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

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