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The phase space of 2+1 gravity Fugleberg, Todd Darwin 1996

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T H E P H A S E S P A C E O F 2+1 G R A V I T Y B y Todd Darwin Fugleberg B . Sc., The University of Saskatchewan, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF SCIENCE in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA December 1995 © Todd Darwin Fugleberg, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Br i t i sh Columbia , I agree that the Library 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 The University of Br i t i sh Columbia 2075 Wesbrook Place Vancouver, Canada V 6 T 1W5 Date: Abstract In recent years there has been a resurgence of interest in 2+1 gravity and there have been claims that 2+1 gravity is quantizable. In order to understand and evaluate these claims the classical phase space on which quantization is attempted must be understood. This thesis is an attempt to understand the phase space of 2+1 gravity in terms of physical models. We write the action of 2+1 gravity in the connection formalism entirely in terms of the holonomies of a genus g surface. We apply this formulation to the genus one and two surfaces. We analyze the structure of the genus two constrained configuration space in detail to show that it consists of five disconnected components. Relat ing our results to a more mathematical analysis we show that only two of these regions are physically relevant and these two are identified wi th one another. Final ly, we discuss the phase space of the genus one and two surfaces including the effect of large diffeomorphisms. We conclude that the theory does not lead to a well defined quantization. i i Table of Contents Abstract ii Table of Contents iii List of Figures vi Acknowledgement viii 1 Introduction 1 1.1 Mot iva t ion for 2+1 Dimensional Gravi ty 1 1.2 Overview 3 2 T h e Hilbert Act ion 5 2.1 Hilbert Ac t ion in Terms of Dreibein and Connection 5 2.2 Hilbert Ac t ion in Terms of Holonomies 10 3 Specific Examples of the Action 20 3.1 Appl ica t ion to the Genus One Surface 20 3.2 Appl ica t ion to the Genus Two Surface 26 3.3 Summary 29 4 Construction of Solutions to the Configuration Constraint 31 4.1 Construction of Solutions for the Genus One Surface 34 4.2 Construction of Solutions for the Genus Two Surface 34 4.3 Construction of Solutions for the Genus g surface 52 i i i 5 Structure of the Reduced Configuration Space 53 5.1 The Structure of the Configuration Space for the Genus Two Surface . . 53 5.2 Summary 60 6 Realization of Solutions 62 6.1 Real izat ion of Solutions for the Genus Two Surface 63 6.2 Ti l ings of the Genus Two Surface 65 6.3 Interpretation of the Ti l ings 73 6.4 Summary 78 7 T h e Phase Space 79 7.1 The Phase Space of 2+1 Gravi ty on the Genus One Surface 79 7.2 The Phase Space of 2+1 Gravi ty on the Genus Two Surface 81 7.3 Large Diffeomorphisms 85 7.4 Quantizing the Theory? 90 8 Conclusions and Areas for Future Research 92 8.1 Conclusions 92 8.2 Future Research 93 Bibliography 96 Appendices 98 A SL(2,R) 98 A . l The SL(2 ,R) Representation of 5 0 ( 2 , \ ) \ 98 A . 2 The Adjoint Ac t ion on SL(2,R) 102 A . 3 The Universal Covering Space of SL(2,R) 103 iv B Hyperbolic Geometry List of Figures 2.1 The one holed torus as a fundamental region wi th identified sides 11 2.2 The two holed torus as a fundamental region with identified sides 12 4.3 The set of unit spacelike vectors (ba) for which the Case 1(a) constraint equation is solvable 37 4.4 The set of unit spacelike vectors (ba) for which the Case 1(b) constraint equation is solvable 39 4.5 The set of unit timelike vectors (b ) for which the Case 2(a) constraint equation is solvable 40 4.6 The curves on the null cones define the sets of null vectors (ba) for which the Case 3(a) constraint equation is solvable 41 4.7 The set of unit spacelike vectors (b ) for which the Case 4(a) constraint equation is solvable 42 4.8 The set of unit spacelike vectors (6°) for which the Case 4(b) constraint equation is solvable 43 4.9 The intersection of these surface defines the curve (b ) for which the Case 7(a) constraint equation has solutions 46 4.10 The set of unit spacelike vectors (ba) for which the Case 7(b) constraint equation is solvable 47 5.11 The paths in the covering space of SL(2,R) for —tt and f J - 1 such that both go to negative trace representatives 60 v i 6.12 The regular tile 67 6.13 The regular tile and a Lorentz transformed copy 68 6.14 A tile in the negative trace region 69 6.15 A tile in the negative trace region 70 6.16 A tile in the t r iv ia l region 71 6.17 Illustration of the path across an identified edge for overlapping patches corresponding to solutions in the t r iv ia l region of the configuration space. 72 7.18 The action of a Dehn twist 85 7.19 Another large diffeomorphism 86 7.20 The configuration space in the genus one case. The regions between the solid lines are a l l identified. The dotted lines are identified and each has periodic identifications wi th period e along their length 88 A.21 The positive trace Lorentz spinor matrices are associated wi th the vectors between or on the hyperbolas wi th identification of the hyperbolas. The identification is indicated by the line connecting points on the hyperbolas. The diagram is rotated around the vertical (t) axis 100 A . 22 The universal covering space of SL(2 ,R) . Note that the parabolic regions meet at infinity 104 B . 23 Geodesies on the hyperbolic plane 108 B.24 Branch Point of arcsin 109 v i i Acknowledgement I would like to thank my supervisor, W . Unruh, for his help and guidance which have greatly benefitted this work. I am grateful to N . Weiss for his cri t ical reading of this thesis. The idea for this line of inquiry arose from discussions wi th M . Blencowe. I would also like to thank K . Schleich for her help and suggestions, and R . Szabo and L . Paniak for many helpful discussions. I am also grateful to S. Car l ip for helpful email correspondence. I would like to acknowledge the financial support of N S E R C over the last two years. The Department of Physics at U . B . C . has provided a stimulating and positive working environment. Thanks to a l l my friends here at U . B . C . for helping make Vancouver a friendly place to live. M y family has my gratitude for their support and encouragement which have given me the confidence to pursue my studies, including this research. This work would not have been possible without the consistent moral and emotional support of my wife L i s a which has been essential to al l of my endeavors including the present work. v i a Chapter 1 Introduction 1.1 Motivation for 2+1 Dimensional Gravity "2+1 gravity" is the name applied to theories of gravity formulated in a spacetime con-sisting of two spatial dimensions and one temporal dimension rather than the usual "3+1" spacetime. The justification for studying gravity in 2+1 dimensions is that it provides a simpler, exactly solvable system that may teach us some things about the more complex problem. The equations of motion of General Relat ivi ty obtained from the Hilbert action form a system of non-linear partial differential equations which constitute a formidable problem. In 2+1 dimensions a new form of this well known action can be derived which leads to a linear system of equations of motion that is actually easily solvable. Another simplification that comes into play is the result that the local geometry of the spacetime becomes unimportant. The topology of the spacetime plays a more fundamental role. Another way of saying this is that there are no local degrees of freedom, only global ones. Therefore the theory wi l l have a finite and actually very small number of degrees of freedom rather than a twofold "infinity of degrees of freedom. Further motivation for this line of study was provided by the encouraging result first reported by W i t t e n [1] that gravity in 2+1 dimensions can be formulated as a Chern-Simons gauge theory which offers hope that the theory might be quantizable. Indeed Car l ip has stated "quantization is straightforward" [2] and Peldan that "one can now say 1 Chapter 1. Introduction 2 that gravity has been successfully quantized in 2+1 dimensions" [3]. Further, others [4] [5] have assumed the existence of a well defined quantization procedure in their discussion of quantum theories of 2+1 gravity. Understanding these claims requires that we first understand the dynamical variables and the structure of the phase space. The goal of this thesis is to develop understanding of the phase space of 2+1 gravity from a physicist's perspective. In order to accomplish this goal physical realizations of solutions to the equations of motion are constructed following Unruh and Newbury [6]. These realizations are then compared and related to previous more mathematical statements about the structure of the phase space. F ina l ly we examine the claims of quantizabili ty of 2+1 gravity in the context of these physical models. Throughout most of this thesis we have examined the genus one and two cases in parallel. This is because the genus one case differs considerably from the higher genus cases. There is a more unified description of the higher genus cases and so the genus two surface w i l l be studied in detail. One of the obstacles to developing a quantum theory of gravity can be phrased as a lack of good dynamical variables. There are currently three methods of formulating 2+1 gravity wi th different sets of dynamical observables. The first formulation is the conventional metric formulation in which the dynamical variables are the components of the metric [2] [7]. The second main formalism is the connection formalism in which the dynamical variables are the components of a basis field (dreibein) and the associated connection coefficients. The final formalism is the loop formulation where the dynamical variables are defined in the loop algebra of traces over holonomies of the spacetime[8]. We w i l l use the connection formalism, but the action w i l l be reformulated in terms of the, holonomies of the surface since there is a correspondence between flat connections on a surface and its holonomies. This differs from the loop formulation but the two seem Chapter 1. Introduction 3 to be related. 1.2 O v e r v i e w The Hilbert action on a 2D surface wi th non-trivial topology can be written exclusive-ly in terms of the holonomies of the surface. The holonomy of a closed curve is the transformation that a vector undergoes when parallel transported around the curve. The holonomies of a surface are the holonomies around the non-contractible loops implied by the non-tr ivial topology. Curves which are homotopic - continuously deformable to one another - w i l l have the same holonomy. Thus there is one holonomy for each homotopy class of non-contractible loops of the surface. We w i l l see that the holonomies are Poincare transformations. The Lorentz part of these holonomies can be treated as configuration variables and we can examine the configuration space by solving the configuration constraint in the SL(2,R) representation. We can use these solutions to construct a geometric picture of the spacetime. The goal is is to develop understanding of the phase space of 2+1 gravity in order to evaluate the claims of quantization of the theory. In Chapter 2 the Hilbert action wi l l be developed in the general connection formalism and then formulated in terms of the holonomies of the surface. In Chapter 3 this action w i l l be applied to the genus one surface and then to the genus two surface. Chapter 4 illustrates how solutions to the constraint equations can be constructed and in Chapter 5 the details of the construction are used to make some comments about the structure of the genus two configuration space. In Chapter 6 geometric realization of the solutions to the genus two constraint equation are presented. The relation between these realizations and the Goldman classification of regions of the configuration space w i l l also be discussed. Chapter 7 w i l l be concerned wi th the structure of the full phase space for genus one Chapter 1. Introduction 4 and two and possible paths to quantization including a discussion of the status of large diffeomorphisms. Chapter 8 w i l l discuss conclusions and areas for future research. Chapter 2 The Hilbert Act ion 2.1 Hilbert Act ion in Terms of Dreibein and Connection The Hilbert A c t i o n of general relativity in 2+1 dimensions is given by: I = J yJ\g\Rdz, X (2.1) where g is determinant of the metric and R is the scalar curvature. One of the difficulties of using this action principle is the fact that the square root of \g\ makes the equations of motion nonlinear and nonpolynomial and thus increases the difficulty of solving them. This problem can be avoided by introducing [1] the dreibein, e^ a, a collection of three vectors in the spacetime which can be viewed as a basis. The upper La t in indices label the 3 vectors while the lower Greek indices label components of these vectors. (Note: The L a t i n letters i,j,k w i l l be used to indicate spatial components not vector labels which w i l l be taken from the start of the alphabet.) The dreibein w i l l be assumed to be an orthonormal basis. The dreibein can be thought of as a vector space isomorphism from the space wi th the orthonormal basis, which is referred to as the internal space, to the coordinate basis spacetime, which is referred to as the Lorentz space or external space. A s well as labelling the vectors of the dreibein, the La t in indices can be thought of as internal space component labels. V/3 = vaep a (2.2) 5 Chapter 2. The Hilbert Action 6 The inverse (if it exists) of this vector space isomorphism, e^ a, can be thought of as a covector basis: e " „ e / = 6%. ( 2 . 3 ) The metric on the Lorentz space can now be written using the vector space isomor-phism and the Minkowski metric on the internal space: 9,tu = ?7a&e/e„ 6 . ( 2 . 4 ) Implicit in this is the assumption that the internal metric is constant. The dreibein are assumed to be orthonormal in the Lorentz spacetime: e^e^ = rf. ( 2 . 5 ) Define the interior covariant derivative to act on internal vectors, pa, as: D»pa = p % + < V 6 p 6 ( 2 . 6 ) where we have introduced the internal connection coefficients u ^ V O n external vectors define it to be: £ > X = (2-7) and therefore: D^eva = eva,ll+uflabevb. ( 2 . 8 ) The full covariant derivative of objects in this mixed space is: V . e / E e / ^ - r ^ + ^ V ; ( 2 . 9 ) where T" are the usual connection coefficients of the Lorentz space. The spacetime connection is metric compatible i f we assume that the beins are co-variantly constant ( V M e „ a = 0 ) and that the spacetime connection, T ^ , is torsion free: V r < ^ , = V r (p^efe/) = V r ( ? ? a 6 ) e / e „ 6 + r]abVT ( e / ) e „ 6 + f]abe^aVr (ej3) = 0 . ( 2 . 1 0 ) Chapter 2. The Hilbert Action 7 The dreibein w i l l be used as fundamental variables. A s in the Pala t in i [10] formal-ism, the components of the internal connection, u^b, w i l l also be used as fundamental variables. Related to the connection is the spin connection, u^a = ^s^co^bc which w i l l be mentioned briefly. The internal metric was assumed to be covariantly constant, which means that the connections must be antisymmetric in their internal indices[9]: The internal curvature tensor, Wab, can be defined using the inverse drebein as a vector space isomorphism from the cotangent space to the internal "cotangent" space. The dreibein maps vectors from the internal space to the external space and thus the curvature tensor can be defined on vectors in the internal space: R^ap = e\e\Rahap (2.12) where Rap1$v13 = 2 V [ 7 V ^ ? ; Q . M u l t i p l y this result by eaa to obtain: eaaRah6ePbeabv'T = e Q a 2 V [ 7 V ^ t ; Q = 2VbVg]eaava = 2DbDs]eaavQ = R \ 6 e a b v a (2.13) where vanishing of the covariant derivative of the dreibein and the fact that the spacetime connection is torsion free has been used. Notice that: yj\g\ = y/\detg^v\ = yj\det(r]abe^eub)\ = yj\ - det(e^ a)det(e I / 6)| = </[ctet(e/)] 2 = de t (e / ) = e (2.14) where e = ^etJ,VT£a})CeliaevbeTc and as a result: e e ^ A = ea^eabce7 c . (2.15) Chapter 2. The Hilbert Action 8 Now return to the Hilbert action: yfc\R = ee^aeP\RabaP = ea^sabce,cRaba0. (2.16) The internal curvature tensor is: Rab-fSPb = 2 (o;ab[s,7] +uac[7ujcbg]j pb. (2.17) Therefore the action can be written: .ab [a,/3] +Uad[atJdb(3]) dx° A dxP A dx\ (2.18) Notice that at this point and for the rest of this discussion that the Lorentz space indices are lower. The absence of Lorentz indices means that this action allows degenerate metrics since g^v need not be calculable. This is crucial to the derivation in the next section. We w i l l see that a l l solutions wi th nondegenerate metrics are gauge equivalent to degenerate metrics. This prospect seems disturbing and may have some very important implications but we w i l l not discuss them. Vary ing this action wi th respect to the e^a gives an equation of motion that is e-quivalent to the equation of motion, — 0, which is an equation of motion for the original formulation. Varying wi th respect to the connection leads to the equation of motion, D^e^f = 0, which is the torsion free condition of the connection assumed in the Hilbert action. Therefore the two action principles lead to the same physics and must be considered equivalent. This "new" action can be written in the form: The last term is a total derivative term and thus contributes only boundary terms which are usually ignored. However, on a closed spacetime these boundary terms w i l l lead to the new form of the action described in the next section. 1 = e2a6jb\ + 2cubcQ(D2ea1 - Diea2) + ea0Rbc12 - 2 t 9 [ 1 ( e ^ c ) ] dzx. (2.19) Chapter 2. The Hilbert Action 9 Thi s action is now seen to be in Hamil tonian form wi th configuration variables u i and u>hc2 wi th conjugate momenta — e<f and ei° respectively (Note the crosslabeling). Further the Hamil tonian is simply a sum of constraints wi th Lagrange multipliers eo a and ubcQ. There seems to be no evolution in this theory. Vary ing wi th repect to the Lagrange multipliers produces the constraint equations: D2ela - I W = 0 (2.20) i ? 6 c i 2 = 0. (2.21) The second constraint indicates that the spatial curvature on a spatial slice is zero and thus the interior of the patch is flat. If the entire space can be covered by a single patch then there are no dynamics and the theory is t r iv ia l . Assuming the fields are well behaved as you approach spatial infinity we can ignore the boundary terms and the Hamil tonian is s imply a sum of constraints. For a manifold wi th nontrivial topology, however, the introduction of patch boundaries allows the boundary term to contribute. We w i l l consider spacetimes consisting of R<S> S where E is a closed compact 2-manifold. This means that S is a g-holed torus which is also called a genus g surface. These S have the non-tr ivial topology desired. Returning now to the constraint equations we examine the gauge transformations in the fundamental variables that they generate through Poisson brackets. The first constraint (2.20) generates a local Lorentz transformation of the dreibein. The infinitesimal form of this variation is: be? = eabceicrb (2.22) 1 1 ,, fated = ^£aderan + -£ade£f "WifU. (2.23) The first equation has exactly the form of a infinitesimal Lorentz transformation. The generators of the matr ix group 50(2 ,1)+ 1 are given by [ Ja]bc = £ a b (where Ja is a 3 x 3 xThis is the portion of the Lorentz group connected to the identity. The restricted Lorentz group. Chapter 2. The Hilbert Action 10 matr ix wi th components labelled by b and c). A n infinitesimal Lorentz transformation is therefore: v'a &va + Tcecbavb (2.24) where the second term has the same form as in (2.22). These local variations can be integrated to the equations: el = U\e\ (2.25) ufb = UacUbdujicd + UadUbd,i. The second constraint (2.21) generates a gauge transformation of the form: <5e/ = p ^ + ^ a d P d (2.26) defied — 0 which can immediately seen to be a local form of: e / = e / + Z V " (2-27) while the connection is left unchanged. This transformation includes infinitesimal coordi-nate tranformations[6], but is more general. It allows us to set e^a to zero in some finite region i f the constraints are satisfied. This results in the degenerate metric discussed previously. No coordinate transformation can result in a degenerate metric. 2.2 Hilbert Act ion in Terms of Holonomies Now starting from the action (2.19) we derive a form of the action written entirely in terms of the holonomies of the spacetime. This derivation follows that of [6] and [9] closely but is a hybrid of both. Chapter 2. The Hilbert Action 11 4 C 3 D B 1 A 2 Figure 2.1: The one holed torus as a fundamental region wi th identified sides. A genus g surface can be described by taking a 4g sided "tile" and identifying pairs of sides. See Figures 2.1 and 2.2 for the genus 1 and 2 cases respectively. Notice that a l l the vertices of the tile are identified. Assume that we have a continuous non-vanishing dreibein field on the interior of the tile which produces a positive definite metric on the tile. The dreibein field is not assumed to be continuous across the identified boundaries as we cannot have a continuous nonvanishing vector field on a surface of genus g > 1[6]. Assume that there exists a choice of gauge such that the metric g^ is well defined and unique everywhere. This means that in the overlap regions of the coordinate charts of the manifold, the dreibein must be related by an S O ( 2 , l ) 2 transformation, U. where X and Y refer to two overlapping patches. For the purposes of this thesis we restrict our attention to SO(2,1)+ - the restricted Lorentz group. A s we cross a boundary of the fundamental tile the dreibein must be related by the transformation that relate the coordinate charts on the patches that contain the boundary. The connection transforms 2The 2+1 Lorentz group. (g^)x = r)abeXilaeXvh = nab(UaceYllc)(UbdeYl/d) = (UacUbdrjab)eY)1ceYl/d (2.28) Chapter 2. The Hilbert Action Figure 2.2: The two holed torus as a fundamental region wi th identified sides. Chapter 2. The Hilbert Action 13 such that it is the connection of the transformed dreibein: ex,a =WXYabeYbtl COX^ = WXYacWXYBDLUYTLCD + WXYadWxY bd (2.29) (2.30) where the X and Y refer to two adjacent overlapping patches. Notice that this is a gauge transformation of the fundamental variables as well as a transformation of the Lagrange multipliers. The constraint equation (2.21) states that the curvature components for an internal vector transported parallel to the surface of the tile are zero. Thus, the parallel transport of a vector around a closed contractible curve in the interior of the tile w i l l be the same as the original untransported vector. Notice that for a genus g surface we have 2g noncontractible loops 3 on the surface - one surrounding each hole and each handle. These loops correspond to curves across the tile between identified points on identified sides. The dreibein across the boundaries are related by (2.29) and so an internal vector crossing the boundary w i l l undergo a transformation by the holonomy of that curve. The genus g surface has 2g holonomies. Assuming the constraints are satisfied, a local gauge transformation of the form (2.27) can be used to set the spatial components of the connection (<£;%) to zero. Using local gauge transformations of the form (2.26) the spatial components of the dreibein field (e; a) can then also be locally set to zero. Refer to [6] [9] for details. Notice this means that our metric is spatially degenerate, but these simplifications are crucial to the derivation. The action is now: 3 Loops which can be deformed to one other are equivalent for our purposes. Chapter 2. The Hilbert Action 14 eabceaaRbcPldxa A dx13 A dx1 (2.32) + / eabcpaRbc(j-fdx^ A dx1 £abcpaD[aRbc^]dxa A dx13 A dx1 where integration by parts has been used. The Bianchi identity causes the last term to vanish. The constraint (2.21) and the fact that the spatial components of the dreibein have been gauged away means that the first term also vanishes leaving only the integral over the boundary. The boundary includes the ini t ia l and final hypersurfaces and the identified spatial edges of the evolving tile. According to the definition of the Hamil to-nian ac t ion 4 no variation of the fundamental variables is allowed on the in i t ia l and final hypersurfaces and therefore only the spatial boundaries need be considered. The contri-bution of the identified spatial boundaries give rise to the terms that we are interested in . where the vanishing of the spatial components of the connection and the fact that the boundary of a boundary vanishes have been used. The spatial part of the integral is an integration over the edges of the polygonal tile. These edges are identified so take the difference of the integrand across each pair of identified edges: I (2.33) edges edges (2.34) 4Modified Hamilton's Principle. Chapter 2. The Hilbert Action 15 where the sum only includes one of each pair of identified sides. The local gauge transformation within the tile is: ha =U{x)\e\ (2.35) Uiab = U(x)acU(x)bdUicd + U{x)adU{x)bdn . (2.36) The Lorentz identification across the edges is: ex„a =WXYabeYbi (2.37) ux„ab = WXYacWXYbdojYtlcd + WXYadWXYbd^ (2.38) where X and Y label the the two sides of an identified edge. The transformation of the intermediate dreibein (e 8 a) across the identified edges is given by: ~eXia = Ux(x)\eXib = Ux(x)abWbceYcfl = Ux(x)abWbc[UY{x)-l]cdeYdl = W(x)adeYd. (2.39) Using the Lorentz transformation W(x)ad to relate the UJ across the edge, the spatial components give the result: 0 = 0 + W(x)adW(x)bd,i. (2.40) This tells us that the W(x) = W are constant along each spatial edge. The local gauge field U(x) transforms across identification boundaries as: Ux{x) = W X Y U Y ( X ) W X Y . (2.41) Returning now to the action (2.34) and using (2.36) we obtain: / = E / £abc [pxa,i (Ux(x)bdUx(x)ceuxode + Ux(x)bdUx(x)cd,Q) edges Jed9es - pYa,i (UY(x)bdUY(x)ceuYOde + UY(x)bdUY(x)cd,0)} dx'dt. (2.42) Chapter 2. The Hilbert Action 16 We also have: ^ x / 6 = WXYacWXYbdcuYflcd + W X Y a d W X Y b d „ . (2.43) Substituting for cux and Ux(x) makes the first term in the action: E / z«hcPxan ([WXYUY(x)]bd[WXYUY(x)}ceujYode (2.44) edges Jed9es ^WXYUY(x)]bf[WXYUY(x)WXY]ceWXYef,o + [WXYUY(x)WXY}be[WXYUY(x)W^Y]ce,Q) dx'dt. The first term above can easily be combined wi th the first part of the second term in the action: h = E / eabc(Px%iWbXYfUY(xydWXYgUY(xyeuYOde (2.45) edges Jed9es -pYa,i UY(x)bdUY(x)ceojYOde) dx'dt = E / eabc(pXd,iWXYda - PY'^UYixffUyW^UYof'dx'dt (2.46) edges Jed9es using the fact that: £-abcU xU yU z — £xyz- (2-47) Remember that u>oab is a Lagrange multiplier and since the other terms do not contain it, I\ produces the constraint equation: Pxb,iWXYba-pYa,i=0. (2.48) WXY is constant along the spatial edge from (2.40) and therefore this constraint is equiv-alent to: Pvb = PxaWXYba + U B (2.49) where the I I 6 are arbitrary constants of integration. Interpreting pb and II 6 as vectors in the internal space and recalling that WXY is a Lorentz transformation this equation Chapter 2. The Hilbert Action 17 tells us that the holonomies of the noncontractible loops are, in general, Poincare trans-formations. Note that this constraint allows the simple solution that the holonomy is a Lorentz transformation, but the general solution of the constraint allows translations as well . The remaining terms in the action (2.42) are: h = E / ^bc [pxa,i ([WXYUY(x)]bf[WXYUY(x)WXY]ceWXYef,0 (2.50) edges Jed9es + [WXYUY{x)WXY)be\WXYUY{x)WXY]c\Q) - pYanUY(x)bdUY(x)cd,0) dxUt. Use (2.49) to substitute for pxa,i. Make use of (2.47) and expand out the derivatives producing: h = Y , l £-bA PYa» ( [ / y l ^ V f / H i ) 0 ^ ^ ^ ^ " ^ ( ^ ^ y l x ) ^ ) _ j Jedaes v ' + Px\i ({WXYUY(x)W^Y}be[WXYUY(x)W^Y]ce (2.51) + . [WXYUY(x)W^Y]be[WXYUY(x)W^Y}ce + [WXYUY(x)WXY]be[WXYUY(x)W^Y]ce)] dx'dt h = Y , l £-bc[ PYa,i (UY(x)bfUY(xydW^YdeWXY^-UY(x)bdUY(x)cd) j J edaes ' + Px\i ( [ ^ x y C / K ( x ) ^ ] 6 e [ ^ y - [ / y ( a ; ) ^ ] c e ) (2.52) + pya,i (uY(x)bdWXYdeUYf(x)Wx1Yfe edges edges + dxldt edges UY{x)bdWXYdeUY{x)cfWXYfe)\ = E / e « 6 c PY°» (UY(x)bfUY(x)cdWXYdeWXYef - UY(x)bdUY(x)cd) (2.53) _ j Jedaes v ' + Pxa,i ([WXYUY(x)W^Y]be[WXYUY(x)W^Y]ce) + PYa,i {uY{x)bdU^d{x) + UY(x)bdUY(x)cfW^YdeW^Yfe) dx'dt Chapter 2. The Hilbert Action 18 = £ / eabcPYaJuY(x)bfUY(x)cdWXYedWXYEF + UY(xfdUY(x)cfWXYedWXYef) edges Jed9es + Pxan (\WXYUY(x)WXY]be[WXYUY{x)WXY)ce) dx'dt. (2.54) The first two terms are antisymetric wi th respect to interchange of b and c so they vanish leaving finally: / 2 = £ / ea^^ilW^U^W^eiWxyU^W^dx'dt (2.55) edges Jed9es = £ / ea^px^iWxY^UyixYfW^eWxY^UYix^WxWdx'dt (2.56) _ j Jedoes £ / £ « 6 c t e a , i ^ F ^ r d ( i i ' A . (2.57) i Jedoes edges edges ' The W are constant along the edge so doing the spatial integration along each edge in this term gives the action: / = 7 1 4 - 7 2 = £ • / sabc (Pxd,i WXYda - pYa,i) UY(x)bfUY(x)ceuYOfedt edges Jed9es + £ / eabcAPxaWXYbdWXYcddt (2.58) edges where Apxa is the difference between pa at opposite ends of side X . A s was already mentioned the first term is a constraint term. The second term can be written: / = £ /eabcAPxaWXYbdWXYcddt = £ / APxa{W}adt (2.59) edges edges by defining: {W}a = eabcWXYbdWXYcd (2.60) following Unruh and Newbury [6]. Therefore we have finally the action: I = £ / eabc (pxd„;WXYda - pYa,A UY(xffUY(x)ceuYQfedt edges Jed9es £ / Apxa{W}adt (2.61) edges ' Chapter 2. The Hilbert Action 19 where this action is writ ten almost entirely of the holonomies of the arbitrary genus surface. In the next chapter we w i l l apply this action to the particular cases of the genus one and two surfaces and obtain an action completely in terms of the holonomies of these surfaces. Chapter 3 Specific Examples of the Act ion 3.1 Application to the Genus One Surface A torus can easily be visualized as a 4 sided tile (Figure 2.1) wi th opposite sides identified as discussed in the previous chapter 1 . Across each pair of identified boundaries we wi l l have a Poincare transformation that relates internal vectors across the boundary. We wi l l denote these transformations by ( W C A I ^ I C A ) and ( W D B ^ D B ) - The constraint term in the action (2.61) obtained in the previous chapter forces internal vectors, pa, on opposite sides of the boundary to be related by the equation: WCAab Pch + KCAa = pAa (3.62) where W Q A is a Lorentz transformation and HCA is a translation vector. A similar equation applies across identified edges B and D . This equation is valid along each of the edges and in particular is valid at the vertex. The non-constraint terms of the action (2.61) only involve the difference ( A p a ) between pa at opposite ends of each identified edge. Therefore we only need concern ourselves with the internal vectors, pa, at the vertices. Note that pAa and psa must agree at the point where sides A and B meet. We can thus unambiguously label the value of pa at vertex 2 as p2a and similarly for the other vertices. 1This application to the genus one and two surfaces was also shown in [9]. 20 Chapter 3. Specific Examples of the Action 21 We now have the 4 equations: W C A \ PAB + n C A a = P l a (3.63) WCAabPz+KcAa = P2a (3-64) WDBab p4b + I W = p3a (3.65) WDB\ P l b + I i D B a = p2a. (3.66) Now using these relations we can circle the vertex and write an internal vector in terms of itself: Pi(p4(p3(p2(pi)))) = Pi (3.67) Pi = ( W C A W E 1 B W S 1 A W D B ) P 1 + W C A W ^ W C A J I D B - WCAW^BWcAnCA - W C A W B I U D B + U C A . (3.68) In order for this to be consistent we need the Lorentz component of this transformation to be the identity and the translation component to vanish. These conditions produce the following constraints: W C A W S I W C A W D B = 1 (3-69) (l-WcA)nDB=il-WDB)UCA (3.70) The consistency of this equation is equivalent to the fact that in traversing a small closed loop circling the vertex (see Figure 2.1), a parallel transported vector is the same as the original vector. This is a necessary condition for the surface to be flat at the vertex as stated in [6]. However, this is not a sufficient condition. If the vertex has a conical singularity wi th defect/excess angle of 27rn the vector parallel transported around the vertex w i l l s t i l l agree wi th the original, but the vertex w i l l not be flat. We insist that the spacelike slices, E , are manifolds and therefore the vertex cannot have a conical singularity. This point w i l l be crucial later in this thesis. Chapter 3. Specific Examples of the Action 22 The action (2.61) without the constraints imposed then reduces to: / = - P I { W C A W O 1 B W C \ W D B } - ( W ^ U C A ) {WDB} + ( W B I U D B ) { W E B W C A W D B } (3.71) where the reader should recall (2.60). This reduction is carried out using the following relations derived in [9]: (UU){UW} = Il{W} (3.72) n{ww2} = n f W i W s l + nW} (3.73) n ( { W i } - { T y 2 } ) = (W^n) { W J T 1 ^ } . (3.74) Note that U is a Lorentz transformation and that n{VF} is no index notation for I I a { i y } a . If we define: n = W C A W B I W C A W U B (3.75) and impose the constraints v i a Lagrange mutipliers the action now reads: i = -Pl{n} + ( W E B U D B ) { S I - W C A } - (WcAnCA) {wDB} + Ca[(l-WCA)TlDB-(l-WDB)TicA}a. ' (3.76) Var ia t ion wi th respect to p\ imposes the additional constraint that that the time deriva-tive of the first constraint vanishes. This contributes nothing to the action so this term can be dropped: / = ( W B B U D B ) { O - W C A } - ( W C A ^ C A ) {WDB} + Ca[(l-WCA)UDB-(l-WDB)UCA]A. (3.77) Chapter 3. Specific Examples of the Action 23 Note that the form of this action is suggestive of a canonical form wi th the Lorentz components of the holonomies playing the role of the configuration variables and the translation playing the role of the conjugate momenta. However, this in not in explicit canonical form because {W} depends not only on W, but also on W . The canonical form w i l l be made explicit in Chapter 7. The first constraint (3.69) is the representation of the fundamental group 2 of the torus in the group 50(2 ,1)+ and is equivalent to the condition that the Lorentz transformations are commuting. We w i l l refer to this constraint as the configuration constraint since the Lorentz components of the holonomies are the "configuration" variables. This constraint generates gauge transformations[6] of the form: n W a = nXY + ( i - w X Y ) a b \ b (3.78) where A a is some arbitrary vector. We can see from (3.63) and (3.64) that this corresponds to adding the same vector to p\ and p2 and thus the difference p2 — p\ is unaffected. The same arguments apply to the difference p3 — p2- The action (2.61), however, only depends on these differences, Ap, along each edge. Therefore the action is invariant under this transformation which demonstrates that it is a gauge transformation. The second constraint (3.70) w i l l be referred to as the momentum constraint because the translation vectors, II, are like momentum variables. It generates gauge transforma-tions of the form: W'XY = LWxyL-1 (3.79) U.'XY — LUXY 2The fundamental group of a surface is the set of equivalence classes of loops that can be deformed to one another. On the sphere all curves can be deformed to one another so the fundamental group is trivial. On the torus the fundamental group consists of all the sets of loops that can be obtained by combinations of the noncontractible loops. Chapter 3. Specific Examples of the Action 24 for an arbitrary Lorentz transformation L . The first equation transforms the axis of W X Y by L . The action is invariant under this transformation so it is a gauge transformation. There are some very important things to note about the constraints and the gauge transformations that they generate. In counting degrees of freedom, we have three for each of the Lorentz transformation WQA a n d WpB and three for each of the translations TicA a n d TIDB for a total of 12 degrees of freedom. We have one matr ix constraint (3.69) on the W ' s , which would seem to imply three constraints, and one vector constraint (3.70) on the ITs for another three constraints. Furthermore each constraint would be expected to produce one gauge freedom. Thus the 12 degrees of freedom should have a total of six constraints and six gauge freedoms. Thus we might expect the theory to be completely t r iv ia l . However, these expectations are not correct because some of the constraints are degenerate as we show below. The configuration constraint, which requires the two Lorentz transformations to com-mute, is degenerate (eg. Tr [WCAWDB — WDBWCA] = 0) and corresponds to only two contraints and two gauge transformations.. Similarly the momentum constraint is also degenerate. Thus instead of the expected 0 degrees of freedom we are left wi th two configuration and two momentum degrees of freedom for a total of four. We can also see this by direct calculation. The fact that the Lorentz transformations commute requires their axes to be parallel 3 . Thus, pick the axis of one (ie. pick 0 and <ft) and the axis of the other is fixed. However, the Lorentz parameters of both are completely unconstrained. The gauge freedom allows us to rotate the axes into a fiducial direction, leaving the two Lorentz parameters as the two configuration degrees of freedom. The presence of operators of the form (1 — W) in the momentum constraint (3.70) eliminate the components of the translation along the axis (fixed vector) of the Lorentz transformation from the constraint equation. For example, we can add any vector in the 3The axis is the fixed vector of the transformation. See Appendix A. Chapter 3. Specific Examples of the Action 25 direction of the axis of W C A to HDB without affecting the constraint equation. The axes of both Lorentz transformations are in the same direction which means that the arbitrary vectors that we can add to each of the LTs are parallel. The constraint equations do not involve these degrees of freedom. Therefore there are only two constraints on the transverse degrees of freedom. Note that in the special case that one of the Lorentz transformations, W , is the identity the (1 — W ) operator w i l l eliminate the II that it acts on from the constraint equation. However, there are s t i l l two constraints on the other II. The gauge transformations (3.78) include operators of the form (1 — W ) which means we can only gauge transform the degrees of freedom of the HXY in the plane perpendicular to the axis of W . We can use this gauge transformation to eliminate two of the tranverse degrees of freedom of Tic A , f ° r example, and the momentum constraint then forces two of the degrees of freedom in the HDB to zero as well. If one of the Lorentz transformations is the identity this w i l l s t i l l work. If W C A is nontrivial then we can eliminate the transverse degrees of freedom of the Tic A- The constraint equation is (1 — W C A ) ^ - D B — 0 which wi l l s t i l l eliminate two degrees of freedom from TIUB-If both Lorentz transformations are identities, however, neither the gauge transforma-tions nor the constraints w i l l remove any degrees of freedom and there are a full six degrees of freedom in the translation vectors. However, we can use the Lorentz gauge transforma-tion (3.80) to remove three degrees of freedom, leaving three physical degrees of freedom in the translations. The fact that there is an extra degree of freedom in the momentum variables for this particular choice of configuration suggests that W C A — WDB = 1 is a singular point in the phase space. Therefore, except for the singular point of the phase space, the four constraints and the fixing of their corresponding gauge freedoms removes eight of the original twelve Chapter 3. Specific Examples of the Action 26 degrees of freeedom leaving a four dimensional phase space. In the higher genus cases, however, the naive expectations are borne out. The con-straint equations corresponding to (3.69) and (3.70) w i l l remove 6 degrees of freedom and w i l l generate 6 gauge transformations. The remaining degrees of freedom after fixing the gauge w i l l be thus be 12g-12. The genus one case thus has an anomalous 4 degrees of freedom. Final ly , note that the configuration constraint is totally independent of the transla-tions (momenta). This means that we can examine the constrained configuration space independently of the momentum space. The momentum constraints are linear in the translations and are thus easily solved. We w i l l therefore concentrate our efforts in the next chapter on examination of the solutions to the configuration constraint. 3.2 Application to the Genus Two Surface A 2 holed torus can be represented as an 8 sided tile wi th identifications as shown in Figure 2.2. Across each pair of identified boundaries we w i l l have the Poincare transformations relating internal vectors across the boundaries: WCAab P4b + UCAA = Pia w H F \ P l b + I W = P6a WGEab p7b + I W - P6 WHF\ P 8 b + r w — pi WGE\ p8b + I W * - P5 WDB\ p5b + I W „ a - P2 W C A \ Pzb + n C A a — Pi w D B \ p4b + r w - Pz (3.80) Chapter 3. Specific Examples of the Action 27 Ci rc l ing the vertex these relations can again be used to write an internal vector in terms of itself: Pi = {WCAWElWBAWDBWGEWHlFWalEWHF)Pl (3.81) + W C A W C I W C A W D B W G E W H P W G I I I H F (3.82) - W C A W ^ B W C A W D B W G E W H I W G I I I G E (3.83) - W C A W B I W ^ A W D B W G E W ^ I H F (3.84) + W C A W E B W C A W D B T I G E (3.85) + W C A W E B W C A ^ D B - W C A W ^ I W C A ^ C A - W C A W E B ^ D B + ^ C A . (3.86) The Lorentz part of the transformation produces the configuration constraint: W C A W B B W C I W D B W G E W ^ W G E W H F = = 1 (3.87) where we have defined: « i = W C A W E B W S \ W D B (3.88) n2 = W Q E W ^ W G I W H E . (3.89) Note that the constraint requires 0.2 — i l f 1 . The momentum constraint is: WEB(1 - W C A ) H D B - WEBUcA + ^ r ' H c A (3.90) + ^ 2 [WHF(1 - WGE)UHF - W B F U G E + ^ U Q E ] = 0. A s in the previous section (3.87) is a necessary, but not sufficient, condition for the surface to be flat at the vertex. This is a very important point for the genus two surface as we w i l l see in later chapters. Chapter 3. Specific Examples of the Action 28 The action without the constraints imposed reduces to: I = p i { f ) 2 L l } - p i { r i i } (3.91) - ( W E B I L D B ) { W C W I } + ( W B I T I C A ) {wshni} - n C £ { i w f } + (wGEWs\nHF) { w y ? } . Imposing the constraints v i a Lagrange multipliers: i = n r V i i n r 1 ^ 1 } + { W E B K C A ) TCJfii} - ( W E 1 B T I D B ) { W ? M - n G , { i r f } + ( W G E W E F U H F ) { W G E ^ 1 } + Ca [ W E B ( 1 - W C A ) U D B - W E B T I C A + ^r'ncA + n2 ( W E \ { I - w G E ) n H F - W E F I I G E + ^ n ^ ) ] " • Varia t ion wi th respect to p\ again tells us that the time derivative of the first constraint vanishes which adds nothing new to the action so the first term can be dropped. Notice that the terms either involve the holonomies across the A<-*C and B<->D identifications or those across the E<-+G and F«-*H identifications but not both. The two new cycles we have added in going from the torus to the two holed torus are decoupled from the first two. This bodes well for generalization to an arbitrary genus surface. The form of this action is, as for the genus one case, suggestive of a canonical form wi th the W X Y P l a y i n g the role of configuration variables and the HXY playing the role of the conjugate momenta. The action w i l l have similar gauge invariances as for the action of the genus one surface: nW = n ^ y ( i - w X Y ) a b \ b (3.93) where A" is some arbitrary vector and: Chapter 3. Specific Examples of the Action 29 W ' X Y = L W X Y L ' 1 (3.94) H ' X Y — L U X Y where L is some Lorentz transformation. We can use a gauge transformation of the type (3.94) to Lorentz transform the axis of W C A to some fiducial direction. We can further apply another Lorentz transformation which keeps the axis of WCA fixed. This second gauge transformation can be used to transform the axis of WDB to some fiducial direction wi th respect to the axis of WCA-For example, i f W C A and WDB a r e generic boosts we can transform the axis of one to the x direction and then boost the other axis in the y direction so that it lies in the x-t plane. Thus we expect to be able to use the Lorentz gauge freedom to remove three degrees of freedom. The crucial difference between the genus one case and the genus two case is that in genus two, the WCA a n d WDB a r e n ° t constrained to have the same axis, which means that a transformation which leaves invariant the axis of one can s t i l l transform the axis of the other. The configuration constraint equation is independent of the translations as in the genus one case. Also the momentum constraint is linear in the translations and easily solved when the Lorentz transformations are known. Thus we w i l l concentrate our efforts on solving the configuration constraint. 3.3 Summary In this chapter we have written the action (3.77) of the genus one surface and the action (3.92) of the genus two surface entirely in terms of the holonomies of the surfaces. Fur-thermore, these actions are both suggestive of a canonical form wi th the W X Y a n d the HXY P a y i n g the role of configuration and momentum variables respectively. The next Chapter 3. Specific Examples of the Action 30 step is to solve the configuration constraint and try to identify the configuration variables and conjugate momenta that w i l l make this canonical form explicit. Chapter 4 Construction of Solutions to the Configuration Constraint M u c h of the structure of the phase space is determined by the configuration space alone, since the configuration constraints are independent of the translations (or momenta). Therefore we w i l l focus on the configuration space for the next few chapters and then return to the full phase space in Chapter 7. The previous chapters consisted mostly of reviews of previous work in this area. The first new results of this thesis are contained in this chapter. The configuration space of the g-holed torus in this formulation consists of the 2g 50(2 ,1 )+ transformations which are the Lorentz portions of the holonomies of the sur-face reduced by the previously mentioned constraint. Constructing solutions to the con-straint equations can provide a great deal of insight into the structure of the reduced configuration space. One of the key questions to examine is which solutions to the constraint equation can be continuously deformed into one another. In other words, we want to determine the connected components of the configuration space. For genus one we wi l l show that there is only one connected component of the configuration space, while for genus two there appear to be five. The constraint equation can be solved most easily in the SL(2,R) representation. The reader should refer to the appendix for full details, but we w i l l discuss it briefly here. SL(2 ,R) denotes 2 x 2 real matrices wi th determinant 1. P S L ( 2 , R ) 1 denotes the group 1 Projective Special Linear group. 31 Chapter 4. Construction of Solutions to the Configuration Constraint 32 obtained from SL(2 ,R) by identifying members which differ only by an overall sign . We w i l l use the sign of the Trace of the SL(2,R) matrix when we want to distinguish between identified elements of P S L ( 2 , R ) . There is a unique member of P S L ( 2 , R ) corresponding to each member of 50(2 ,1)+ . The P S L ( 2 , R ) representation of 50(2 ,1)+ is a representation over spinors: X = x a E a (4.95) where E a are a set of 2 x 2 matrices defined in (A. 198) and xa is a vector. The action of 50 (2 ,1 )+ on xa is obtained using an adjoint action: X = AX A'1 (4.96) on the spinors where A is a P S L ( 2 , R ) matrix. For the details of this thesis we actually use a different representation of SL(2 ,R) in terms of complex 2-dimensional matrices. This is because it was not unti l after a lot of the work was completed that it became clear that the matrix group being used was actually isomorphic to SL(2 ,R) . It is similar to SU(1,1), the matrix group used by Louko and Maro l f [5], but it is not the same. The important point is that the algebra of the generators is independent of the representation which means that the results w i l l be independent as well . For the purposes of this thesis we wi l l refer to the matr ix group used as SL(2 ,R) since they are isomorphic to one another. The representation we use consists of matrices of the form (Appendix A ) : L[u,ua] = cosh[u] + i sm/i[w]w a E a R[u,ua] = cos\u] +i sin[u]uaT,a (4.97) N[ua] = l + i w a E a where ua is a unit vector along the axis of the Lorentz transformation and ' u ' is the SL(2 ,R) parameter. The first type generates a Lorentz transformation wi th a spacelike Chapter 4. Construction of Solutions to the Configuration Constraint 33 fixed vector or axis, ua, wi th Lorentz parameter 2u. Lorentz boosts are of this type so we w i l l refer to these transformations as generic boosts (alternatively hyperbolic trans-formations [11]). The second type generates a Lorentz transformation wi th a timelike fixed vector or axis, ua, wi th parameter 2u. Lorentz rotations are of this type so we w i l l refer to these transformations as generic rotations (alternatively elliptic transformations[ll]). The last type generates a Lorentz transformation wi th a null fixed vector or axis, ua. Note that a nul l vector cannot be a unit vector. We w i l l refer to these as null transformations (alternatively parabolic transformations[ll]). Note that each type of transformation depends on three parameters and these param-eters can be represented as vectors as in Figure A.21 . The vector ua in (4.97) w i l l be a unit vector in the same direction and the parameter, u, w i l l be the Minkowski length of the vector in Figure A .21 . The universal covering space of PSL(2 ,R) is shown in Figure A.22 . The segments of Figure A.22 are copies of Figure A.21 wi th alternating overall sign of the corresponding SL(2,R) matrix. Note that the type of an SL(2,R) matr ix can be determined by its trace. The S a are traceless matrices and therefore the trace of an SL(2,R) matrix w i l l be 2 cosh[u], 2 cos[u] or 2 depending on whether the axis is timelike, spacelike or null . Therefore a nontrivial generic boost w i l l have trace > 2, a nontrivial generic rotation wi l l have trace < 2 and a nul l transformation w i l l have trace exactly 2. F ina l ly note that applying the adjoint action of the matrix group to members of the group (A —> BAB-1) w i l l have the affect of transforming the axis of A by the Lorentz transformation corresponding to B (see E q . A.226). The reader should refer to Appendix A for details. Chapter 4. Construction of Solutions to the Configuration Constraint 34 4.1 Construction of Solutions for the Genus One Surface The constraint equation for this situation is actually easy to solve in both 50(2 ,1)+ and SL(2 ,R) representations. The constraint equation, (3.69), in the SL(2,R) representation, is: ABA~XB~X = ± 1 (4.98) where we use the notation A = W C A a n d B = W ^ B f ° r simplicity. If the right side of the equation is —1, as we might expect to be possible in the P S L ( 2 , R ) representation, this constraint would be equivalent to the condition that the SL(2,R) matrices A and B anticommute. However, there are no anticommuting SL(2,R) matrices (see Appendix A ) . Notice that for the right hand side equal to 1 we have the t r iv ia l solution wi th A = B = 1. This constraint equation tells us that the Lorentz transformations must commute. The Lorentz transformations must have the same axis, but their Lorentz parameters are completely unconstrained. The resulting configuration space is connected since al l solutions can easily be transformed to the t r iv ia l one by simply allowing the Lorentz parameters to go to zero for both A and B . Thus for genus one there is only one connected component of the contrained configuration space. 4.2 Construction of Solutions for the Genus Two Surface The configuration constraint equation (3.87) for a genus two surface in the P S L ( 2 , R ) representation i s 2 : ABA-lB~lCDC~lD'1 = ± 1 (4.99) where we can assume that a l l matrices have positive trace since changing their signs w i l l not alter the equation. Notice that the sign is arbitrary on the right side of the equation 2A = WCA, B = W^B, C = W G E and D = W ^ . Chapter 4. Construction of Solutions to the Configuration Constraint 3 5 since the positive trace matrices do not form a subgroup of SL(2 ,R) . The arbitrary sign is also related to the fact that P S L ( 2 , R ) , not SL(2 ,R) , is isomorphic to SO(2,1)+. The constraint equation, however, is an equation involving SL(2,R) matrices which we must solve for both signs. The solutions depend on the sign of the right hand side. This immediately suggests that there w i l l be at least two components to the config-uration space (although as in genus one, one of these could be empty). We can write the constraint (4.99) as: ABA^B'1 = ±n (4.100) CDC^D'1 = i J T 1 (4,101) where Q and f 2 _ 1 have positive trace and the signs in the two equations are independent. Note that ft is also allowed to have zero trace. The zero trace SL(2,R) matrices are generic rotations wi th parameters | . If the axis of this generic rotation is ua = (1,0,0) this w i l l correspond to a pure rotation by 7r. A l l solutions to the constraint equation can be constructed by solving these last two equations. Unruh and Newbury[6] demonstrated the existence of solutions to (4.100) wi th positive sign on the right hand side in certain cases. They examined the cases where £1 and B are both generic boosts, where f£ and B are both generic rotations and the two cases where one of Q and B is a generic rotation and the other is a generic boost 3 . We generalize this construction to the complete set of solutions to the constraint equation and illustrate the allowed solutions which was not done in [6]. We w i l l solve (4.100), and (4.101) w i l l follow the same procedure. Equat ion (4.100) can be written in the form: ABA'1 = ±QB (4.102) 3Cases 1(a), 2(a), 4(a) and 5(a) below. Chapter 4. Construction of Solutions to the Configuration Constraint 36 which because of the cyclic nature of traces gives: Tr(B) = Tr(±QB). (4.103) This trace constraint is the key equation contraining B and ft. There are 9 different cases according to the nature of ft and B . Each case has different subcases according to the "sign of ft" (ie. the sign in front of ft in E q . 4.102). From (4.97) and using the fact that the generators, E„ , are traceless, the trace con-straint becomes in this representation: T r [ ± f t S ] = ± 2 [C(w)C(6) + S(u)S(b)uaba] = 2C(b) = Tr[B] (4.104) where C(b) refers to Cosh[b], Cos[b] or 1 and S(b) refers to Sinh[b], Sin[b] or 1 depending on whether B is a generic boost, a generic rotation or a null transformation respectively. In what follows, lowercase Greek or La t in letters (eg. b, u>) are the parameters of the SL(2 ,R) matrices represented by the uppercase Greek or La t in letters (eg. B , ft). A s well , the lowercase letters wi th superscripts (eg. b , ua) represent the unit norm axes of the associated SL(2 ,R) matrix. In each case, (4.104) gives us an equation involving the parameters of ft and B and their axes. The equations are Lorentz invariant so we w i l l fix the axis of ft in t rying to solve the equation. 1. ft generic boost and B generic boost (a) Tr(B) = Tr(QB) Tanh[b] = - T a n * [ * ] . (4.105) uaba First notice that since the range of Tanh[b] is (-1,1) the right side of the constraint equation must lie in the same range. This means that the dot product of the two vectors cannot be zero and in fact must have absolute value Chapter 4. Construction of Solutions to the Configuration Constraint 37 Figure 4.3: The set of unit spacelike vectors (ba) for which the Case 1(a) constraint equation is solvable. greater than |Tan/ i[^] | . Without loss of generality we can assume that u> and b are positive since a Lorentz transformation by a negative parameter is identical to one wi th positive parameter and opposite axis. W i t h this assumption we see that the dot product must be less than zero. The set of spacelike unit vectors is a hyperboloid of one sheet. Choosing u)a = (0,1,0), the allowed region for b for which the constraint is solvable is shown in Figure 4.3. The region on the left is the allowable region. The region on the right is a copy of the region on the left corresponding to the redundant solutions where b < 0. Both regions are pictured to illustrate the region of forbidden vectors which corresponds to a timelike slice centred on the plane where uiaba — 0 and thickness determined by the value of Tanh[^]. Notice that as u —• 0, Tan / i [ | ] —• 0 and the forbidden slice shrinks unti l the entire hyperboloid is the allowable region for ba. Different choices for toa can er 4. Construction of Solutions to the Configuration Constraint 38 be obtained by Lorentz transforming tia, and the allowed regions for b are obtainable from the allowed region in this simple case using the same Lorentz transformation. W i t h different choices of u)a the allowed region and its copy w i l l shift around on the hyperboloid, but they w i l l always be separated by a forbidden slice between two parallel planes. (b) Tr{B) = T r ( - O B ) Tanh[b] = _ ^ S l . (4.106) uaba The reasoning 1 in this case is very similar to the previous case. The important difference is that |Cor7i[*|]| > 1. This means that the dot product of the vectors must also have magnitude strictly greater than 1. Notice that the two vectors cannot span a spacelike plane or this condition w i l l be violated. Given the same assumptions as in the previous case the allowed regions for ba for which the constraint is solvable are the left two regions in Figure 4.4. The two regions on the right are again redundant regions included to illustrate the forbidden region. Notice that there are now two disconnnected regions on the left of the figure. A s u goes to zero Coth[^] —> oo and the forbidden region expands causing the two allowed regions to shrink and stay disjoint. The allowed regions for different choices of ua are Lorentz transformations of these regions. Notice that i f the unit spacelike vector moves toward a nul l vector the plane orthogonal to it t ilts toward the same null vector. The forbidden region centred on the plane orthogonal to ua t i l ts toward the null vector and shrinks one of the regions off to infinity and allows the other region to expand. Chapter 4. Construction of Solutions to the Configuration Constraint 39 Figure 4.4: The set of unit spacelike vectors (b°) for which the Case 1(b) constraint equation is solvable. 2. ft generic boost and B generic rotation (a) Tr(B) = Tr(+ftB) Tan[b] = Tanh[\ Cjabn (4.107) In this case, since the range of Tan[b] includes al l the reals, the dot product of the vectors can take any value. The set of timelike unit vectors is a hyperboloid of two sheets. If we assume without loss of generality that u, b > 0 and again choose uja — (0,1,0) the allowed region for ba for which the constraint equation has a solution is shown in Figure 4.5 No redundant regions are included. However, the satisfaction of the trace constraint, (4.103), does not guarantee a solution to (4.102). A Lorentz transformation cannot take a future pointing timelike vector to a past pointing timelike vector. Therefore the adjoint action on the SL(2 ,R) representative of a generic rotation cannot change its sign of rotation. Thus the sign of rotation of ftB and B must be the same. Chapter 4. Construction of Solutions to the Configuration Constraint 40 Figure 4.5: The set of unit timelike vectors (b) for which the Case 2(a) constraint equation is solvable. Checking the sign of rotation of VLB shows that the sign of its Eo term (see E q . 4.97) is determined by the sign of Sin[b]b°. The sign of the E 0 term is determined by the sign of b°. Therefore the sign of rotation of QB agrees (without loss of generality assume b < | ) wi th the sign of rotation of B and thus wi th that of ABA-1. Therefore in this case the solutions to (4.103) are solutions to (4.102) as well. (b) Tr(B) = Tr(-ttB) Tan[b] = Coth[%] toak (4.108) The same logic applies except that the sign of rotation of Q.B is determined by —Sin[b]b° which is opposite to the sign of rotation of B as determined by b° and there is no solution in this case. Chapter 4. Construction of Solutions to the Configuration Constraint 41 Figure 4.6: The curves on the null cones define the sets of null vectors (ba) for which the Case 3(a) constraint equation is solvable. 3. ft generic boost and B N u l l (a) Tr(B) = Tr(+SIB) Tanh[%\ uabn (4.109) In this case the dot product of the vectors must be exactly —Tanh[^]. The set of nul l vectors form a pair of cones meeting at the origin. The two regions have a sign of "rotation" associated wi th them. A future pointing null vector cannot be Lorentz transformed to a past pointing null vector and therefore the adjoint action on a null transformation cannot change its sign of "rotation". The allowed region for ba for which the trace constraint has a solution is a pair of curves on the cones shown in Figure 4.6. Checking the sign of rotation of VLB shows that the sign of the Eo term is determined by the sign of b° which agrees wi th that of B . Therefore the regions illustrated does represent solutions to the trace constraint. Chapter 4. Construction of Solutions to the Configuration Constraint 42 Figure 4.7: The set of unit spacelike vectors (ba) for which the Case 4(a) constraint equation is solvable. (b) Tr(B) = Tr(-nB) 1 = -Coth{%] Cjabn (4.110) The same logic applies except that the sign of rotation of VLB is determined by —b° which is opposite to that of B and so there is no solution in this case. 4. Q, generic rotation and B generic boost (a) Tr(B) = Tr(+{IB) Tanh[b) = Tan[%] Cjabn (4.111) Assume 0 < u < | and that Cja = (1,0,0). The allowed region for ba on the spacelike hyperboloid is the upper region in Figure 4.7. Aga in notice the forbidden slice about the plane orthogonal to ua. If |c<;| —> 0 the entire upper half of the hyperboloid is available. If ua = (—1,0,0) instead, the allowed region w i l l be the lower region of Figure 4.7. Chapter 4. Construction of Solutions to the Configuration Constraint 43 Figure 4.8: The set of unit spacelike vectors (b) for which the Case 4(b) constraint equation is solvable. (b) Tr(B) = Tr(-nB) Tanh[b] = -Cot[%] uaba (4.112) Assuming 0 > uJ > — | . 4 the allowed region of the hyperboloid is the lower region in Figure 4.8. A s uJ —• 0, Co t [y ] —> oo and the forbidden slice expands. Note that as uJ goes to — | , u> goes to | . A t this point: Cot[-]=Tan[-\. (4.116) 4Rotation by a positive angle, a, is the same as a rotation by a — 2ir. A PSL(2,R) parameter b implements a rotation by 2b. Therefore b' = b — TT implements the same Lorentz rotation as b. Cos[b]= COS[TT + b'} = -Cos[b'] (4.113) Sin[b}= Sin[ir + b']= -Sin[b'\ (4.114) B' = Cos[b'] + iSin[b']Cjaba = -B (4.115) As 6 —» 7r ,b'and ->0B and B' go to the negative trace identity. This primed notation simplifies some of the logic and makes the double covering nature of the SL(2,R) explicit. See Figure A.22. Chapter 4. Construction of Solutions to the Configuration Constraint 44 In this case (4.111) and (4.112) agree. Figures 4.7 and 4.8 w i l l look exactly the same. This is the case where f2 has zero trace and tells us that we could have handled case 4(a) and (b) together since the allowed regions are the same at this point. 5. ft generic rotation and B generic rotation (a) Tr(B) = Tr(+flB) Tan[b] = I ^ M ( 4 . H 7 ) uaba \co\ < I which need not be positive in this case. Checking the sign of rotation, the sign of the Eo term is determined by the sign of: Cos[u]Cos[b] (Tan[u} - T a n [ | ] ) . (4.118) This has the same sign as UJ which is opposite to the sign of b i f toaba < 0 (which we can assume without loss of generality). Therefore there is no solution in this case. (b) Tr(B) = Tr(-flB) T a n[ b] = _ ^ M . ( 4 n 9 ) u)aba \co'\ < | and the sign of the Eo term is determined by: - Cos[oj']Cos[b] (ran[uj'} + Cotf^]j . (4.120) This must be opposite to the sign of b°Sin[b]. Therefore there is no solution in this case either. 6. Q generic rotation and B N u l l (a) Tr(B) = Tr(+£IB) r J ~ uab, 1 = - n r 1 - (4-i2i) Chapter 4. Construction of Solutions to the Configuration Constraint 45 Assume without loss of generality that \u\ < | and u>a — (1,0,0). In this case: Tan[^A = -b° (4.122) and the Eo term has sign determined by Cos [a;] (Tanfw] — T a n [ | ] ) which has the same sign as u>. The sign of u> must be opposite to that of b°. Therefore there are no solutions in this case. (b) Tr{B) = Tr(-ttB) 1 = r - ^ 1 (4.123) Assume without loss of generality that \u>'\ < | and Cja — (1,0,0). In this case: - C o * [ | ] = -b° (4.124) and the Eo term has sign determined by: - Cos[J] (ran[u'\ + Cot[j^J (4.125) which has opposite sign to u . u> must have the same sign as b°. Therefore there are no solutions in this case. 7. f2 nul l and B generic boost (a) Tr(B) = Tr(+QB) coaba = 0. ' (4.126) Assume that b > 0 without loss of generality. Look at the l imi t of Case 1(a) as u) —> 0: Tanh[b] « = ^ . (4.127) uaba 2coaba In order for there to be a nontrivial solution as u> —• 0, cuaba —» 0 as well . A s ft to a null transformation, u 0,(see Appendix A ) and so (4.126) is the Chapter 4. Construction of Solutions to the Configuration Constraint 46 Figure 4.9: The intersection of these surface defines the curve (ba) for which the Case 7(a) constraint equation has solutions. l imi t of 1(a) as ua .goes to a null vector. Note that most of the solutions of 1(a) as u —> 0 are B —• ± 1 which is a solution to this case also. The only nontrivial solutions are in the plane orthogonal to oja which gives us two curves of intersection of this plane with the hyperboloid of one sheet. (Figure 4.9). (b) Tr(B) = Tr(-QB) uaba = -2Coth[b]. (4.128) Assume that b > 0 without loss of generality. Look at the l imit of Case 1(b) as oj —» 0: 2 o Tanh[b] « -• - = ^ . (4.129) uaba uaba Since u —> 0 as tba goes to a null vector (4.128) is the l imit of Case 1(b). If we assume that coa = (1,1,0) then the allowed region for ba is the left region in Figure 4.10. The other region is a region of redundant solutions included to illustrate the forbidden slice. Comparing Figure 4.10 to Figure 4.4 we see that the lower allowed region in Case 1(b) has vanished as u)a goes to a future pointing nul l vector. The diagram would be nipped upside down if we had let Chapter 4. Construction of Solutions to the Configuration Constraint 47 Figure 4.10: The set of unit spacelike vectors (ba) for which the Case 7(b) constraint equation is solvable. Cja goes to a past pointing null vector. The important thing to note is that one of the allowed regions vanishes as uia goes to a null vector. 8. ft nul l and B generic rotation (a) Tr(B) = Tr(+QB) ooaba = 0. (4.130) A timelike vector, b , cannot be orthogonal to a null vector, u>a. Therefore there is no solution. (b) Tr(B) = Tr(-flB) uaba = -2Cot[b]. (4.131) Assume without loss of generality that uja = (1,1,0). The sign of the So term is determined by the sign of: - [Sm[&](6° + b2) + Cos[b]] = -Cos[b] [Tan[b](b° + b2) + l ] (4.132) Chapter 4. Construction of Solutions to the Configuration Constraint 48 or: Cos [b] (4.133) ( - 6 ° + 6 1) Now recall that > \b2\ which means that the first term in brackets has the same sign as b°. If ujaba < 0 then 6° > 0 and Cot[b] > 0. This means that 0 < u < | . Therefore the sign of the Eo term is negative which is opposite to the sign of b°Sin[u] which determines the sign of rotation for B . The analysis is similar i f we assume cuaba > 0 and/or i f we assume that coa — (—1,1,0). Therefore there is no solution in this case. Chapter 4. Construction of Solutions to the Configuration Constraint 49 9. fl nul l and B null (a) Tr{B) = Tr(+flB) toaba = 0. (4.134) ba can be any multiple of uja. Assume that ba = cu>a QB = 1 + i(ua + cuja) - ieabcuja(aub)Zc = 1 + i ( l + c)coa (4.135) A Lorentz transformation can't take a future pointing null vector to a past pointing null vector and thus (1 + c) > 0. The S 0 term of flB is + c) and that of B is b° = cw°. Therefore there wi l l be a solution as long as c > 0. (b) Tr(B) = Tr(-flB) uaba = - 2 . (4.136) Assume that ua = (1,1,0). The sign of the E 0 term is determined by the sign of - [ 6 ° + 1 + b2]. Now 16° | > |6 2 | . b° > 0 must be true in order for the dot product to have the right sign, so the sign of the E 0 term is opposite to that of b°. Similar ly i f ua = ( - 1 , 1 , 0 ) . Therefore there is no solution in this case. 10. il = ±l (a) Tr(B) = Tr(+flB) In this case and the next it is easier to return to (4.100) and see that: [A,B]=0. (4.137) There are no restrictions placed on, ba, the axis of B . (b) Tr(B) = Tr{-VlB) {A,B} = 0. (4.138) There are no anticommuting matrices in SL(2 ,R) . Therefore there are no so-lutions in this case. Chapter 4. Construction of Solutions to the Configuration Constraint 50 Summary Using the results obtained above we make the following general statements: 1. For any ft we can always find a solution wi th B a generic boost. 2. For ft a generic boost and B a generic rotation or null there is a solution for positive trace fl. 3. For fl nul l and B nul l there is a solution for positive trace ft. 4. A l l other cases have no solution. 5. For a l l negative trace fl the only solutions are wi th B a generic boost. Once the trace constraint has been satisfied it remains to find A such that: ABA'1 = ±fiB. (4.139) The adjoint action rotates the axis of the transformation B (Appendix A ) and this equa-t ion amounts to: Ab^A'1 = bZa (4.140) zo, where ba is the axis of B and b is the axis of ftB. This equation says that we are looking zo. for the transformation A that transforms ba to b . It is fairly easy to see that there is always a solution. The equality of the traces of B and ftB means that their axes are both of the same type (spacelike, timelike or null) since, as we mentioned, the SL(2 ,R) matrices can be classified by their traces. A n y spacelike vector can be taken to any other spacelike vector by means of a Lorentz transformation. A s well , any timelike (null) vector can be taken to any other timelike (null) vector as long as they are both future pointing or both past pointing - a condition taken care of by requiring that the sign of "rotation" was the same. Chapter 4. Construction of Solutions to the Configuration Constraint 51 A solution can easily be constructed as the product of a pure rotation and a pure ZO boost which takes ba to b , but it is not unique. Assume that A is another: A&ZaA-1 = b \ a . " (4.141) This implies that: Ab^A-1 = AbaHaA~l (4.142) 6 a E a = A-lAbai:aA-lA. (4.143) Therefore ba is a fixed point of A~lA. The most obvious solution to this is that A = A, but this is not the only solution. In fact there is a whole line of fixed points for each Lorentz transformation defined by its axis. Thus: A~lAB = BA~lA (4.144) or: [ i - 1 A , B ] = 0 (4.145) Therefore we have: A'1 A = B(a) (4.146) where B(a) refers to a transformation wi th the same axisoas that of B , but arbitrary Lorentz parameter, a. Thus a l l solutions for A(a) can be generated from a single solution (A) by: A(a) = A[B(a)}-\ (4.147) Therefore we have a one parameter family of solutions for A such that ABA-1 = VLB. Fina l ly having shown how a solution can be constructed, something further can be said about the nature of A . Taking the inverse of the constraint equation we get: BAB^A'1 = ±tt~1 (4.148) Chapter 4. Construction of Solutions to the Configuration Constraint 52 which must have a solution whenever (4.100) does. We can use exactly the same argu-ments as above and obtain the same constraints on the allowable axes for A as we did for the axes of B . Therefore the summary of the allowed types of solutions on page 50 applies i f we substitute A for B . Furthermore the same arguments w i l l apply to solutions of (4.101) for C and D . The results of this section w i l l be used in the next chapter to discuss the structure of the constrained configuration space. 4.3 Construction of Solutions for the Genus g surface The constraint equation in the SL(2,R) representation is[9]: f[ ABiA-'Bi1 = ± 1 (4.149) which we suggest wri t ing as: f[ AiBiA-lB-1 = i f T 1 (4.151) A construction similar to that in the previous section could be carried out for higher genus solutions, but the details are much more complicated even at g=3. We w i l l not construct solutions for higher genus surfaces. AxBxA^B^1 = ±fl (4.150) Chapter 5 Structure of the Reduced Configuration Space The reduced configuration space of the one holed torus is, as mentioned in the last chapter, a s imply connected set. It has been argued that the constrained configuration space of the genus g surface has 4g-3 disconnected components [12] [11]. This chapter and the next w i l l be concerned wi th understanding the existence and status of the components of the configuration space for the higher genus surfaces. The structure of the phase space of the genus one surface w i l l be discussed in the Chapter 7. 5.1 T h e Structure of the Configuration Space for the Genus Two Surface The configuration space consists of <g>4SO(2,1)+ subject to the constraint that 1 : A B A - 1 B _ 1 C D C - 1 D - 1 = 1. (5.152) The set of Lorentz transformations are the Lorentz components of the holonomy group of the closed spacelike hypersurface, S. Recall from Chapter 2 that the constraint equation is an attempt to ensure that S is a smooth 2 : manifold by ensuring that a vector parallel transported around a closed curve surrounding the vertex wi l l be unchanged (see Figures 2.1 and 2.2). However, as pointed out there, this constraint may not be sufficient as it s t i l l allows the vertex to be singular with a deficit angle ±2n7r. Our purpose in this chapter is to determine the structure of the constrained configu-ration space using the results of the last chapter. 1 Where we use the notation of the previous chapter: A = WCA, B = W^B, C = WQE, and D = WgF. 53 Chapter 5. Structure of the Reduced Configuration Space 54 The group P S L ( 2 , R ) is isomorphic to SO(2,1)+ and thus given the constraint on the Lorentz transformations: A B A - 1 B - 1 C D C - 1 D - 1 = 1 (5.153) the corresponding constraint in the PSL(2 ,R) representation is: ABA^B^CDC^D-1 = ± 1 . (5.154) Now since switching the sign of one of A , B , C , or D w i l l not affect the overall sign we can, for the purposes of the solution, assume that A, B, C and D are the positive trace representatives. This does not mean that their product is a positive trace matr ix as this set is not closed under multiplication. Therefore it is possible that this product of Lorentz matrices results in the negative trace representative of the identity. In order for the constraint equation to be satisfied the following must be true: ABA-lB~l = ± 0 (5.155) CDC~lD-1 = ±fT1 (5.156) where Q and Q.~l are assumed to have positive trace. Note that the arbitrary signs in these equations are independent. Also note that the inverse is in the SL(2,R) sense and thus H _ 1 is also a positive trace matrix. The possible solutions to these equations were discussed extensively in the previous chapter. The question we now ask is whether or not the solutions can be deformed into each other. Case I Take the positive sign in each of (5.155) and (5.156). The summary on page 50 tells us that there are many solutions. Examining a l l the positive sign cases in the last chapter shows that in a l l cases where there are solutions, except one, there are s t i l l Chapter 5. Structure of the Reduced Configuration Space 55 solutions i f u> —*• 0 and thus ft goes to an identity transformation. In these cases the solutions for B easily go to the identity transformation. For example, in Case 1(a) the constraint equation is: Tanh[b] = - I ^ m . (5.157) waba A s u —> 0, b w i l l automatically be forced to zero and the axis of B can remain constant. If B and ft go to the identity transformation then (5.155) becomes AA~l = 1. A t this point we can reduce the parameter of A to zero, taking A to the identity transformation. The same arguments apply to C and D as cu —> 0. In the exceptional case, Case 3(a), there are solutions for Cja going to a null vector which takes us to Case 9(a). In this case there are solutions as the null vectors coa and ba both go to the zero vector which means that ft and B go to identity transformations. A is then taken to the identity as in the last case. The same logic applies to C and D as well . Th i s means that we can find a path through the t r iv ia l solut ion 2 between any two points in the configuration space corresponding to positive trace ft solutions. Therefore al l these types of solutions are in a single connected component of the configuration space. This region w i l l be referred to as the trivial component. Case II Now take the negative sign in both (5.155) and (5.156). Examining the nega-tive sign cases in the previous chapter shows that Case II is different from Case I. There are s t i l l solutions to these equations, but in this case in the l imit that u> —> 0 the solutions for B do not go to the identity transformation. For example in Case 1(b) the constraint equation is: -Tanh[b] = - ^ M . (5.158) wab„ Ja 2A,B,C,D=1. Chapter 5. Structure of the Reduced Configuration Space 56 Note that Coth[^] —> oo as u> —> 0. The right side of the equation must remain less than 1 for there to be a solution, so ba must go to a null vector 3 . Things do not work as easily as they did Case I. This can also be seen as a result of the fact that there are no anticommuting SL(2 ,R) matrices since i f u> goes to zero, ft goes to -1, and this is case 10(b) which has no solutions. Note that a l l solutions in this Case II must have A, B, C and D a l l be generic boosts. A s well , these boosts must al l be nontrivial (ie. not 1). If, for example, A was the t r iv ia l boost then (5.155) would be, in this case, BB~l = 1 = — ft which is contradictory to our assumption that the right hand side is a negative trace generic boost. A similar argument applies to B, C and D. We have already noted that A, B, C and D such that: ABA~lB~l = ft (5.159) CDC~lD-1 = ft'1 are in the connected component of the configuration space referred to as the t r iv ia l region. If we can smoothly deform A, B, C and D satisfying: ABA~lB~l = -ft (5.160) CDC^D'1 = -ft-1 for positive trace ft, to A, B, C and D in the t r iv ia l region the path must have the trace of the right hand side of the equations go from negative to positive and thus pass through zero trace (see Figure A.22) . The zero trace ft is a generic rotation by 7r. In order to determine i f a path exists in the constrained configuration space between solutions to Eqs. (5.160) and solutions to Eqs. (5.159) we w i l l try to take ft to a rotation by 7r and determine i f continuous solutions for A , B , C and D exist along this path. 3 Spacelike unit vectors very near null vectors are very long in the Euclidean sense and thus have large components. The dot product of these unit vectors with ui = (0,1,0) will become very large. Chapter 5. Structure of the Reduced Configuration Space 57 If —ft is originally a generic rotation it can be taken to a positive trace generic rotation while remaining a generic rotation (see Figure A.22). This is because a generic rotation by 7T corresponds to a vanishing trace SL(2,R) matrix. The construction in Case 4(a) and 4(b) shows that there are continuous solutions for A, B, C and D along this path. This means that there is a continuous path from solutions to the constraint equation (5.160) to solutions to (5.159) which are in the t r iv ia l region of the constrained configuration space. If — ft is originally a null transformation it can be taken to a negative trace generic rotation by taking the null axis, w a , t o a timelike axis, Cba. The regions of allowed solutions for ba i f —ft is null (Figure 4.10) overlaps that for —ft a generic rotation (Figure 4.8). Therefore solutions for B can have the same axis in both cases. The constraint equation for — ft a generic rotation (Case 4(b)) is: Tanh[b] = - (5.161) A s — ft becomes a nul l transformation, u> —> 0, and the constraint is: 1 2 Tanh[b] « ^ - = (5.162) waba waba which approaches the constraint equation for — ft a null transformation (Case 7(b)): waba = -2Coth[b\. (5.163) Therefore, there are continuous solutions for A , B , C , and D for which —ft starts as a null transformation and becomes a negative trace rotation. Using the previous result we can thus deform these solutions to the constraint equation to solutions in the t r iv ia l region of the constrained configuration space. If — ft is originally a generic boost the situation is more complicated. The only continuous way for a negative trace generic boost, — f t , to become a positive trace entity Chapter 5. Structure of the Reduced Configuration Space 58 is by passing through nul l transformations and becoming a generic rotation (see Figure A.22) . There are solutions to the constraint equation in this case, for A , B , C and D (points in the reduced configuration space) such that there is no path to the solutions where — fl is a null transformation. In order to see this examine Figure 4.4. Note that the axes of fl (wa) and fl-1 (wfnv) lie in opposite directions. If we assume that u>a = (0,1,0) then u>fnv = (0, —1,0). In this case we can reinterpret the left and right regions in Figure 4.4 as the allowable regions for ba and da (axis of D) respectively. If ua goes to a null vector the forbidden slice for both ba and da w i l l t i l t toward the same null vector. In the process one of the allowed regions for each vector w i l l be sent off to infinity. There are no solutions when uja actually becomes nul l that have b anywhere near this region, (compare Figures 4.4 and 4.10) If our starting point corresponds to a solution wi th b in its upper allowed region and da in its lower allowed region (or vice versa) there are st i l l solutions as Cja goes to a null vector. Therefore solutions of this type can be smoothly deformed to solutions of (5.160) where — fl is a negative trace null transformation. Aga in the previous results show that these solutions are therefore connected to the t r iv ia l region. If, however, our starting point corresponds to a solution wi th both ba and da in their lower allowed regions then taking Coa to a null vector means that either b or da w i l l be in a region that vanishes. Therefore — fl and — fl-1 w i l l always be negative trace generic boosts for smooth deformations of A , B , C, and D . Therefore there is no path from this starting point in the reduced configuration space that w i l l lead to the t r iv ia l region of the configuration space. These two regions are disconnected. If the starting point corresponds to a solution wi th both ba and da in their upper allowed regions the same logic applies. This is another disconnected region of the config-uration space. The two new disconnected components of the constrained configuration space arise in Chapter 5. Structure of the Reduced Configuration Space 59 a similar way and thus both w i l l be referred to as regular components of the configuration space for reasons that w i l l become clear in the next chapter. Case III Now look at the solutions which produce a negative trace identity represen-tation: ABA-lB~l = - f t (5.164) CDC~lD~l = ft-1 where we assume that —ft has negative trace and ft-1 has positive trace without loss of generality. In this case B must be a generic boost. In order to connect these solutions to solutions in the t r iv ia l region, —ft must go to a positive trace transformation while ft-1 stays a positive trace transformation. In order to connect these solutions to one of the regular components of the constrained configuration space, ft-1 must go to a negative trace transformation while —ft stays a negative trace transformation. The allowed regions for ba are the left regions in Figure 4.4. The allowed region for d is the left region in Figure 4.3. Due to the less restrictive nature of the allowed regions for da the axis of —ft (cua) can go to nul l vector in such a way that the allowed region for solutions to each of the equations do not vanish. If b is in its upper allowed region then taking Qja to a future pointing null vector w i l l allow solutions in which —ft and ft-1 can become nul l transformations and then generic rotations. However, at this point it is worth noting that in order for —ft (or ft-1) to change the sign of its trace (ie. change its overall sign) it must pass through a point where its trace vanishes. A t this point the trace of ft-1 (—ft) w i l l also vanish. The paths followed by ft-1 and —ft meet at this solution. These paths are shown in the universal covering space of SL(2,R) in Figure 5.11. If —ft continues on this path to become a positive trace transformation, ft-1 becomes a negative trace transformation (and vice versa). A t every point on the path the product —f t f t - 1 has to Chapter 5. Structure of the Reduced Configuration Space 60 J I Positive )" Trace Represenation I Negative > Trace [ Represenation Figure 5.11: The paths in the covering space of SL(2,R) for — fl and fl 1 such that both go to negative trace representatives. be the negative trace identity matr ix (see Eqs. 5.164 and 5.154) and there is no way to reach the other three regions of the configuration space. This logic defines two regions amounts to choosing the original fl lying above or below fl~l in the universal covering space. These final two components of the configuration space wi l l be referred to as the negative trace components because the negative trace of the identity representation they produce distinguishes them from the other three components. 5.2 Summary In this chapter we have shown that there are apparently five disconnected components of solutions to the configuration constraint for the genus two surface in this formulation of 2+1 gravity. This means that we have solutions that cannot be deformed to one another. Spacetimes corresponding to a solution in one of these components cannot evolve to any of the configuration space depending on which allowed region b was ini t ia l ly in . This Chapter 5. Structure of the Reduced Configuration Space 61 of the other components. However, only two of these regions w i l l turn out to be physically relevant. The next chapter w i l l discuss why the constrained configuration space has these disconnected components and which regions are physically relevant. Chapter 6 Realization of Solutions The purpose of this chapter is to examine the solutions in different disconnected regions of the genus two configuration space. A s was mentioned in Chapter 3, the constraint equations are not sufficient to ensure the spatial slice, E , is smooth at the vertex. We w i l l show that that the t r iv ia l and negative trace components of the constrained configuration space correspond to E wi th conical singularities at the vertex and that only the regular components of the constrained configuration space correspond to E a smooth manifold at the vertex. Thus the regular components are the only physically relevant components. Furthermore, these regular components differ only in the labelling of the vertices and sides of the "tile" and are thus physically equivalent. Thus the physical configuration space has only one component. We also show how these results can be related to a previous more abstract classification of the regions of the constrained configuration space in terms of the Euler class of the mapping from the fundamental group of the surface to 5 0 ( 2 , 1 ) + . These goals are accomplished through the construction of a realization of the solutions to the constraint equations. We w i l l concentrate on the solutions to the configuration constraint as we did in the previous chapter since, as mentioned there, the configuration constraint is independent of the momenta. We w i l l examine the specific solutions to the constraints such that the momenta f l are a l l zero. The holonomies in this case are Lorentz transformations, W X Y -62 Chapter 6. Realization of Solutions 63 Our representation w i l l be based on the fact that every compact 2-dimensional Rie-mann surface, E , has a simply connected universal cover, £ [17]. The uniformisation theorem [17] tells us that there are only three simply connected Riemann surfaces and and they each correspond to a metric of constant Gaussian curvature. The realization of the solutions w i l l consist of fundamental tiles representing £ in the hyperbolic plane (H2), the universal covering space of surfaces of genus g > 1. 6.1 Realization of Solutions for the Genus Two Surface We construct a physical representation of spacetimes corresponding to solutions of the configuration constraint suggested by Unruh and Newbury [6]. This representation con-sists of a "cylinder" of tiles on hyperbolic sheets. We take this realization one step further by i l lustrating the spacetimes in terms of fundamental tiles and classifying the spacetimes according to the region of the constrained configuration space the holonomies of the tiles are in . Start wi th a 3 dimensional Minkowski space - the internal space. Choose an arbitrary path, X ° ( r ) , through the spacetime and consider this as the in i t ia l vertex of a 4g sided figure at each value of r . Use a solution to the configuration constraint as holonomies of the surface to obtain the other 4g- l vertices for each value of r . The holonomies are Lorentz transformations, so a l l the vertices for each value of r lie on a spacelike hyperboloid (H2) when X^ir) is timelike separated from the origin. When X^r) is spacelike separated from the origin, al l the points lie on a timelike hyperboloid. When X" (r) is on the nul l cone al l the vertices lie on the null cone. We w i l l only consider the situation where X^ir) is always timelike separated from the origin. A l l the vertices at each r w i l l lie on a spacelike hyperboloids, labelled by r , in the forward and past light cones of the internal space. This is referred to in the literature Chapter 6. Realization of Solutions 64 as the spacelike sector. Choose the edges of the 4g sided figure on each r slice to be the unique geodesies joining the pairs of adjacent (according to their vertex label - see Figure 2.2) vertices. Geodesies on the hyperbolic plane are discussed in Appendix B . Take the interior of the 4g sided tile to lie on the same spacelike hyperboloid and identify the edges as in Figure 2.2. This tile, after identification of the edges, represents a genus g manifold (E) . The edges chosen automatically have the correct holonomies 1 and the identification at the edges is smooth 2 . Ac tua l ly to be accurate, only solutions in the regular region of the configuration space produce smooth orientable genus g manifolds. Solutions in the other regions produce conifolds or non-orientable manifolds. We can reparameterize r to be the intersection of the spacelike hyperboloid that a tile is on wi th the time axis of the internal space. Spacelike coordinates can be defined on the spacelike tiles producing coordinates ( T , X % ) on the internal space. The dreibein as described in Chapter 2 can be constructed from this representation. Define an internal vector field to be the internal coordinates of each point. pa(r,xi)=XA(T,xi) (6.165) and define the dreibein: e / = X a ( r , . r V (6.166) The metric associated wi th the dreibein is: 9^ = e /e„„ = riabXa^ X \ v . (6.167) The e^ thus constructed are related across the identification edges by the holonomies of the surface. aThe vertices were generated using the holonomies. The holonomies are Lorentz transformations and thus will take the unique geodesic through any two points to the unique geodesic through any other two points. 2 A Lorentz transformation of the whole tile will produce another tile on the hyperbolic plane that shares an edge with the original. See Figure 6.13. Chapter 6. Realization of Solutions 65 Note that there appears to be no evolution of the theory as the holonomies - only W X Y since we have assumed that the TLXY — 0 - on each slice are the same. The shape of the tile on each r-slice w i l l be exactly the same. The spacetime itself, however, appears to have an evolution in r as the tiles w i l l be singular at r = 0 and w i l l increase in size in the forward light cone as r increases. We could obtain different holonomies on each r-slice from the original holonomies by: W X Y ( T ) = L(r)WXYL-l{r) (6.168) and thus the new dreibein is: eva = L{r)ahevb. (6.169) Th i s is a r dependent gauge transformation (see 3.94) and so this spacetime is gauge equivalent to the original and we need not introduce this complication. The spacetime constructed thus has the required Lorentz holonomies and provides a physical model of solutions to the configuration constraint. The tiles on each hyperbolic slice are identical so in what follows we w i l l only deal with a tile on one slice, but the results w i l l be applicable on every slice. Note that because the hyperbolic plane is the universal covering space of £ (see next section), the hyperbolic plane can be tiled with Lorentz transformations of the original tile. The Lorentz transformations are products of the holonomies, WXy > which relate the edges of the tiles. The Lorentz transformed tiles completely cover the hyperbolic plane wi th no overlapping and 8 tiles w i l l meet at every vertex as in Figure 2.2. 6.2 Tilings of the Genus Two Surface It was assumed in the last section that the hyperbolic plane is the universal covering space of £ and thus can be ti led by copies of E related by the Lorentz holonomies, W X Y -Chapter 6. Realization of Solutions 66 If this is not true, we can st i l l represent solutions as tiles on the hyperbolic plane wi th edges identified, but they w i l l correspond to S wi th conical singularities at the vertex. Solutions of the constraint equation from each of the five regions of the configuration space described in Chapter 5 w i l l produce tiles wi th different characteristics. The results in the rest of this chapter were arrived at by developing Mathemat ica programs to produce actual solutions to the configuration constraint and then using these solutions to construct diagrams of the tiles of the type shown in the figures to follow. When unifying characteristics of tiles in each region were noticed after examination of many tiles, the second program was altered in order to measure the internal angles of the tile on the hyperbolic surface. The results then led to a clearer understanding of the nature of the regions of the configuration space and how they arose. 1. Ti les for the regular regions of the configuration space Th i s region is called the regular region of the configuration space because it con-tains the set of holonomies that produce the regular tile (Figure 6.12), so called because of its symmetry. Other sets of holonomies in this component of the reduced configuration space produce tiles that share the property wi th the regular tile that the internal angles of the tiles sum to 27r. Discussion of "measurement" of the internal angles of tiles on the hyperbolic plane is contained in Appendix B . The fact that the internal angles sum to exactly 27r is important because the vertices of the tile w i l l meet at a point after identifications of the boundaries as in Figure (2.2). The sum of these internal angles must be 27r in order for the object produced to be smooth at the vertex and not have a conical singularity. This can perhaps be seen better in the universal covering space, H2. We can use the holonomies of the spacetime to produce Lorentz transformed copies of the original tile. The hyperbolic plane can be completely tiled wi th the copies of the original Chapter 6. Realization of Solutions 67 Figure 6.12: The regular tile. t i le. This requires that the sum of the interior angles, which a l l meet at the vertices, must be 27T since H2 is locally like the plane. Figure (6.13) illustrates the original tile and one of the Lorentz transformed copies. The configuration constraint ensures that the sum of the internal angles is always a multiple of 27T and since the region is connected, we expect the sum of the internal angles to be the same for a l l solutions in this region. This result shows that these solutions in the regular region of the configuration space reproduce the original smooth manifold which was used in the derivation of the holonomy action. However, the manner in which the sum of internal angles was determined does not appear to lend itself to a general proof. In the next section we w i l l show how the components of the configuration space can be classified in a more mathematical way. In terms of the new classification we wi l l show that this means that a l l solutions in the regular regions of the constrained configuration space correspond to smooth manifolds and produce tiles wi th the sum of their internal Chapter 6. Realization of Solutions 68 Figure 6.13: The regular tile and a Lorentz transformed copy, angles equal to 2ir. The reader w i l l recall that there is another region of the configuration space that is in some sense "symmetric" to this region. This is borne out by the the types of tiles they produce. The tiles are exactly the same as those described above except in the direction of the numbering of the vertices around the closed polygon (see Figure 6.12). In this region the vertices are numbered counterclockwise rather than clockwise. The direction of "rotation" of the vertices does not really matter to the spacetime as long as it is consistent. One must therefore identify these two regions as the same region under the relabelling of the vertices. This is an example of physical equivalence of distict solutions to the constraint equations under "large diffeomorphisms" which we w i l l discuss in the next chapter. 2. Tiles for the negative trace regions of the configuration space A t first glance there appear to be two different types of tiles produced by solutions Chapter 6. Realization of Solutions 69 - 2 4 2 0 - 1 0 -5 0 5 Figure 6.14: A tile in the negative trace region. in this region of the configuration space. Both tiles, however, are shown to have the property that the sum of their internal angles is 4 7 T . This means that the objects produced after identification of the edges have a conical singularity at the vertex wi th an excess angle of 2ir. H2 cannot be tiled wi th Lorentz transformed copies of these tiles because the tiles surrounding a vertex overlap each other. Notice that a vector parallel transported along a closed loop around the vertex w i l l s t i l l return to its original state as required by the configuration constraint (3.87). A vector parallel transported around a conical singularity w i l l undergo a rotation by the excess angle - in this case 2-7T - and the vector w i l l thus return to itself. Therefore the configuration constraint is s t i l l satisfied but the object is not a manifold. The first type of tile, shown in Figure 6.14, looks simple enough but the presence of two interior angles obviously greater than 7r indicates the presence of a conical singularity at the vertex. The second type of tile, shown in Figure 6.15 is immediately seen to be "perverse" in that the original tile overlaps itself. Consider the overlapping sections of the Chapter 6. Realization of Solutions 70 2 Figure 6.15: A tile in the negative trace region. tile as ly ing on different branches of the surface wi th a branch cut along one of the edges containing the interior vertex - vertex 7. The two sides that meet at this vertex w i l l be on different branches of the surface and this w i l l mean that the angle at this vertex w i l l itself be greater than 2TT. This again indicates a conical singularity wi th an excess angle at the vertex. These objects have the topology of a 2-holed tori and are almost smooth manifolds, but have conical singularities and thus .are conifolds. A s in the previous section, the two negative trace regions of the configuration space produce identical tiles up to vertex relabelling, meaning that they must be physi-cally identified. These do not represent smooth manifolds and are thus physically irrelevant. 3. Tiles for the t r iv ia l regions of the configuration space This region is named for the fact that the t r iv ia l solution to the constraint equation, Chapter 6. Realization of Solutions 71 - 0 . 5 0 0.5 1 Figure 6.16: A tile in the t r iv ia l region. i n which a l l the holonomies are identities, lies i n this region. A tile produced by a solution in this region, shown in Figure 6.16, is immediately seen to be perverse. The sum of the internal angles for this tile vanishes. The important detail to notice here is that the direction of rotation of the vertices is not consistent. Further, in construction of the tile we take the face of the tile to be the interior of the polygon. In the previous cases we had a consistent way to define the interior of the tile - the region on the left as the boundary is traversed counterclockwise. In this case that definition, at first glance, does not appear to work. We can make a consistent definition of the interior of the tile and produce something which is almost a manifold, but to do so involves a reflection of one of the loops so that they have the same orientation. This would be equivalent to taking the normal of the surface to point in opposite directions. We would consider the manifold to consist of the top of the hypersurface (upward pointing normal) on one loop and on the underside of the hypersurface on the other loop. In this Chapter 6. Realization of Solutions 72 Figure 6.17: Illustration of the path across an identified edge for overlapping patches corresponding to solutions in the t r iv ia l region of the configuration space. way the definition of the interior of the manifold would be consistent. However, the holonomies required for this construction are no longer in the 51 / (2 ,1) + . The alternative is to produce something which is a nonorientable manifold. To see this, Lorentz transform the original tile as was done in Figure 6.13. Note that the identification of edges in Figure 2.2 requires edge A to be identified to edge C in such a way that vertex 1 is identified wi th vertex 4 and vertex 2 is identified wi th vertex 3. Looking at Figure 6.16 we see that in order for the vertices to line up in this way the Lorentz transformed tile must overlap the original tile as in Figure 6.17. This is true for al l other identified sides as well. Passing across a boundary amounts to suddenly starting to move backward on the hypersurface. Note that the interior angles can therefore be negative and we st i l l have the parallel transported vector agreeing wi th the original vector because the closed loop is actual a contractible curve. The sum of internal angles calculated in this way is zero. The object has a Chapter 6. Realization of Solutions 73 conical singularity wi th excess angle of — 2TT 3 . We cannot prove that the sum of the interior angles for every tile in each region is as described above using these methods. However, the insight provided by this analysis illustrates how the constraint equation (3.87) is insufficient because it allows the vertex to be a conical singularity with excess/deficit angle of 2im where n is an integer. This means that the sum of the interior angles must be constant in each disconnected region or it must have discontinuities. Using Goldman's classification of the components of the configuration space we wi l l see that the latter possibility is eliminated. 6.3 Interpretation of the Tilings The contents of the previous section illustrate that the constraints allow not only solutions corresponding to the smooth manifolds that we desire, but also cases which correspond to objects which are almost manifolds except for a conical singularity at the vertex. The path that most people in the field have taken is simply to use the regular region of the configuration space and discount a l l the others. The justification is usually a reference to the classification of the regions of the configuration space here attributed to Car l ip [2] based on a paper by Goldman [11]. This reasoning is not particularly transparent. The above realization of solutions shows that the negative trace regions and the t r iv ia l region must be discarded as they correspond to S with conical singularities. Also , this realization illustrates that the extra solutions came into the picture as a result of the constraint (3.87) being insufficient to ensure that, S, is flat at the vertex. However, in order to be able to prove these results for a l l solutions to the configuration constraint, we need to demonstrate the connection between our regions of the constrained configuration 3This corresponds to a deficit angle at the vertex of 27T. Chapter 6. Realization of Solutions 74 space and those obtained using Goldman's theorem. If we let S be a closed oriented surface of genus g > 1, and we let 7r denote its fundamental group 4 , Goldman's Theorem[ll] is: Theorem B 1 Let G=PSL(2,R). The connected components of Hom(ir,G) are the preimages e _ 1 ( n ) where n is an integer satisfying \n\ < 2g — 2. Hom(7r ,G) means the space of Homomorphisms from the fundamental group of the surface to the group G . This is exactly the space of solutions to the configuration con-straint. The Euler class (e) is a characteristic class of flat principal bundles 5 which determines invariants of representations of the fundamental group Hom(ir,G) over E . The Euler class is actually the second obstruction, o2, of the t r ivial i ty of the principal bundle. The Euler class is calculated as a function of the curvature 2 form 1Z (and thus of the connection 2 form) on the associated vector bundle: e{(f,)=L i ~ i b T r n 2 ( 6 - 1 7 0 ) where 0 is a member of Hom(7r,G). Using hyperbolic coordinates on the tile: \--^Tr1l2 = ^-sinhp dp A d9 (6.171) V OTTZ ZTT and integrating over the tile we find the Euler class for the regular tile shown in Figure 6.12 to be 2. This satisfies the Gauss-Bonnet theorem: 6 e(4>) = \X\ (6.172) 4 The fundamental group of a surface consists of equivalence classes of loops that cannot be deformed to one another. The fundamental group of S2 is trivial while the fundamental group of the torus is generated by the two noncontractible loops. 5 Principal bundles are bundles where the fibres are the same as the structure group. Basically to every element in the manifold corresponds a copy of the group. 6The theorem is taken from [14] using the notation of [11]. Chapter 6. Realization of Solutions 75 for a compact orientable manifold since the Euler characteristic of the genus two surface is x = — 2. This is consistent wi th the above results. The integration over a tile wi th relabelled vertices in the other region of the configuration space w i l l have the same value, but wi th reversed sign because of the reversed orientation of the vertices. Therefore the two regular regions of our constrained configuration space are the components wi th Euler class ± 2 . Integrating (6.171) over the tile in the negative trace region shown in Figure 6.14 we obtain the Euler class 1. This does not satisfy the Gauss Bonnet theorem and thus the object is not a manifold, which is again consistent wi th our above results. The two negative trace regions of our reduced configuration space are the components wi th Euler class ± 1 . This means that the t r iv ia l region of the configuration space must be the component wi th Euler class 0. Th i s classification scheme, along wi th the Gauss-Bonnet theorem, tells us that the regular regions are the only regions corresponding to solutions which produce manifolds and these two regions are identified. The requirement we had in the previous section for S to be a manifold is that the sum of the internal angles of the representative tile is 27T. Our results in this section show that this applies generally to a l l tiles in the regular regions of the configuration space. A t this point it is worth mentioning that the term connection has many usages. A n affine connection [14] is a map V : X{M)®X(M) —> X(M)7 which satisfies some Liebnitz like properties. This map tells us the covariant derivative of one vector field along the other vector field. This basically defines the notion of parallel transport in an associated vector bundle (usually the tangent bundle!). The connection coefficients to^b are often simply referred to as the connection since 7X(M) is the set of smooth vector fields on the manifold M. Chapter 6. Realization of Solutions 76 they implement the mechanics of the covariant derivative described above. Connection two forms (mentioned above just before 6.170) on the manifold can be used as a shorthand way of calculating the components of the Riemann tensor. F ina l ly the connection one form on a principal bundle is a Lie algebra valued one form which separates the vertical and horizontal components of the principal bundle. A s well, there are local and global definitions of this form. A l l of these notions are equivalent, but the presence of non-trivial vector bundles can lead to confusion. In the development of the action in Chapter 2 it was implied that the internal connec-t ion coefficients, oj^b, define the connection on the tangent bundle. It is implici t because this is how the connection is defined in most cases in General Relativity. However, the statement 8 : "a flat connection is determined, uniquely up to gauge transformations, by its holonomies around the nontrivial loops..." refers to the local connection two form on a principal bundle and this relation can be made explicit by the formula 9 [1]: where Pa and Ja are the generators of 750(2,1)+. Connections on principal bundles induce covariant derivatives on their associated vector bundles. In particular one principal bundle is the frame bundle which consists of 8 Taken from [9] which quotes Carlip [2]. 9P stands for path ordering. (6.173) where the local connection one form is defined as: All = ellaPa + uallJa (6.174) Chapter 6. Realization of Solutions 77 orientations of the basis at each point in space. A connection on this principal bundle induces a covariant derivative on the tangent bundle. This returns us to our original notion of connections. However, i f we take the holonomies that solve the constraint equation as fundamental, we cannot assume that they al l correspond to connections on the frame bundle because they do not. Some solutions of the constraint equation induce covariant derivatives on vector bundles other than the tangent bundle. W h a t this means, in short, is that the assumption in the derivation that the connection coefficients define a covariant derivative on the tangent bundle is not respected by the constraints on the action principle. The shows us again in a different way that the constraint equations are insufficient. Goldman's theorem can be used in this way to classify the regions of the constrained configuration space of this formulation for the case of surfaces wi th any genus g > 1. In particular the components of the configuration space wi th absolute value of Euler class equal to 2g-2 (which is 2 in this case) is exactly the Teichmuller space of E [15]. The Teichmuller space of E is the quotient space of al l constant curvature metrics on E by the diffeomorphisms connected to the identity: M e f c o n , t ( E ) / f i i y r ^ = D .y / , (E) ' ( f U 7 5 ) The uniformisation theorem tells us that every metric on the genus g > 1 surface is conformally equivalent to a constant curvature metric[17]. This is the natural configuration space of gravity i f we neglect large diffeomorphisms. This relation between the regular regions of the constrained configuration space and the Teichmuller space is further justification that they are the only physically relevant components. Including large diffeomorphims in the gauge group produces the M o d u l i space which we w i l l discuss in the next chapter. Chapter 6. Realization of Solutions 78 6.4 S u m m a r y The first thing we have accomplished in this chapter is to relate the disconnnected regions of the reduced configuration space obtained here to the Goldman classification scheme. The second thing we have accomplished is to illustrate that only the regular regions of the configuration space are really applicable to 2+1 gravity and that these regions are identified wi th one another. The thi rd thing we have accomplished is to illustrate how the extra regions of our configuration space make their way into this theory. Chapter 7 The Phase Space This chapter uses the knowledge of the structure of the constrained configuration space developed in previous chapters to draw some conclusions about the structure of the full phase space. We w i l l first discuss the phase space in the genus one case which has gone unmentioned since Chapter 5. The genus two case is then discussed. F ina l ly we discuss how the inclusion of large diffeomorphisms affects the phase space and the implications for a quantum theory of 2+1 gravity in this formulation. 7.1 The Phase Space of 2+1 Gravity on the Genus One Surface Louko and Maro l f [5] analysed the phase space of 2+1 gravity on the genus one surface. In this section we repeat part of their analysis which is necessary for our purposes. We w i l l use these results in later sections to draw our own conclusions about the phase space of the genus one surface and also to illustrate some problems that the genus two surface does not have. We have already seen in section 4.2 that the configuration space of the genus one sur-face in the connection formalism consists of pairs of commuting Lorentz transformations. A s well , we mentioned that the translation constraints allow the translations complete freedom in the direction of the axis of the Lorentz transformations. There are four types of solutions, the first of which are solutions where the Lorentz part of both holonomies are identity transformations and the holonomy is purely trans-lat ional . This region w i l l be referred to as Mo- A s shown in Chapter 3, this region has 79 Chapter 7. The Phase Space 80 a 3-dimensional momentum space rather than the two dimensional space at any other point of the configuration space and is a singularity of the phase space. The other three regions have Lorentz part of at least one of the holonomies nontrivial and they are classified according to the timelike, spacelike or null character of the Lorentz part. They are labelled as A4S, J\4.t and M . N . The union of al l four types of solutions w i l l be denoted by j \ A . Note that M Q is separated from the rest. Louko and Marolf[5] claim that M — M Q is a manifold while M is not, which is consistent wi th our above discussion. A n analysis of j\4s is a l l that is necessary for our purposes. The holonomies in this case can be gauge transformed to the following form: Wi ( coshXi sinhXi 0 ^ sinhXi coshXi 0 0 0 1 (7.176) o (7.177) where a l l variables can take arbitrary values except that the A; are not both equal to zero. Note that the translations are in the same direction as the axis of the Lorentz part of the holonomy. The topology of the phase space is thus R 2 ® { R 2 minus the origin}. The origin in A-space corresponds to both Lorentz transformations being the identity, which we have excluded as part of M.Q. (Note that the subscripts take the values i = l , 2 ; i = l refers to the C A holonomy and i=2 refers to the D B holonomy as in Section 3.1) There is a further redundancy in that a gauge transformation of a rotation by ir rotates A; —• — A; and a, —> —a,. This rotation is a gauge transformation so we must identify the points according (A,, a,) and (—Aj, — a;). The origin (A,, a,) would be a fixed Chapter 7. The Phase Space 81 point of this identification and would lead to a conical singularity. However, since the origin is absent in this case, the resulting space is st i l l a manifold wi th the topology of R3 ® S1. This structure is just the cotangent bundle of the punctured plane. Substituting (7.176) and (7.177) into the action (3.77) we obtain the action: 1 = J £ijaj(t)\i(t)dt (7.178) since the constraints are solved. This is in canonical form. The symplectic form fts = e^ddjAdXi on the manifold thus gives it the structure of a phase space - in particular that of the cotangent bundle over the punctured plane. The symplectic form w i l l map the Poisson bracket to the commutator during quantization. Louko and Marolf[5] c la im to have a parameterization that combines the manifolds j\4s, j\4t and j\An into a single manifold that has the structure of a phase space. They then discuss quantization on this object. We only discuss Ais because we cla im that the action of the large diffeomorphisms on the phase space wi l l produce identifications in the momentum space of this manifold which w i l l make quantization on j\43 and on any larger manifold containing it i l l defined. 7.2 The Phase Space of 2+1 Gravity on the Genus Two Surface In this section we use our results from Chapter 4 to discuss the phase space of the genus two surface. The configuration space of 2+1 gravity on the genus two surfaces w i l l be taken to be a regular region of the full configuration space as has been suggested in the literature and justified in the previous chapter. The regular region consists of W C A , W D B , WQE and WHF a l l nontrivial generic boosts wi th the property that: - f t = W C A W E I W C I W D B (7.179) - f t " 1 = WGEWalFWalWHF (7.180) Chapter 7. The Phase Space 82 where fl is a generic boost as well (see Eqs. 5.160). The construction of solutions described in Chapter 4 provides the parameterization of the configuration space. We can use a gauge transformation of the form (3.94) to rotate the axis of fl to a fiducial direction such as ua — (0,1,0) since: ( L W C A L - 1 ) (LWEhL-1) (LWc\L-1) ( L W D B L - 1 ) = L (WCAWEBWcAWDB) L'1 = L(-rt)L-K (7.181) Now we can apply a second gauge transformation of this form that leaves the axis of —ft invariant but transforms the axis of W^B m * ° the x-t plane 1 . After these gauge transformations, the spacelike unit vector which is the axis of W^B o n l y n a s o n e degree of freedom left. The vector must lie in the intersection of Figure 4.4 wi th the vertical x-t plane. The trace constraint equation in this case is from section 4.2, Case 1(b): Tanh[b] = - ^ M (7.182) waba where b refers to the parameter of W^B m this case and ba refers to its axis. Th i s constraint equation becomes: Tanh[b] = (7.183) We have two degrees freedom in picking u and &1, but picking these fixes the parameter b. There is another degree of freedom in the construction of W Q A since it must be of the form (4.147): WCA(a) = A[WDB{a)\-\ (7.184) Similar ly for W^p and W Q E the trace constraint equation is: Tanh[d] = _ ^ S ! (7.185) xNote that Wpg and fl cannot have the same axis or the trace constraint equation (4.105) could not be satisfied. Chapter 7. The Phase Space 83 where d refers to the parameter of W ^ F and da refers to its axis, to has already been fixed. We have two degrees of freedom in picking da in one of the regions of Figure 4.4. This fixes the value of d. F ina l ly WQE has an additional degree of freedom analagous to that of WQA-These six degrees of freedom clearly form a six dimensional manifold which is the configuration space. The momentum space is the space of solutions to the trace constraint which can be writ ten in the form: WEB(1 - W C A ) ^ D B + ^ r 1 ( l - W C A W E B W C I ) TICA (7.186) + Q 2 [ W ^ ( l - W G E ) U H F + Q 2 1 ( l - W G E W ^ W S E ) U G E ] = 0. We can gauge away three of the degrees of freedom using the gauge transformations (3.93) leaving a homogenous system of 3 equations in 9 variables. Actua l ly the presence of (1 — W ) operators again means that each vector has a degree of freedom in the direction of the axis of W . This leaves three constraint equations in five variables. This is a homogeneous system of equations with fewer equations than variables. There is always a solution and in this case the solution space is a two dimensional vector space. Thus the momentum space forms a six dimensional vector space as required. Note that a singularity of the type that occurs in the genus one case, where both the W Q A a n d W D B are identity transformations, cannot occur in this case. A l l the W X Y must be nontrivial (ie. not 1) in this case and the constraints and their corresponding gauge freedoms w i l l always reduce the degrees of freedom of the translation by six. Thus we have the structure of a cotangent bundle over a six dimensional manifold. Now we need to define a symplectic form on the manifold which wi l l give it the structure of a phase space. Unfortunately, the parameterization used does not lend itself naturally to this purpose. The way in which W G A and W Q E are constructed means that Chapter 7. The Phase Space 84 they each depend on three of the parameterization variables and they do so in a very complicated way. The terms of the action which would determine the symplectic structure are: I - ( W E B T I D B ) {WclAfli} + ( W E B K C A ) {WbWi} - U G E { W B ~ F W } + ( W G E W ^ U H F ) { W G ~ E W } . recalling (2.60): {W}a = £abcWXYbdWXYcd (7.187) The matr ix WxYcd can be written as a sum of matrices involving the time derivative of each of the configuration variables. Therefore a term of the form: Tla{W}a (7.188) w i l l be linear in the time derivative of each of the configuration variables and linear in the momentum variables. The general form of this action w i l l be: 6 . / = £ 7 1 - 2 h(qk)ij qj (7.189) where qi are the six configuration space variables and 7T,- are the six momentum space variables which are components of the IPs. The components of h{qi)1- are complicated functions of the configuration variables. Note that this is not in canonical form and thus the 7T? are not conjugate momenta to the However, i f we define new variables: Pi = (7.190) Qi = Qi (7.191) this action w i l l be in canonical form. W i t h these new variables we have a two form: fl = dPi A dQi (7.192) Chapter 7. The Phase Space 85 Figure 7.18: The action of a Dehn twist. The phase space does not appear to have any singularities as in the genus one case. Therefore it appears that Q defines a symplectic form on the phase space manifold. For the purposes of this thesis we w i l l assume at this point that the mentioned sym-plectic structure exists and discuss the large diffeomorhisms. 7.3 Large Diffeomorphisms The large diffeomorphisms (also called the mapping class group and the modular group) are diffeomorphisms that cannot be smoothly deformed to the identity (small diffeomor-phisms). For example, on the genus one surface some of them correspond to cutting the surface along one of the basic nontrivial loops and twisting along that loop by an integer number of rotations and rejoining the surface so that al l points match up exactly the same as before (Dehn Twist - see Figure 7.18). This procedure is not actually carried out, but it describes a change of coordinates where the coordinates are twisted as you travel around the other noncontractible loop. Notice this means that traveling along the other noncontractible loop w i l l now involve the holonomies of both the original noncon-tractible loops. This description can be generalized to an arbitrary number of cuttings and twistings and also to an arbitrary genus surface. Chapter 7. The Phase Space 86 Figure 7.19: Another large diffeomorphism. The rest of the large diffeomorphisms on the genus one surface are those of relabelling the two noncontractible loops (Figure 7.19). The loop about the hole of the torus becomes the loop about the body of the torus and vice versa. These transformations are simply coordinate transformations and thus should not constitute different physical solutions. We should physically identify a l l solutions that differ by large diffeomorphisms. The reduction of the Teichmiiller Space by the large diffeomorphisms (modular group) produces what is called the M o d u l i space of E . The action of the mapping class group on the configuration space is known to be that of the outer automorphisms [12]. Outer automorphisms are defined to be transformations of the type[12]: W X Y - * W X Y G (7.193) where W X Y is one of the 2g Lorentz holonomies (assuming al l f l = 0) of the E and G is Chapter 7. The Phase Space 87 an arbitrary product of the Lorentz holonomies: G = n ( W r y ) < (7.194) G is said to be a member of the holonomy group generated by the holonomies of the noncontractible loops. In our discussion of the Dehn twists we argued that its effect on the holonomies of the spacetime is that of taking the composition of the original holonomy of a loop wi th the holonomy of the loop along which we cut and twist. For example, in Figure 7.18 a curve around the outside of the torus wi l l be twisted around the torus by the pictured Dehn twist. Because of the twisting of this curve around the torus, the holonomy of the curve w i l l be the composition of the original holonomy wi th the holonomy of the curve along which the twist is made. We can see that (7.193) is the generalisation of this effect for arbitrary combinations of Dehn twists in the special case where the II's are zero and the holonomies are the Lorentz transformations, W X Y -However, this only looks at the effect of the mapping class group on the configuration space. We want to look at the effect on the whole phase space. The effect of these maps on the translations are nontrivial as well. Returning to the idea of the Dehn twist, the action of large diffeomorphisms is seen to redefine the holonomies of one of the noncontractible loops by mult iplying it by the holonomy of another. This corresponds to: and we can see that the action of the large diffeomorphisms will involve the full phase space. In the genus one surface this seems to be particularly simple since the the translation vectors lie on the axis of the Lorentz transformations and the Lorentz transformations ( W C A , R C A ) - ( W D B W C A , W D B U C A + ^ D B ) (7.195) Chapter 7. The Phase Space 88 ^2 = 3 A, -| Figure 7.20: The configuration space in the genus one case. The regions between the solid lines are a l l identified. The dotted lines are identified and each has periodic identifications wi th period e along their length. commute. This means that a large diffeomorphim corresponds to simply adding nX\ to A 2 and na i to a 2 or vice versa. Therefore we identify the points: (nAi + 171X2, nXi + 171X2, nai + ma2, na\ + ma 2 ) 2L -J- A. m ' rh. (7.196) (7.197) in the phase space. Notice that this involves identifications in the momentum space and the configuration space. Even in the configuration subspace alone, these identifications make things very com-plicated. After identification, the space does not have a fundamental region - a region of distinct points which corresponds to al l points in the original space. Assume Ai is very small - e. A long the line X\ — e (see Figure 7.20) we indentify al l the points that differ in A 2 by e. A s s goes to zero the identified points get arbitrarily close together. This whole line must also be identified wi th the line A i = e + n A 2 as well. In order to contain Chapter 7. The Phase Space 89 a representative of al l these points, a fundamental region must contain an e-region of one of these lines and thus must also contain a small region above this line (or a small region to the left of the vertical line). This region w i l l contain multiple copies of points on the line A i = e — 6 unless the width of the region along the line A i = n A 2 + (e — 6) line decreases faster than (e — 6). The same argument, however, applies i f we reverse the roles of Ai and A 2 . The same region must have vanishing width in two independent directions. This illustrates why there is no fundamental region of the configuration space reduced by the large diffeomorphisms. This intuitive picture of what goes wrong w i l l be useful in what w i l l follow, but Peldan [3] has offered a proof that there is no fundamental region of the configuration space. The same argument applies to the momentum space also since the action of the large diffeomorphisms is the same. The implicat ion of this is that the phase space reduced by the large diffeomorphisms is not a manifold - furthermore the compactification produced by these periodic identi-fications means that the phase space is not a cotangent bundle. The action of the large diffeomorphisms on the genus two theory at this point is difficult to describe in terms of our parameterization of the phase space. The action of a special class of the large diffeomorphisms can be easily illustrated in terms of the configuration space parameters, a. Comparing the forms of (7.184) and (7.193) we can identify W C A wi th W C A W B D by identifying the parameter a in E q . (7.184) wi th period of the Lorentz parameter of WpB. Similar arguments apply to WQE- However, the rest of the large diffeomorphisms cannot be handled so easily. It is difficult to demonstrate that invariance of the theory under the large diffeomor-phisms requires identifications in the momentum space as in the genus one case. However, the form of (7.195) would seem to suggest that identifications in the momentum space are required and we can see no reason that this should not be so from our parameterization. There are some differences between the two cases. Another result of Goldman's [15] Chapter 7. The Phase Space 9 0 is that Lorentz holonomies ( W X Y ) corresponding to solutions in the regular region of the constrained configuration space generate a properly discontinuous subgroup of the 2+1 Lorentz group. Wi thout going into details, this means that a fundamental region of the configuration space can be argued to exist under the action of the mapping class group[16]. Unfortunately, large diffeomorphisms do not act freely on the phase space [5] which means that there are fixed points of the phase space. These fixed points w i l l mean that the phase space under identifications by the large diffeomorphisms is not a manifold, but an orbifold. The properly discontinuous nature of the Lorentz holonomy group has been suggested to be a crucial fact that makes the process of defining a quantum theory of 2+1 gravity easier on the genus two surface than the genus one surface[5] [2]. However, because this disregards the action of the large diffeomorphisms on the momentum space, we w i l l argue in the next section that there is a more basic barrier to quantization that affects both the genus one and two cases. The arguments used here in genus two apply to arbitrary genus surfaces. 7.4 Quantizing the Theory? The genus one case has now been formulated in terms of a set of canonical variables that form a symplectic manifold before large diffeomorphisms are taken into account. The genus two case has been formulated in terms of canonical variables wi th the structure of a cotangent bundle before large diffeomorphisms are taken into account. This forms a symplectic manifold i f we assume the existence' of a symplectic form on the manifold. Given the canonical variables quantization would theoretically proceed by promoting Chapter 7. The Phase Space 91 the canonical variables to the status of operators acting on a Hilbert space of wave functions, \I>. Reducing the phase space by the large diffeomorphisms before quantization w i l l mean that both the configuration variables and their conjugate momenta w i l l have periodic identifications. If we try to promote these variables to operators we expect the operators to be i l l defined. If only the configuration space suffered from this problem, as for a particle on a circle, quantization might be possible. However, because there are identifi-cations in the momentum space as well (the phase space is not a cotangent bundle) we do not know how to proceed. Another possible route to quantization is that of quantizing the theory on the original phase space and identifying the wave functions related by a representation of the mapping class group. Th i s does not make sense either since the wave functions, \1>, are functions of the configuration variables only and the action of the mapping class group involves the momentum space variables as well as the configuration space. The quantum theories of 2+1 gravity in the connection formalism [2] [3] only include the action of the mapping class group on the configuration variables and ignore its action on the momentum variables. Therefore these theories are not complete. These results suggest that the process of quantization of 2+1 gravity is not well defined. This result would seem to apply to the case of arbitrary genus. Chapter 8 Conclusions and Areas for Future Research 8 .1 Conclusions The purpose of this thesis has been to develop an understanding of the phase space of 2+1 gravity and the status of claims of quantization. We have shown following [6] [9] how the action of 2+1 gravity in the connection formalism can be written entirely in terms of the holonomies of the genus g surface. We have solved the configuration constraint equation in this formulation for the genus two surface in a l l possible cases. The solutions to the constraint have enabled us to place restrictions on the types of solutions that exist. Using these restrictions we showed that the genus two constrained configuration space consists of five disconnected components. We constructed physical models of solutions to the configuration constraints in each of these five regions of the constrained configuration space. These models were used to illustrate that these disconnected regions occur because the naive configuration constraint is not restrictive enough to ensure a smooth manifold. In particular, three of these five regions have conical singularities. We have demonstrated the correspondence between this classification of the constrained configuration space in terms of physical models wi th a more abstract classification using a theorem by Goldman. This correspondence enables us to draw the conclusion that only two of the regions of the constrained configuration space are physically relevant and these two are equivalent under relabelling. We discussed the phase space for genus one drawing on a discussion by Louko and 92 Chapter 8. Conclusions and Areas for Future Research 93 Marolf . The phase space of the genus two surface was discussed in terms of an implici t parameterization based on our construction of solutions to the configuration constraint. The effect on the phase space of physical identification by the large diffeomorphisms is discussed for both genus one and genus two. In genus one, we have shown that the inclusion of the large diffeomorphims involve periodic identifications in the momentum space as well as the configuration space. We argue that this should also happen in genus two and in higher genus cases. Based on these results we conclude that this formulation of 2+1 gravity does not have a well defined quantization. 8.2 Future Research Further study of the genus two surface would be greatly facilitated by the development of a better parameterization of the phase space. One could hope that this would lead to the determination of the symplectic form on the manifold. The other important area where a better parameterization would be helpful is in the investigation of the effect of the large diffeomorphisms on the phase space in order to determine i f their inclusion in the gauge group w i l l lead to periodic identifications in the momentum space. It is not clear at this point why other authors have chosen to only consider the action of the large diffeomorphisms on the configuration space when our results suggest that this is an incomplete characterization of the large diffeomorphisms. The quantum theories they have developed may provide useful insight, but it seems they do not completely describe a quantum theory of 2+1 gravity. There have been some suggestions that the gauge group of 2+1 gravity, and gravity in general, should not include the large diffeomorphisms. Since such large diffeomorphisms correspond to relabelling of the boundaries, or to coordinate transformations, it is unclear what this would mean. Wha t kinds of observables could differentiate between them? Chapter 8. Conclusions and Areas for Future Research 94 Generalization of the results to arbitrary genus surfaces would, of course, be useful. Based on our results, however, we believe that the conclusions we have reached are applicable to the higher genus surfaces. In this thesis we only considered the spacelike sector of the internal space which corre-spond to spacetimes wi th spacelike hypersurfaces, E , in the forward light cone. However, Unruh and Newbury [6] have shown that these solutions are gauge equivalent to solutions which enter the timelike sector and thus become timelike surfaces. In addition to which is the fact that even without entering the timelike sector the translation components of the holonomies allow identification of points which are timelike separated. These two situations imply the presence of closed timelike curves. The implications of this to a quantum theory are not well understood. The fact that this formulation allows degenerate metrics means that it is not com-pletely equivalent to the metric formulation. A s well, the fact that a l l solutions are gauge equivalent to solutions wi th degenerate metrics means that any spacetime can be gauge transformed to a degenerate spacetime. This prospect is confusing and disturbing. Knowledge of the set of solutions corresponding to degenerate metrics in the higher genus cases could provide insight into the relation between the connection formulation and the metric formulation in the higher genus surfaces. The presence of degenerate metrics also may indicate the possibility of topology changing quantum theories. The formulation of a 2+1 gravity in Ashtekar loop variables has shown promise in recent years although it s t i l l is an incomplete theory. Relating the formulation presented here to the loop formulation could provide insights to both theories. F ina l ly note that the paper by Wit ten[ l ] , which initiated this renewed interest in 2+1 gravity, showed that it was equivalent to a Chern Simons gauge field theory. Thus it seems that a path integral formulation of this theory could be very useful. This might also give an insight into the effect of large diffeomorphisms on the quantum theory. Chapter 8. Conclusions and Areas for Future Research 95 W i t h so many unanswered questions, 2+1 gravity is likely to be a topic of interest for some time to come. One can only hope that insights gained in this simplified model of 3+1 gravity w i l l eventually aid in the development of a quantum theory of gravity. Bibliography Wit ten , E . 1988, "2+1 Dimensional Gravi ty as an Exact ly Soluble System", Nucl. Phys. B311, 46-78 Car l ip , S. 1991, "Observables, Gauge Invariance and Time in (2+l)-Dimensional Quantum Gravi ty" , Phys. Rev. D42 2647-2654 Peldan, P. 1995 "Large Diffeomorphisms in (2+l)-Quantum Gravi ty on the Torus" Preprint CGPG-95-1-1 (gr-qc/9504014) G i u l i n i , D . and Louko, J . 1995 "Diffeomorphism Invariant Subspaces in Wit ten 's 2+1 Quantum gravity on R x T 2 " , Class. Quant. Grav. 12, 2735-2746 Louko, J . and Marolf, D . M . 1994, "The Solution Space of 2+1 gravity on R x T 2 in Wit ten ' s Connection Formulation" Class. Quant. Grav., 11 311-330 Unruh, W . G . and Newbury, P., 1993, "Solution to 2+1 Gravi ty in DreBein Formal-ism" , Phys. Rev. D 48, 2686 Moncrief, V . 1989 "Reduction of the Einstein Equations in 2+1 Dimensions to a Hamil tonian System over Teichmuller Space", J. Math. Phys. 30 2907-2914 Marolf, D . 1993 "Loop Representations for 2+1 Gravity on a Torus", Class. Quant. Grav. 10 2625-2647 Newbury, P., 1993 (2+1)-Dimensional Gravity over a Two-Holed TorusM.Sc. Thesis (Vancouver: U . B . C) Pala t in i , A . 1919 "Deduzione invariantiva delle equazioni gravitaxionali dal principio d i Hami l ton" , Rend. Circ. Mat. Palermo 43 203-212 Misner, C . W . , Thorne, K . S . , and Wheeler, J . A . 1973 Gravitation (New York, Free-man) 491ff Goldman, W . M . 1988, "Topological Components of Spaces of Representations" In-vent. Math, 93, 557-607 Car l ip , S. 1991, "Measuring the Metr ic in (2+l)-Dimensional Quantum Gravi ty" , Class. Quant. Grav. 8 5-17 Mess, G . 1990 "Lorentz Spacetimes of Constant Curvature" Institutes des Hautes Etudes Scientifiques preprint I H E S / M / 9 0 / 2 8 96 Bibliography 97 [14] Nakahara, M . 1990 Geometry, Topology and Physics (London: IOP) [15] Goldman, W . M . 1985, "Representations of Fundamental Groups of Surfaces" in Geometry and Topology, Lecture Notes in Mathematics, vol 1167, ed. J . Alexander and J . Harer (Berlin:Springer) [16] Ito, K . 1987 "Discontinuous Groups" in Encylopedic Dictionary of Mathematics, ed. Ito, K . [17] Nash, C . 1991 Differential Topology and Quantum Field Theory (London: Academic Press) Appendix A SL(2,R) A . l T h e SL(2,R) Representation of 5 0 ( 2 , 1 ) \ The group SL(2 ,R) is the double covering group of 5 0 ( 2 , 1 ) ^ i n the same way that SL(2 ,C) is the double covering group of 5 0 ( 3 , Using the generators: S 0 = <73 E i = io~ 1 £ 2 = »V2 (A . 198) the action of 50(2,1)+ can be obtained using the matrices obtained from: exp[iuaZa] (A.199) as a representation over spinors: X = xaZa = x° ix1 + x2 • 1 2 IX — xz -xv (A.200) using an adjoint action: X = AX A'1. (A.201) The parameters of the SL(2,R) transformation can be thought of as a vector in a Minkowski space and represent the SL(2,R) transformation. Taking a Minkowski norm gives: ua exp[i\ua\ j ^ | S a ] = exp[i\u\ « a £ a ] . (A.202) If the vector is null leave it in the form of (A.199) 98 Appendix A. SL(2,R) 99 These give: L(ua) — cosh[u] + isinh\u]uaY,a R{ua) — cos\u\ + isin\u\uaY,a N(ua) = l + m a E a (A.203) (A.204) (A.205) L(ua) = R(ua) = cosh[v] + ivPsinh[u] — ulsinh[u] + iu2sinh[u] —ul sinh[u] — iu2sinh[u] cosh[u] — iu°sinh[u] cos[u] + iu°sin[u] — ulsin[u] + iu2sin\u] —ulsin\v\ — iu2sin[u] cos[u] — iu°sin[u] N(ua) = 1 + iu° 1 • 2 —ul — vur -u1 + iu2 1 IU (A.206) (A.207) (A.208) Transformations generated by a spacelike 3-vector w i l l be referred to as generic boosts (hyperbolic transformations[11]). Those generated by a timelike 3-vector w i l l be referred to as generic rotations (elliptic transformations[11]). This terminology is used because a boost is generated by a 3-vector wi th no timelike component and a rotation is generated by a 3-vector wi th no spacelike component. These w i l l be referred to as pure boosts and pure rotations. The vector, ua, defines the axis of the transformation in that a l l multiples of this vector are fixed points of the transformation. Transformations generated by a nul l vector w i l l be called null transformations (parabolic transformations[ll]) The adjoint action (A.201) of L(ua) on a spinor, X , w i l l produce a spinor, X , corre-sponding to a vector, £ a," (A.200) which is a generic boost of the original vector, xa, by the Lorentz parameter 2u about the axis ua. For example, i f ua = (0,u,0), xa w i l l be boosted by 2u in the +y direction. Appendix A. SL(2,R) 100 Figure A .21 : The positive trace Lorentz spinor matrices are associated wi th the vectors between or on the hyperbolas wi th identification of the hyperbolas. The identification is indicated by the line connecting points on the hyperbolas. The diagram is rotated around the vertical (t) axis. The adjoint action R(ua) produces generic rotation of xa by angle — 2u. Due to the cyclic nature of the rotations, many different vectors w i l l produce the same Lorentz transformations. Restricting the vectors to lie on or between the hyperbolas at u = ± | (Figure A.21) w i l l ensure one vector is associated with one Lorentz transformation after identification of the bounding hyperbolas. The identified vectors are reflections of each other through the origin. These vectors produce the positive trace spinor representatives of the 2+1 Lorentz group. These do not form a group, but including the negative trace spinor representatives (the negative of these matrices) makes a group. It is well known that SL(2 ,C) is a double covering group of SO(3 , l ) (ie. two 2 x 2 matrices are associated wi th the same Lorentz transformation). SO(2 , l ) is easily seen to be a subgroup of SO(3 , l ) defined by the property that it keeps vectors in the 2+1 Appendix A. SL(2,R) 101 dimensional subspace of the full spacetime. The portion of SL(2,C) associated wi th SO(2 , l ) can also be shown to be a group. The matrices (A.206), (A.207), and (A.208) and their negatives make up this subgroup, which is isomorphic to SL(2 ,R) . The form of the matrices are: U = (A.209) a + ib —c + id —c — id a — ib The type of transformation is determined by the Tr[U]. If = a < 1, then it is a generic rotation. If = a = 1, then it is a null transformation. Otherwise, it is a generic boost. This is because 'a ' is cosh[u],cos[u] or 1 in (A.206), (A.207), and (A.208) respectively. Using the fact that the determinant is equal to 1, we have: a 2 + b2 _ c2 _ d2 = j_ which in the 3 cases becomes: b 2 - c 2 - d 2 = - ( s inh [ U ] ) 2 b 2 - c 2 - d 2 = +(sin[u]) 2 b 2 - c 2 - d 2 = 0 (A.210) (A.211) (A.212) (A.213) which means: (-b2 - c 2 -d2)/(smh[u})2 = 1 (-b2 - c 2 -d2)/(sm[u])2 = - 1 (-b2 d2) = 0 (A.214) (A.215) (A.216) These equations then force a matr ix of the form U to be one of (A.206), (A.207), or (A.208). Appendix A. SL(2,R) 102 The form of U is invariant under multiplication and we know that inverses and an identity are part of this set. Therefore, this set of matrices is a group. Since SL(2 ,C) is a double covering group of SO(3 , l ) , this group is a double covering group of SO(2 , l ) . The generators obey the algebra: [ E a , E 6 ] = 2 i e a 6 c E c . (A.217) Thei r mult ipl icat ion rule is: S a S f , = -nab + ieabcZc (A.218) and they also have the anticommutation relation: { S a , E 6 } = -2Vab. (A.219) The centre of SL(2 ,R) , the SL(2,R) matrices that commute wi th every matr ix in SL(2 ,R) , is: Z ( SX(2 , R)) = {+1, -1} = Z 2 . (A.220) The set of transformations that commute with a specific Lorentz transformation con-sists of the centre and the set of a l l transformations with the same axis. The set of transformations that anticommute wi th an arbitrary Lorentz transformation is empty. A.2 The Adjoint Act ion on SL(2,R) The adjoint action on the spinor has the effect of Lorentz transforming the vector: AX A'1 = A(xaT,a)A~1 = xaZa = X. (A.221) Notice that the overall sign of A does not matter which leads to the double covering property of SL(2 ,R) over SO(2, Appendix A. SL(2,R) 103 Taking the axis to be a — (0, a, 0) the adjoint action on the spinor leads to: X = o x • ~1 ~2 IX — X ix1 + x2 —x x°cosh[2a] + x2sinh[2a] ix1 + (x2cosh[2a\ + x°sinh[2a\) ix1 — (x2cosh[2a\ + x°sinh[2a]) — (x°cosh[2a\ + x2sinh[2a]) The vector, xa, undergoes a Lorentz boost in the y direction. Taking the axis to be a = (a, 0,0) the adjoint action on the spinor leads to: o (A.222) x° = z ° x1 = cos^a jx 1 + sin[2a]x2 x2 = — sin[2a\xl + cos[2a]x2 (A.223) (A.224) (A.225) It is easy to see that the vector xa undergoes a rotation by 9 = —2a. The adjoint action applied to another SL(2,R) matrice transforms it by Lorentz trans-forming it 's axis. ABA'1 = A(exp[i\b\ 6 a S a ] ) A _ 1 = A(l + i^Afr^A-1 - ^bfAfrXafA-1 + = l + i\b\9aXa-±\b\2(b>aXa)2 + ,.. = B' A.3 T h e Universal Covering Space of SL(2,R) (A.226) The stucture of the universal covering space of SL(2,R) is as shown in Figure A . 2 2 [ l l ] . The entire diagram should also be rotated about its vertical axis of symmetry. Each region corresponds to the portion of 3D Minkowski space shown in Figure A . 2 1 . The hyperbolic regions are a l l copies of the set of generic boosts. The zeroth hyperbolic Appendix A. SL(2,R) 104 Figure A.22: The universal covering space of SL(2 ,R) . Note that the parabolic regions meet at infinity. Appendix A. SL(2,R) 105 region is a region of positive trace representatives and the 2 x 2 identity is at its ori-gin. The two other hyperbolic regions pictured correspond to regions of negative trace representatives and the negative trace 2 x 2 identity representative is at their origins. Each elliptic region corresponds to a copy of the generic rotations and each region contains both positive and negative trace elements wi th the boundary between them being generic rotations wi th parameter u> — ±|. Fina l ly the parabolic regions correspond to copies of the null transformations. The pattern outlined here repeats infinitely. If we are only concerned wi th P S L ( 2 , R ) , we can take any one of the individual copies with identification at the top and bottom to represent i t , as in Figure A . 2 1 . A representation of SL(2,R) itself is better pictured as two adjacent copies identified at the top and bottom. F ina l ly we mention that since these copies a l l identical, using the primed generic rotation parameter, to' = 7r — u mentioned in Chapter 4 corresponds to a parameter wi th respect to one of the negative trace identities matrices which are at the intersections of the parabolic regions in a negative trace components. Appendix B Hyperbolic Geometry Constructing a representation of solutions to the configuration constraint equations in Chapter 5 produces tiles bounded by geodesies that lie on a spacelike hyperboloid . In order to determine i f the solutions produce a manifold, the interior angles at the vertices (where geodesies meet) must be determined. This appendix is a discussion of geodesies in hyperbolic geometry and measuring angles between intersecting geodesies. The Minkowski metric on a three dimensional vector space is: ds* = -dtl + dRz + R2d62. (B.227) M a k i n g the definitions: t = r c o s h p (B.228) R = r s i n h p (B.229) d = 6 (B.230) the resulting metric is: ds2 = -dr2 + r2[dp2 + s inh 2 p d62). (B.231) This is the usual flat metric in hyperbolic coordinates. Now restrict the metric to a single hypersurface (ie. r = 1) and the metric on the curved surface is: ds2 = dp2 + s inh 2 p d92. (B.232) 106 Appendix B. Hyperbolic Geometry 107 The Christoffel symbols for this metric are: Tee = - ^ s i n h p p6 coth p (B.233) (B.234) (B.235) The metric is independent of the the 0 coordinate which means that £ = dg is a k i l l ing vector. Using the fact that the.dot product of the tangent vector wi th a k i l l ing vector is constant along a geodesic (wa£a — k), solve for the components of the tangent vector to a geodesic: wa = ppa + eea 6 = k csch2 p p = zL\Jl — k2csch2p. Using these results: solve for the geodesies: Of — 9i = d6 d6 dX k csch2p dp dX dp ± \ / l — k'2csch'2p ' , 1 . fl -k2 -2k2 csch2p' ± - arcsm — 2 I 1 + A;2 PS (B.236) (B.237) (B.238) (B.239) (|*| ^ 0) (B.240) p = X 9 = ±9t (k = 0). (B.241) There is a unique geodesic (up to direction) going through any two given points specified by the value of 'k' . Corresponding to -k is the same geodesic wi th opposite parametrization. Ini t ial and final coordinates can be substituted into the first geodesic and the expres-sion solved for k. The solution w i l l give two unique values for k plus their negatives. Only one of these values wi l l produce the desired geodesic. Appendix B. Hyperbolic Geometry 108 10)2.5-Figure B.23: Geodesies on the hyperbolic plane. The affine parametrization of these geodesies is obtained from: — ± ^ 1 — k2 csch2p. dX The solution of this equation is: A - A , - = PS (B.242) M • which can be inverted to give: 2^1 + (1 - k2)csch2p - k2cshc4p + (1 - k2)csch2p + 2 csch2p (B.243) (B.244) p = csch 1 4g(Pi)e ±2A \\ (9(Pi)e±2X - 1 + k2)2 + 4k2 The argument of the square root has a local maximum at: (B.245) A„ = ± - l n _(k2 +1)2_ (B.246) Appendix B. Hyperbolic Geometry 109 Figure B.24: Branch Point of arcsin. which means that p has a local minimum. Substituting into the solution for p we obtain that csch[p] = Using this value for csch[p\, we obtain p = 0 and also that the argument of arcsin in the geodesic (B.239) is -1 at this point. This value of the argument is at a branch point of arcsin between the principle branch and the branch just below it(see Figure B.24). The geodesic is well defined i f values for arcsin in (B.239) are taken from the lower branch for A > A 0 . Such care has been taken wi th this detail in order to show that the geodesic is well defined and also because i t is crucial to producing pictorial representations of the tiles. These equations plus the solution for 6 as a function of p, (B.240), allow the plott ing of the unique geodesic between two points as a function of the affine parameter once the value of k has been determined from the in i t ia l and final points. The angle, a , of intersecting geodesies can be determined from the coordinates of the point of intersection, the value of k for each geodesic and by the sign of p for each Appendix B. Hyperbolic Geometry 110 according to the formula: cos a = P1P2 + M 2 (B.247) ± \ / l — k2csch2pj ±^\A — k2csch2p^j + s inh 2 p(kicsch2p){kicsch2p). 

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