount]; > else { OrderedRoundDen[Dcount]=RoundDen[Dcount]+1; > > /* Sort the array RoundDen i n descending order. */ qsort(OrderedRoundDen, DMAX, sizeof(OrderedRoundDen[0]), longcompare); /* Computes f i r s t NMAX elements of FinalArray: Takes r a t i o of fa 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, Unity); while(OrderedRoundNum[Fcount] -Zl> OrderedRoundDen[Fcount]) { Z2=Z1++; UnsignedLongl=(OrderedRoundNum[Fcount]-Z2); mpf_mul_ui (Intermediate1, Intermediate 1, UnsignedLongl); } mpf_set(FinalArray [Fcount], Intermediate1); } else { Z1=0; mpf_set (Intermediate1, Unity); while(OrderedRoundDen[Fcount]-Zl> OrderedRoundNum[Fcount]) { Appendix 135 Z2=Z1++; UnsignedLongl=OrderedRoundDen[Fcount] -Z2; mpf_mul_ui (Intermediate1, Intermediatel, UnsignedLongl); } mpf_div (Intermediatel, Unity, Intermediatel); mpf_set(FinalArray[Fcount], Intermediatel); > > /* Computes the remaining DMAX-NMAX elements of FinalArray: 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 function for the f a c t o r i a l : Returns the value (OrderedRoundDen[Fcount])! as a multiple precision integer.*/ /* Declare and i n i t i a l i z e variables. */ long BiggestArrayIndex=0; long OrderedRoundDenValue=0; /* Set GMP variables 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 (mpArray[3], 6); mpz_ set. . s i (mpArray[4] , 24); mpz_ set. . s i (mpArray[5] , 120); mpz_ set. . s i (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]; /* If OrderedRoundDenValue i s 0,1,...,10,11 then take f a c t o r i a l from mpArray variables. */ i f (OrderedRoundDenValue ==0) { mpz_set (mpFinalArrayValue, mpArray[0] ); > else i f (OrderedRoundDenValue == 1) { mpz_set (mpFinalArrayValue, mpArray[l]); } else i f (OrderedRoundDenValue == 2) { mpz_set (mpFinalArrayValue, mpArray[2]); > else i f (OrderedRoundDenValue == 3) { mpz_set (mpFinalArrayValue, mpArray[3]); > else i f (OrderedRoundDenValue == 4) { mpz_set (mpFinalArrayValue, mpArray[4]); } else i f (OrderedRoundDenValue == 5) { mpz_set (mpFinalArrayValue, mpArray[5]); > else i f (OrderedRoundDenValue == 6) { mpz_set (mpFinalArrayValue, mpArray[6]); } else i f (OrderedRoundDenValue == 7) { mpz_set (mpFinalArrayValue, mpArray[7]); } else i f (OrderedRoundDenValue == 8) { mpz_set (mpFinalArrayValue, mpArray[8]); } else i f (OrderedRoundDenValue == 9) { mpz_set (mpFinalArrayValue, mpArray[9]); > else i f (OrderedRoundDenValue == 10) { mpz_set (mpFinalArrayValue, mpArray[10]) } else i f (OrderedRoundDenValue == 11) Appendix 137 { mpz_set (mpFinalArrayValue, mpArray[11]); > /* If OrderedRoundDenValue > 11 then calculate 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, Unity, mpFloatFinalArrayValue); mpf_set (FinalArray[Fcount], mpInverseFinalArrayValue); > /* Computes sixjsum from the elements of FinalArray. */ mpf_set (sixjsumfactorl, Unity); for(Fcount=0; Fcount<DMAX; Fcount++) { mpf_mul (sixjsumfactorl, sixjsumfactorl, FinalArray[Fcount]); } mpf_sqrt (sixjsumfactorl, sixjsumfactorl); zmin3factor=pow(-l, zmin3); mpf_set_si (mpzmin3factor, zmin3factor); mpf_mul (sixjsumfactorl, mpzmin3factor, sixjsumfactorl); mpf_add (sixjsumfa c t o r 2, sixjsumfa c t o r 2, sixjsumfactorl); sixjsum=mpf_get_d (sixjsumfa c t o r 2 ) ; > } else { Appendix 138 sixjsum=0; } / * Clear a l l GMP var iables . * / mpf_clear (mpzmin3factor); mpf_clear (s ixjsumfactorl ) ; mpf_clear (sixjsumfactor2); for (Fcount=0; Fcount<DMAX; Fcount++) { mpf_clear (FinalArray[Fcount]) ; } mpf_clear (Intermediate1); mpf_clear (Unity); mpz_clear (mpArrayIndexCount); mpz_clear (mpArrayValue); mpz_clear (mpFinalArrayValue); mpf_clear (mpFloatFinalArrayValue); mpf_clear (mpInverseFinalArrayValue); for (FFcount=0; FFcount<12; FFcount++) { mpz_clear (mpArray[FFcount]); > return sixjsum; / * L i s t i n g of function longcompare (compares sizes of two type long var iab le s , and reports the resu l t to qsort) whose prototype i s l i s t e d i n funct ion sixjsymbol. * / long longcompare(const void * p o i n t l , const void *point2) { r e turn( - *(long *)point l + *( int *)point2); } 

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