the task of quantizing general relativity is one of the outstanding problems of modern theoretical physics .
attempts to reconcile quantum theory and general relativity date back to the 1930s ( see   for a historical review ) , and decades of hard work have yielded an abundance of insights into quantum field theory , from the discovery of dewitt - faddeev - popov ghosts to the development of effective action and background field methods to the detailed analysis of the quantization of constrained systems . but despite this enormous effort , no one has yet succeeded in formulating a complete , self - consistent quantum theory of gravity . the obstacles to quantizing gravity are in part technical .
general relativity is a complicated nonlinear theory , and one should expect it to be more difficult than , say , electrodynamics .
moreover , viewed as an ordinary field theory , general relativity has a coupling constant g with dimensions of an inverse mass , and standard power - counting arguments  finally confirmed in 1986 by explicit computations    indicate that the theory is nonrenormalizable .
general relativity is a geometric theory of spacetime , and quantizing gravity means quantizing spacetime itself . in a very basic sense
for example : 
 ordinary quantum field theory is local , but the fundamental ( diffeomorphism - invariant ) physical observables of quantum gravity are necessarily nonlocal.ordinary quantum field theory takes causality as a fundamental postulate , but in quantum gravity the spacetime geometry , and thus the light cones and the causal structure , are themselves subject to quantum fluctuations.time evolution in quantum field theory is determined by a hamiltonian operator , but for spatially closed universes , the natural candidate for a hamiltonian in quantum gravity is identically zero when acting on physical states.quantum mechanical probabilities must add up to unity at a fixed time , but in general relativity there is no preferred time - slicing on which to normalize probabilities .
ordinary quantum field theory is local , but the fundamental ( diffeomorphism - invariant ) physical observables of quantum gravity are necessarily nonlocal .
ordinary quantum field theory takes causality as a fundamental postulate , but in quantum gravity the spacetime geometry , and thus the light cones and the causal structure , are themselves subject to quantum fluctuations .
time evolution in quantum field theory is determined by a hamiltonian operator , but for spatially closed universes , the natural candidate for a hamiltonian in quantum gravity is identically zero when acting on physical states .
quantum mechanical probabilities must add up to unity at a fixed time , but in general relativity there is no preferred time - slicing on which to normalize probabilities .
faced with such problems , it is natural to look for simpler models that share the important conceptual features of general relativity while avoiding some of the computational difficulties .
general relativity in 2 + 1 dimensions  two dimensions of space plus one of time  is one such model . as a generally covariant theory of spacetime geometry ,
( 2 + 1)-dimensional gravity has the same conceptual foundation as realistic ( 3 + 1)-dimensional general relativity , and many of the fundamental issues of quantum gravity carry over to the lower dimensional setting . at the same time , however , the ( 2 + 1)-dimensional model is vastly simpler , mathematically and physically , and one can actually write down viable candidates for a quantum theory . with a few exceptions , ( 2 + 1)-dimensional solutions
are physically quite different from those in 3 + 1 dimensions , and the ( 2 + 1)-dimensional model is not very helpful for understanding the dynamics of realistic quantum gravity . in particular , the theory does not have a good newtonian limit [ 107 , 49 , 94 ] .
but for understanding conceptual problems  the nature of time , the construction of states and observables , the role of topology and topology change , the relationships among different approaches to quantization  the model has proven highly instructive . work on ( 2 + 1)-dimensional gravity dates back to 1963 , when staruszkiewicz first described the behavior of static solutions with point sources .
progress continued sporadically over the next twenty years , but the modern rebirth of the subject can be traced to the seminal work of deser , jackiw, t hooft , and witten in the mid-1980s [ 107 , 105 , 106 , 249 , 103 , 277 , 279 ] . over the past twenty years , ( 2 + 1)-dimensional gravity has become an active field of research , drawing insights from general relativity , differential geometry and topology , high energy particle theory , topological field theory , and string theory . as i will explain below , general relativity in 2 + 1 dimensions has no local dynamical degrees of freedom .
classical solutions to the vacuum field equations are all locally diffeomorphic to spacetimes of constant curvature , that is , minkowski , de sitter , or anti - de sitter space . broadly speaking ,
three ways to introduce dynamics have been considered : 
 point particles can be added , appearing as conical 
most of the earliest papers in the field [ 246 , 105 , 106 , 107 , 249 , 103 ] were investigations of the dynamics of such conical singularities.if a negative cosmological constant is present , black hole solutions can be found [ 41 , 40 ] .
for such solutions , dynamics at either the horizon or the boundary at infinity can lead to local degrees of freedom [ 78 , 247 , 59 , 96 , 51 , 24 , 91 ] , although these are certainly not yet completely understood .one can consider nontrivial spatial or spacetime topologies [ 277 , 279 ] .
solutions have moduli  a finite number of parameters that distinguish among geometrically inequivalent constant curvature manifolds  and these can become dynamical . 
 in this paper , i will limit myself to the third case , ( 2 + 1)-dimensional vacuum 
 this review is based in part on a series of lectures in   and an earlier review , and much of the material can be found in more detail in a book .
there is not yet a comprehensive review of gravitating point particles in 2 + 1 dimensions , although [ 65 , 197 , 195 , 37 , 36 , 199 , 63 , 183 ] will give an overview of some results .
several good general reviews of the ( 2 + 1)-dimensional black hole exist [ 75 , 39 ] , although a great deal of the quantum mechanics is not yet understood .
most of the earliest papers in the field [ 246 , 105 , 106 , 107 , 249 , 103 ] were investigations of the dynamics of such conical singularities .
if a negative cosmological constant is present , black hole solutions can be found [ 41 , 40 ] . for such solutions , dynamics at either the horizon or the boundary at infinity can lead to local degrees of freedom [ 78 , 247 , 59 , 96 , 51 , 24 , 91 ] , although these are certainly not yet completely understood .
solutions have moduli  a finite number of parameters that distinguish among geometrically inequivalent constant curvature manifolds  and these can become dynamical .
although string theory is perhaps the most popular current approach to quantum gravity , i will have little to say about it here : while some interesting results exist in 2 + 1 dimensions , almost all of them are in the context of black holes ( see , for example , [ 157 , 170 , 187 , 188 , 189 ] ) .
i will also have little to say about ( 2 + 1)-dimensional supergravity , although many of the results described below can be generalized fairly easily , and i will not address the coupling of matter except for a brief discussion in section 5 . throughout , i will use units 16g = 1 and  = 1 unless otherwise noted .
the first step towards quantizing ( 2 + 1)-dimensional general relativity is to understand the space of classical solutions .
one of the principal advantages of working in 2 + 1 dimensions is that for simple enough topologies , this space can be characterized completely and explicitly .
indeed , there are several such characterizations , each leading naturally to a different approach to the quantum theory ; by understanding the relationships among these approaches , one can gain important insights into the structure of quantum gravity . in any spacetime , the curvature tensor may be decomposed into a curvature scalar r , a ricci tensor r , and a remaining trace - free , conformally invariant piece , the weyl tensor \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${c_{\mu \nu \rho}}^\sigma$\end{document}. in 2 + 1 dimensions , however , the weyl tensor vanishes identically , and the full curvature tensor is determined algebraically by the remaining pieces : 1\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${r_{\mu \nu \rho \sigma } } = { g_{\mu \rho}}{r_{\nu \sigma } } + { g_{\nu \sigma}}{r_{\mu \rho } } - { g_{\nu \rho}}{r_{\mu \sigma } } - { g_{\mu \sigma}}{r_{\nu \rho } } - { 1 \over 2}({g_{\mu \rho}}{g_{\nu \sigma } } - { g_{\mu \sigma}}{g_{\nu \rho}})r.$$\end{document } this means that any solution of the field equations with a cosmological constant  , 2\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${r_{\mu \nu } } = 2\lambda { g_{\mu \nu}},$$\end{document } has constant curvature : the spacetime is locally either flat (  = 0 ) , de sitter (  > 0 ) , or anti - de sitter (  < 0 ) .
physically , a ( 2 + 1)-dimensional spacetime has no local degrees of freedom : there are no gravitational waves in the classical theory , and no propagating gravitons in the quantum theory .
this absence of local degrees of freedom can be verified by a simple counting argument [ 49 , 94 ] . in n dimensions ,
the phase space of general relativity is parametrized by a spatial metric at constant time , which has n(n  1)/2 components , and its conjugate momentum , which adds another n(n  1)/2 components .
but n of the einstein field equations are constraints rather than dynamical equations , and n more degrees of freedom can be eliminated by coordinate choices .
we are thus left with n(n  1 )  2n = n(n  3 ) physical degrees of freedom per spacetime point . in four dimensions ,
this gives the usual four phase space degrees of freedom , two gravitational wave polarizations and their conjugate momenta .
it is instructive to examine this issue in the weak field approximation . in any dimension ,
the vacuum field equations in harmonic gauge for a nearly flat metric g =  + h take the form 3\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\square{\bar h_{\mu \nu } } = { \mathcal o}({h^2}),\quad \quad { \partial _ \mu } { \bar h^{\mu \nu } } = 0,$$\end{document } where \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\bar h_{\mu \nu } } = { h_{\mu \nu } } - { 1 \over 2}{\eta _ { \mu \nu}}{\eta ^{\rho \sigma}}{h_{\rho \sigma}}$\end{document } and indices are raised and lowered with the flat metric . the plane wave solutions of equation ( 3 ) are , to first order , 4\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${\bar h_{\mu \nu } } = { \epsilon _ { \mu \nu}}{e^{ik \cdot x}}\quad \quad { \rm{with}}\,{k^2 } = 0\,{\rm{and}}\,{k^\mu}{\epsilon _ { \mu \nu } } = 0.$$\end{document } choosing a second null vector n with n  k = 1 and a spacelike unit vector with k  m = n  m = 0 , we can construct a ( 2 + 1)-dimensional analog of the newman - penrose formalism ; the polarization tensor  then becomes 5\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${\epsilon _ { \mu \nu } } = a{k_\mu}{k_\nu } + b({k_\mu}{m_\nu } + { k_\nu}{m_\mu } ) + c{m_\mu}{m_\nu},$$\end{document } apparently giving three propagating polarizations .
there is , however , a residual symmetry : a diffeomorphism generated by an infinitesimal vector field  with  = 0 preserves the harmonic gauge condition of equation ( 3 ) while giving a  gauge transformation 
\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\delta { \bar h_{\mu \nu } } = { \partial _ \mu}{\xi _ \nu } + { \partial _ \nu}{\xi _ \mu } - { \eta _ { \mu \nu}}{\partial _
\rho}{\xi ^\rho}$\end{document}. writing 6\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${\xi _ \mu } = ( \alpha { k_\mu } + \beta { n_\mu } + \gamma { m_\mu}){e^{ik \cdot x}},$$\end{document } it is easy to check that 7\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\delta { \epsilon _ { \mu \nu } } = 2i\alpha { k_\mu}{k_\nu } + i\gamma ( { k_\mu}{m_\nu } + { k_\nu}{m_\mu } ) + i\beta { m_\mu}{m_\nu}.$$\end{document } the excitations ( 5 ) are thus pure gauge , confirming the absence of propagating degrees of freedom .
fortunately , while this feature makes the theory simple , it does not quite make it trivial .
a flat spacetime , for instance , can always be described as a collection of patches , each isometric to minkowski space , that are glued together by isometries of the flat metric ; but the gluing is not unique , and may be dynamical .
this picture leads to the description of ( 2 + 1)-dimensional gravity in terms of  geometric structures .  the global geometry of vacuum spacetimes in 2 + 1 dimensions
is described mathematically by the theory of geometric structures [ 256 , 62 , 147 ] ( for examples of geometric structures , see ; for a slightly more detailed version of this section , see ) . for simplicity ,
if the spacetime manifold m is topologically trivial , then by equation ( 1 ) the vacuum field equations imply that ( m , g ) is simply a subset of ordinary minkowski space ( v ,  ) .
if m is topologically nontrivial , it can still be covered by contractible coordinate patches ui , each isometric to v , with the standard minkowski metric  on each patch .
the geometry is then encoded entirely in the transition functions on the intersections ui  uj , which determine how these patches are glued together .
since the metrics in ui and uj are identical , these transition functions must be isometries of  , that is , elements of the poincar group iso(2 , 1 ) .
similarly , if   0 , a vacuum spacetime can be built by patching together pieces of de sitter or anti - de sitter space by appropriate isometries : so(3 , 1 ) for  > 0 and so(2 , 2 ) or sl(2 ,  )  sl(2 , )/2 for  < 0 .
such a construction is an example of a geometric structure , in the flat case a lorentzian or ( iso(2 , 1),v ) structure . in general , a ( g , x ) manifold
is one locally modeled on x , much as an ordinary n - dimensional manifold is modeled on . more precisely , let g be a lie group that acts analytically on some n - manifold x , the model space , and let m be another n - manifold .
a ( g , x ) structure on m is then a set of coordinate patches ui for m with  coordinates  i : ui  x taking their values in x and with transition functions \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${g_{ij } } = { \phi _ i } \circ { \phi _
j}^{- 1}\vert{u_i } \cap { u_j}$\end{document } in g. a fundamental ingredient in the description of a ( g , x ) structure is its holonomy group , which can be viewed as a measure of the failure of a single coordinate patch to extend around a closed curve .
let m be a ( g , x ) manifold containing a closed path . as illustrated in figure 1 , we can cover  with coordinate charts 8\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${\phi _ i}:{u_i } \rightarrow x,\quad \quad
i = 1 , \ldots , n,$$\end{document } with constant transition functions gj  g between ui and ui+1 , i.e. , 9\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\begin{array}{*{20}c}{{\phi _ i}|{u_i } \cap { u_{i + 1 } } = { g_i}\circ { \phi _ { i + 1}}|{u_i } \cap { u_{i + 1}},\ , } \\
{ \;\;{\phi _ n}|{u_n } \cap { u_1 } = { g_n}\circ { \phi _ 1}|{u_n } \cap { u_1}. } \\
\end{array}$$\end{document } let us now try to analytically continue the coordinate 1 from the patch u1 to the whole of . we can begin with a coordinate transformation in u2 that replaces 2 by = 2  2 , thus extending 1 to u1  u2 .
continuing this process along the curve , with \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\phi _ j}^\prime = { g_1 } \circ \cdots \circ { g_j}_{- 1 } \circ { \phi _ j}$\end{document }
if the new coordinate function \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\phi _ n}^\prime = { g_1 } \circ \cdots \circ
{ g_n}_{- 1 } \circ { \phi _ n}$\end{document } agrees with 1 on un  u1 , we will have covered  with a single patch . otherwise , the holonomy h( ) = g1    gn measures the obstruction to such a covering . 

figure 1the curve  is covered by coordinate patches ui , with transition functions gi  g. the composition g1
the curve  is covered by coordinate patches ui , with transition functions gi  g. the composition
it may be shown that the holonomy of a curve  depends only on its homotopy class .
in fact , the holonomy defines a homomorphism 10\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$h:{\pi _ 1}(m ) \rightarrow g.$$\end{document } 
 h is not quite uniquely determined by the geometric structure , since we are free to act on the model space x by a fixed element h  g , changing the transition functions gi without altering the ( g , x ) structure of m. such a transformation has the effect of conjugating h by h , and it may be shown that h is unique up to such conjugation .
the space of holonomies is thus the quotient 11\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\begin{array}{*{20}c}{{\mathcal m } = \hom ( { \pi _ 1}(m),g)/\sim , } \\ { { \rho _ 1}\sim{\rho _ 2}\quad \quad { \rm{if}}\,\,{\rho _ 2 } = h\cdot{\rho _ 1}\cdot{h^{- 1}},h \in g. } \\\end{array}$$\end{document } note that if we pass from m to its universal covering space \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\widetilde m$\end{document } , we will no longer have noncontractible closed paths , and 1 will be extendible to all of \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\widetilde m$\end{document}. the resulting map \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$d:\widetilde m \to x$\end{document } is called the developing map .
at least in simple examples , d embodies the classical geometric picture of development as  unrolling   for instance , the unwrapping of a cylinder into an infinite strip .
the holonomies of the geometric structure in ( 2 + 1)-dimensional gravity are examples of diffeomorphism - invariant observables , which , as we shall see below , are closely related to the wilson loop observables in the chern - simons formulation .
it is important to understand to what extent they are complete  that is , to what extent they determine the geometry .
it is easy to see one thing that can go wrong : if we start with a flat three - manifold m and simply cut out a ball , we can obtain a new flat manifold without affecting the holonomy .
this is a rather trivial change , though , and we would like to know whether it is the only problem . for the case of a vanishing cosmological constant , mess   has investigated this question for spacetimes with topologies   . he shows that the holonomy group determines a unique  maximal  spacetime m  specifically , a domain of dependence of a spacelike surface . mess also demonstrates that the holonomy group h acts properly discontinuously on a region w  v of minkowski space , and that m can be obtained as the quotient space w / h . this quotient construction can be a powerful tool for obtaining a description of m in reasonably standard coordinates , for instance in a time - slicing by surfaces of constant mean curvature .
similar results hold for anti - de sitter structures . some instructive examples of the construction of spacetimes with  < 0 from holonomies are given in .
for de sitter structures , on the other hand , the holonomies do not uniquely determine the geometry .
an explicit example of the resulting ambiguity has been given by ezawa   for the case of a topology   t ( see also section 4.5 of ) .
a similar ambiguity occurs for ( 2 + 1)-dimensional gravity with point particles , where , as matschull has emphasized , it may imply a physical difference between the metric and chern - simons formulations of ( 2 + 1)-dimensional gravity .
we close this section with a partial description of the space of solutions of the vacuum einstein field equations on a manifold    , where  is a compact genus g two - manifold , that is , a surface with g  handles . 
the fundamental group of such a spacetime , 1(m )  1( ) , is generated by g pairs of closed curves with the single relation 12\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${a_1}{b_1}{a_1}^{- 1}{b_1}^{- 1}{a_2}{b_2}{a_2}^{- 1}{b_2}^{- 1 } \ldots { a_g}{b_g}{a_g}^{-
1}{b_g}^{- 1 } = 1.$$\end{document } by equation ( 11 ) , the space of holonomies is the space of homomorphisms from 1( ) to g ( where g is iso(2 , 1 ) for  = 0 , so(3 , 1 ) for  > 0 , or so(2 , 2 ) for  < 0 ) modulo overall conjugation . for g > 1 , this space of homomorphisms has dimension 12 g  12 : 1( ) has 2 g generators and one relation , and the identification by conjugation leaves 2 g  2 choices of elements of a six - dimensional group g1 .
there are two subtleties that prevent the space ( 11 ) from being the exact moduli space of solutions of the vacuum field equations .
first , as noted above , the holonomies do not always determine a unique geometric structure .
in particular , for  > 0 one may need an additional discrete variable to specify the geometry .
second , not all homomorphisms from 1( ) to g give geometric structures that correspond to smooth manifolds .
the space of homomorphisms ( 11 ) is not connected , and , in general , only one connected component gives our desired geometry .
even once these caveats are taken into account , though , we still have a ( 12 g  12)-dimensional space of solutions that can , in principle , be described completely .
the formalism of geometric structures provides an elegant description of vacuum spacetimes in 2 + 1 dimensions , but it is rather remote from the usual physicist s approach . in particular , the einstein - hilbert action is nowhere in sight , and even the metric makes only a limited appearance .
fortunately , the description is closely related to the more familiar first - order chern - simons formalism [ 108 , 277 , 279 , 2 ] , which , in turn , can connect us back to the standard metric formalism .
the first - order formalism takes as its fundamental variables an orthonormal frame (  triad  or  dreibein  )
\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${e_\mu}^a$\end{document } , which determines a metric \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${g_{\mu \nu } } = { \eta _ { ab}}{e_\mu}^a{e_\nu}^b$\end{document } , and a spin connection \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\omega _ \mu}^{ab}$\end{document}. as in the palatini formalism , e and  are treated as independent quantities . in terms of the one - forms 13\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${e^a } = { e_\mu}^ad{x^\mu},\quad \quad { \omega ^a } = { 1 \over 2}{\epsilon ^{abc}}{\omega _ { \mu bc}}d{x^\mu},$$\end{document } the first - order action takes the form 14\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$i = 2\int\nolimits_m { \left [ { { e^a } \wedge
\left({d{\omega _ a } + { 1 \over 2}{\epsilon _ { abc}}{\omega ^b } \wedge { \omega ^c } } \right ) + { \lambda \over 6}{\epsilon _ { abc}}{e^a } \wedge { e^b } \wedge { e^c } } \right]},$$\end{document } with euler - lagrange equations 15\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${t_a } = d{e_a } + { \epsilon _ { abc}}{\omega ^b } \wedge { e^c } = 0,$$\end{document } 
 16\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${r_a } = d{\omega _ a } + { 1 \over 2}{\epsilon _ { abc}}{\omega ^b } \wedge { \omega ^c } = - { \lambda \over 2}{\epsilon _ { abc}}{e^b } \wedge { e^c}.$$\end{document } the first of these implies that the connection is torsion - free , and , if e is invertible , that  has the standard expression in terms of the triad . given such a spin connection , equation ( 16 ) is then equivalent to the standard einstein field equations . the action ( 14 ) has two sets of invariances , the local lorentz transformations 17\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\delta { e^a } = { \epsilon ^{abc}}{e_b}{\tau _
c},\quad \quad \delta { \omega ^a } = d { \tau ^a } + { \epsilon ^{abc}}{\omega _ b}\tau_c,$$\end{document } and the  local translations 
18\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\delta { e^a } = d{\rho ^a } + { \epsilon ^{abc}}{\omega _ b}{\rho _ { c,}}\;\;\delta { \omega ^a } = - \lambda { \epsilon ^{abc}}{e_b}{\rho _ c}.$$\end{document } provided the triad e is invertible , the latter are equivalent to diffeomorphisms on a shell ; more precisely , the combination of transformations with parameters  =  
e and  =    is equivalent to the diffeomorphism generated by the vector field . the invertibility condition for e is important ; if it is dropped , the first - order formalism is no longer quite equivalent to the metric formalism . as first noted by achcarro and townsend   and further developed by witten [ 277 , 279 ] , the first - order action ( 14 ) is equivalent to that of a chern - simons theory .
the relevant gauge group  the group g of the geometric structure  is then the poincar group iso(2 , 1 ) , with standard generators \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\mathcal j^ a}$\end{document } and \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\mathcal p^a}$\end{document } and commutation relations 19\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$[{{\mathcal j}^a},{{\mathcal j}^b } ] = { \epsilon ^{abc}}{{\mathcal j}_c},\quad \quad[{{\mathcal j}^a},{{\mathcal p}^b } ] = { \epsilon ^{abc}}{{\mathcal p}_c}\quad \quad [ { { \mathcal p}^a},{{\mathcal p}^b } ] = 0.$$\end{document } the corresponding gauge potential is 20\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$a = { e^a}{{\mathcal p}_a } + { \omega ^a}{{\mathcal j}_a}.$$\end{document } if one defines a bilinear form ( or  trace  ) 21\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${\rm{tr}}({{\mathcal j}^a}{{\mathcal p}^b } ) = { \eta ^{ab}},\quad{\rm{tr}}({{\mathcal j}^a}{{\mathcal j}^b } ) = { \rm{tr}}({{\mathcal p}^a}{{\mathcal p}^b } ) = 0,$$\end{document } it is straightforward to show that the action ( 14 ) can be written as 22\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${i_{{\rm{cs}}}}[a ] = { k \over { 4\pi}}\int\nolimits_m { { \rm{tr } } } \left({a \wedge da + { 2 \over 3}a \wedge a \wedge a } \right),$$\end{document } with k = 1/(4 g ) .
equation ( 22 ) may be recognized as the standard chern - simons action   for the group iso(2 , 1 ) .
 e/ together constitute an so(2 , 1 )  so(2 , 1 ) gauge potential , with a chern - simons action 23\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$i[{a^{(+)}},{a^{(- ) } } ] = { i_{{\rm{cs}}}}[{a^{(+ ) } } ] - { i_{{\rm{cs}}}}[{a^{(-)}}]$$\end{document } that is again equivalent to equation ( 14 ) , provided we set k = (4 g ) .
if  > 0 , the complex one - form \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${a^a } = { \omega ^a } + i\sqrt \lambda { e^a}$\end{document } may be viewed as an sl(2 ,  ) gauge potential , whose chern - simons action is again equivalent to the first - order gravitational action . for any value of  , it is easily checked that the transformations ( 17 ) are just the gauge transformations of a. vacuum general relativity in 2 + 1 dimensions is thus equivalent  again up to considerations of the invertibility of e  to a gauge theory .
we can now connect the first - order formalism to the earlier description of geometric structures .
the field equations coming from the action ( 22 ) are simply 24\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$f[a ] = da + a \wedge a = 0,$$\end{document } implying that the field strength of the gauge potential a vanishes , i.e. , that a is a flat connection .
such a connection is completely determined by its holonomies , that is , by the wilson loops 25\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${u_\gamma } = p\exp \left({- \int\nolimits_\gamma a } \right)$$\end{document } around closed noncontractible curves . this use of the term  holonomy  is somewhat different from that of section 2.2 , but the two are equivalent .
indeed , any ( g , x ) structure on a manifold m determines a corresponding flat g bundle : we simply form the product g  uj in each patch , giving the local structure of a g bundle , and use the transition functions gij of the geometric structure to glue the fibers on the overlaps .
the holonomy group of this flat bundle can be shown to be isomorphic to the holonomy group of the geometric structure , and for ( 2 + 1)-dimensional gravity , the flat connection constructed from the geometric structure is that of the chern - simons theory
. an explicit construction may be found in section 4.6 of ; see also [ 7 , 263 ] .
the first - order action allows us an additional step that was unavailable in the geometric structure formalism  we can compute the symplectic structure on the space of solutions .
the basic poisson brackets follow immediately from the action : 26\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\{{e_i}^a(x),{\omega _ j}^b(x^{\prime})\ } = { 1 \over 2}{\eta ^{ab}}{\epsilon _
i}_j{\delta ^2}(x - x^{\prime}).$$\end{document } the resulting brackets among the holonomies have been evaluated by nelson , regge , and zertuche [ 210 , 211 ] for  < 0 , for which the two sl(2 ,  ) factors in the gauge group g may be taken to be independent .
the brackets are nonzero only for holonomies of curves that intersect , and can be written in terms of holonomies of  rerouted  curves ; symbolically , 27\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\left\{{\left . { } \right\rangle , \left\langle { } \right . }
\right\ } = \pm { 1 \over { 4\ell}}\epsilon ( p)\left({\left . { } \right\rangle \left\langle { } \right .
\right),$$\end{document } where (p ) is the oriented intersection number at the point p that the curves cross .
the composition of loops implicit in the brackets ( 27 ) makes it difficult to find small closed subalgebras of the sort needed for quantization .
however , nelson and regge have succeeded in constructing a small but complete ( actually overcomplete ) set of holonomies on a surface of arbitrary genus that form a closed algebra [ 213 , 212 ] , and loll has found a complete set of  configuration space  variables . by generalizing a discrete combinatorial approach to chern - simons theory due to fock and rosly   and alekseev et al .
[ 3 , 4 , 5 ] , several authors have further explored the quantum group structure of these brackets , which can be expressed in terms of the quantum double of the lorentz group [ 37 , 36 , 61 , 201 ] .
it is also interesting that the symplectic structure obtained in this way is closely related to the symplectic structure on the abstract space of loops on  first discovered by goldman [ 145 , 146 ] .
we next turn to a more traditional approach to classical general relativity , the conventional metric formalism in the space / time splitting of arnowitt , deser , and misner . as moncrief   and hosoya and nakao  
have shown , this metric formalism can also be used to give a full description of the solutions of the vacuum field equations , at least for spacetimes with the topology   . we start with the adm decomposition of the spacetime metric g , 28\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$d{s^2 } = { n^2}d{t^2 } - { g_{ij}}(d{x^i } + { n^i}dt)(d{x^j } + { n^j}dt),$$\end{document } as illustrated in figure 2 .
the action then takes the usual form2 
 29\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${i_{{\rm{grav } } } } = \int { { d^3}x\sqrt { { - ^{(3)}}g } } { ( ^{(3)}}r - 2\lambda ) = \int { dt } \int\nolimits_\sigma { { d^2}x } ( { \pi ^{ij}}{\dot g_{ij } } - { n^i}{{\mathcal h}_i } - n{\mathcal h}),$$\end{document } with canonical momentum \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${{\rm{\pi}}^{ij } } = \sqrt g ( { k^{ij } } - { g^{ij}}k)$\end{document } and the momentum and hamiltonian constraints 30\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${{\mathcal h}_i } = - 2{\nabla _
j}{\pi ^j}i,\quad{\mathcal h } = { 1 \over { \sqrt g}}{g_{ij}}{g_{kl}}({\pi ^{ik}}{\pi ^{jl } } - { \pi ^{ij}}{\pi ^{kl } } ) - \sqrt g ( r - 2\lambda).$$\end{document } 
 figure 2the adm decomposition is based on the lorentzian version of the pythagoras theorem .
the adm decomposition is based on the lorentzian version of the pythagoras theorem . to solve the constraints
, we can choose the york time - slicing , in which the mean ( extrinsic ) curvature is used as a time coordinate , \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$ - k = { g_{ij}}{\pi ^{ij}}/\sqrt g = t$\end{document}. andersson et al .
have shown that this is a good global coordinate choice for classical solutions of the vacuum field equations .
we next select a useful parametrization of the spatial metric and momentum . up to a diffeomorphism
, any two - metric on  can be written in the form [ 1 , 121 ] 31\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${g_{ij } } = { e^{2\lambda}}{\bar{g}_{ij}}({m_\alpha}),$$\end{document } where ij(m ) are a finite - dimensional family of metrics of constant curvature k ( k = 1 for the two - sphere , 0 for the torus , and 1 for spaces of genus g > 1 ) .
these standard metrics are labeled by a set of moduli m that parametrize the riemann moduli space of . as in section 2.2 , such constant curvature metrics can be described in terms of a geometric structure  for genus g > 1 an (  , psl(2 ,  ) ) structure  with moduli parametrizing the homomorphisms ( 10 ) .
we can count these just as in section 2.2 ; now , since psl(2 ,  ) is three - dimensional , we find that a constant negative curvature surface of genus g > 1 is described by 6 g  6 parameters .
the corresponding decomposition of the conjugate momentum is described in : up to a diffeomorphism , the trace - free part of  can be written as a holomorphic quadratic differential p , that is , a transverse traceless tensor with respect to the covariant derivative compatible with ij .
the space of such quadratic differentials parametrizes the cotangent space of the moduli space , and the reduced phase space becomes , essentially , the cotangent bundle of the moduli space . with the decomposition of ,
the momentum constraints i = 0 become trivial , while the hamiltonian constraint becomes an elliptic differential equation that determines the scale factor  in equation ( 31 ) as a function of ij and p , 32\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\bar \delta \lambda   - \frac{1}{4}({t^2 } - 4\lambda ) { e^{2\lambda } } + \frac{1}{2}[{\bar g^ { - 1}}{\bar g_{ij}}({m_\alpha } ) { \bar g_{kl}}({m_\alpha } ) { p^{ik}}({p^\alpha } ) { p^{jl}}]{e^ { - 2\lambda } } - \frac{k}{2 } = 0,$$\end{document } where p are the momenta conjugate to the moduli , 33\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${p^\alpha } = \int\nolimits_\sigma { { d^2}x{p^{ij}}{\partial \over { \partial { m_\alpha}}}{{\bar g}_{ij}}.}$$\end{document } the theory of elliptic equations ensures that equation ( 32 ) determines a unique scale factor . the action ( 29 ) then simplifies to a  reduced phase space 
action , involving only the physical degrees of freedom , 34\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${i_{{\rm{grav } } } } = \int { dt } \left({{p^\alpha}{{d{m_\alpha } } \over { dt } } - h(m , p , t ) } \right),$$\end{document } with a time - dependent hamiltonian 35\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$h = \int\nolimits_{{\sigma _ t } } { { d^2}x } \sqrt { \bar{g } } { e^{2\lambda ( m , p , t)}}.$$\end{document } the classical poisson brackets can be read off directly from equation ( 34 ) : 36\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\ { { m_\alpha } , { p^\beta } \ }   = \delta _ \alpha ^\beta , \quad \quad \ { { m_\alpha } , { m_\beta } \ }   = \ { { p^\alpha } , { p^\beta } \ }   = 0.$$\end{document } three - dimensional gravity again reduces to a finite - dimensional system , albeit one with a complicated time - dependent hamiltonian
. the physical phase space is parametrized by ( m,p ) , which may be viewed as coordinates for the cotangent bundle of the moduli space of . for a surface of genus g > 1 , this gives us 12 g  12 degrees of freedom , matching the results of section 2.2 .
if  = 0 , this correspondence can be made more explicit : for g = iso(2 , 1 ) and m     , the space ( 11 ) of geometric structures is itself a cotangent bundle , whose base space is the space of hyperbolic structures on . this follows from the fact that the group iso(2 , 1 ) is the cotangent bundle of so(2 , 1 ) . concretely , in the first - order formalism of section 2.3 , the curvature equation ( 16 ) with  = 0 implies that  is a flat so(2 , 1 ) connection ; and if (s ) is a curve in the space of such flat connections , the tangent vector e = d(s)/ds satisfies the torsion equation ( 17 ) . for   0 , i know of no such direct correspondence , and the general relationship between the adm and first - order solutions seems less transparent .
discrete approximations to general relativity have existed for decades . in 2 + 1 dimensions ,
this follows from the peculiar nature of the field equations in three dimensions : as discussed above , any vacuum solution can be patched together from finite pieces of constant curvature spacetime , and the dynamics occurs only in the patching .
the  standard  discrete approach to classical general relativity is regge calculus , initially developed for ( 3 + 1)-dimensional gravity but extendible to arbitrary dimensions .
classical regge calculus in 2 + 1 dimensions was investigated by rocek and williams , who showed that it gave exact results for point particle scattering .
the first discrete formulation designed explicitly for 2 + 1 dimensions was developed by t hooft et al .
this approach has been used mainly to understand point particle dynamics , but recent progress has allowed a general description of topologically nontrivial compact spaces .
t hooft s hamiltonian lattice model is based on the metric formalism , and starts with a piecewise flat cauchy surface tessellated by flat polygons , each carrying an associated frame .
the einstein field equations with  = 0 then imply that edges of polygons move at constant velocities and that edge lengths may change , subject to a set of consistency conditions .
one obtains a dynamical description parametrized by a set of lengths and rapidities , which turn out to be canonically conjugate .
complications occur when an edge shrinks to zero length or collides with a vertex , but these are completely understood .
the resulting structure can be simulated on a computer , providing a powerful method for visualizing classical evolution in 2 + 1 dimensions .
a related first - order hamiltonian lattice model has been studied by waelbroeck et al . [ 266 , 267 , 268 , 270 ] .
this model is a discretized version of the first - order formalism of section 2.3 , with triads assigned to faces of a two - dimensional lattice and lorentz transformations assigned to edges .
in particular , for a spacetime m     , one can choose a lattice that is simply a 4g - sided polygon with edges identified ; the resulting spacetime can be visualized as a polygonal tube cut out of minkowski spacetime , with corners lying on straight worldlines and edges identified pairwise .
this reproduces the quotient space picture discussed by mess in the context of geometric structures . with a different gauge choice ,
waelbroeck s model is classically equivalent to t hooft s , but the two models are related by a nonlocal change of variables , and may not be equivalent quantum mechanically .
much of the recent work on lattice formulations of ( 2 + 1)-dimensional gravity have centered on spin foams and on random triangulations , both inherently quantum mechanical .
it is worth noting here , though , that recent work on diffeomorphisms in spin foam models   may permit a classical description quite similar to that of waelbroeck . up to now , i have avoided discussing an important discrete symmetry of general relativity on topologically nontrivial spacetimes .
the description of a solution of the field equations in terms of holonomies ( sections 2.2 and 2.3 ) or moduli ( section 2.4 ) is invariant under infinitesimal diffeomorphisms , and hence under  small  diffeomorphisms , those that can be smoothly deformed to the identity .
but if the spacetime manifold is topologically nontrivial , its group of diffeomorphisms may not be connected : m may admit 
the group of such large diffeomorphisms ( modulo small diffeomorphisms ) , \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\mathcal d(m)$\end{document } , is called the mapping class group of m ; for the torus t , it is also known as the modular group .
the archetype of a large diffeomorphism is a dehn twist of a torus , which may be described as the operation of cutting t along a circumference to obtain a cylinder , twisting one end by 2 , and regluing .
similar transformations exist for any closed surface  , and in fact the dehn twists around generators of 1( ) generate \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\mathcal d(\sigma)$\end{document } [ 57 , 56 ] .
it is easy to see that the mapping class group of a spacetime m acts on 1(m ) , and therefore on the holonomies of section 2.2 . as diffeomorphisms , elements of the mapping class group
also acts on the constant curvature metrics ij , and hence on the moduli of section 2.4 .
classically , geometries that differ by actions of \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\mathcal d}(m)$\end{document } are exactly equivalent , so the  true  space of vacuum solutions for a spacetime with the topology    is really \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\mathcal m/\mathcal d(m)$\end{document } , where \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\mathcal m}$\end{document } is the moduli space ( 11 ) .
quantum mechanically , it is not clear whether one should impose mapping class group invariance on states or whether one should merely treat \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\mathcal d}(m)$\end{document } as a symmetry under which states may transform nontrivially ( see , for instance , ) . in 2 + 1 dimensions , though , there seems to be a strong argument in favor of treating the mapping class group as a genuine invariance , as follows . using the chern - simons formalism
, one can compute the quantum amplitude for the scattering of a point particle off another particle , a black hole , or a handle . in each case
, it is only when one imposes invariance under the mapping class group that one recovers the correct classical limit .
it may still be that simple enough representations of \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\mathcal d}(m)$\end{document } lead to sensible physical results , but it is at least clear that the mapping class group can not be ignored . the simplest nontrivial vacuum cosmology occurs for a spacetime with the topology   t , where t is the two - dimensional torus .
this case is in some ways exceptional  for example , the standard metric ij of equation ( 31 ) is flat rather than hyperbolic  but it is also simple enough that a great deal can be done explicitly . later in this review , the torus universe will be a canonical test of quantization ; here we review classical aspects .
the problem of finding the classical solutions , as well as an approach to the quantization , was , i believe , first discussed by martinec .
i refer the reader to [ 76 , 81 , 87 , 86 ] for further details
. a similarly detailed analysis may be possible when the spatial topology is that of a klein bottle ( see , for instance , ) but so far , this and other nonorientable examples have been studied in much less detail . for simplicity ,
the group g of section 2.2 , or , equivalently , the gauge group in the chern - simons formalism of section 2.3 , is then so(2 , 2 ) .
the fundamental group 1(  t ) has two generators , [ 1 ] and [ 2 ] , satisfying a single relation similar to equation ( 12 ) : 37\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$[{\gamma _ 1 } ] \cdot [ { \gamma _ 2 } ] = [ { \gamma _ 2 } ] \cdot [ { \gamma _ 1}].$$\end{document } the holonomy group ( 11 ) is therefore generated by two commuting so(2 , 2 ) matrices , unique up to overall conjugation .
it is a bit more convenient to describe the holonomies as elements of the covering group sl(2 ,  )  sl(2 ,  ) .
let [a ] denote the two sl(2 ,  ) holonomies corresponding to the curve a .
an sl(2 ,  ) matrix s is called hyperbolic , elliptic , or parabolic according to whether |tr s| is greater than , equal to , or less than 2 , and the space of holonomies correspondingly splits into nine sectors .
it may be shown that only the hyperbolic - hyperbolic sector corresponds to a spacetime in which the t slices are spacelike [ 117 , 119 , 182 , 209 ] . by suitable overall conjugation , the two generators of the holonomy group in this sector can then be taken to be 38\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${\rho ^ \pm } [ { \gamma _ 1 } ] = \left ( { \begin{array}{*{20}c } { { e^{r_1^ \pm /2 } } } & 0\\ 0 & { { e^ { - r_1^ \pm /2}}}\\ \end{array } } \right),\quad { \rho ^ \pm } [ { \gamma _ 2 } ] = \left ( { \begin{array}{*{20}c } { { e^{r_2^ \pm /2 } } } & 0\\ 0 & { { e^ { - r_2^ \pm /2}}}\\ \end{array } } \right),$$\end{document } where the \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$r_a^ \pm$\end{document } are four arbitrary parameters .
note that this gives the right counting : the riemann moduli space of the torus is two dimensional , so from section 2.4 we expect a four - dimensional space of solutions . to obtain the corresponding geometry , we can use the quotient space construction of section 2.2 .
note first that three - dimensional anti - de sitter space can be represented as the submanifold of flat  ( with coordinates ( x1 , x2 , t1 , t2 ) and metric \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$d{s^2 } = dx_1 ^ 2 + dx_2 ^ 2 - dt_1 ^ 2 - dt_2 ^ 2$\end{document } ) defined by the condition that 39\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\det \vert { \bf{x}}\vert = 1,\quad { \bf{x } } = { 1 \over \ell } \left({\begin{array}{*{20}c } { { x_1 } + { t_1 } } & { { x_2 } + { t_2 } } \\ { - { x_2 } + { t_2 } } & { { x_1 } - { t_1 } } \\ \end{array } } \right).$$\end{document } this gives an isometry between ads3 and the group manifold of sl(2 ,  ) .
the quotient of ads3 by the holonomy group ( 38 ) may now be obtained by allowing the [a ] to act on x by left multiplication and the [a ] to act by right multiplication .
it is straightforward to show that the resulting induced metric is 40\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \begin{array}{*{20}c}{d{s^2 } = d{t^2 } - { { { \ell ^2 } } \over 4}\left [ { { { ( r_1^ + ) } ^2 } + { { ( r_1^ -)}^2 } + 2r_1^ + r_1^ - \cos { { 2 t } \over \ell } } \right]d{x^2 } } \\ { \quad \quad \quad \;\ ; - { { { \ell ^2 } } \over
4}\left [ { r_1^ + r_2^ + + r_2^ - r_2^ - + ( r_1^ + r_2^ - + r_1^ - r_2^ + ) \cos { { 2 t } \over \ell } } \right]dx\;dy } \\ { \quad \quad \quad \;\ ; - { { { \ell ^2 } } \over 4}\left [ { { { ( r_2^ + ) } ^2 } + { { ( r_2^ -)}^2 } + 2r_2^ + r_2^ - \cos { { 2 t } \over \ell } } \right]d{y^2 } , } \\
the triad may be read off directly from equation ( 40 ) , and it is easy to solve equation ( 15 ) for the spin connection . the resulting chern - simons connections a of equation ( 23 ) are flat , and their holonomies reproduce the holonomies ( 38 ) of the geometric structure we began with . to relate these expressions to the adm formalism of section 2.4 , we must first find the slices of constant extrinsic curvature t. for the metric ( 40 ) , the extrinsic curvature of a slice of constant t is \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$t = - { 2 \over \ell}\cot { { 2 t } \over \ell}$\end{document } , which is independent of x and y. a constant t slice is thus also a slice of constant york time .
the standard flat metric on t , the genus one version of the standard metric ( 31 ) , is 41\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$d{\sigma ^2 } = { \tau _ 2}^{- 1}\vert dx + \tau \;dy{\vert ^2},$$\end{document } where  = 1 + i2 is the modulus . comparing ( 40 )
, we see that a slice of constant t has a modulus 42\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\tau = \left({r_1^ - { e^{it/\ell } } + r_1^ + { e^{- it/\ell } } } \right)\left({r_2^ - { e^{it/\ell } } + r_2^ + { e^{- it/\ell } } } \right){.^{- 1}}$$\end{document } the conjugate momentum p = p + ip can be similarly computed from equation ( 33 ) , 43\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$p = - { { i\ell } \over { 2\sin { { 2 t } \over \ell}}}\left({r_2^ + { e^{it/\ell } } + r_2^ - { e^{- it/\ell } } } \right){,^2}$$\end{document } while the adm hamiltonian h of equation ( 35 ) becomes 44\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$h = - { { { \ell ^2 } } \over 4}\sin { { 2 t } \over \ell}\left({r_1^ - { r_2}^ + - r_1^ + { r_2}^ - } \right ) = { \left({{t^2 } + { 4 \over { { \ell ^2 } } } } \right)^{- 1/2}}{[{\tau _ 2}^2p\bar p ] ^{1/2}}.$$\end{document } in the limit of vanishing  , these relations go over to those of . to quantize this system , we will need the classical poisson brackets , which can be obtained from equation ( 26 ) : 45\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\{{r_1}^ \pm , { r_2}^ \pm \ } = \mp { 1 \over \ell},\;\;\;\;\{{r_a}^ + , { r_b}^ - \ } = 0.$$\end{document } these , in turn , determine the brackets among the moduli and momenta  and p , 46\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\{\tau , \bar{p } \ } = \{\bar \tau , p\ } = 2,\;\;\;\;\{\tau , p\ } = \{\bar{\tau } , \bar{p } \ } = 0,$$\end{document } a result consistent with equation ( 36 ) .
it may be shown that the version of hamilton s equations of motion coming from these brackets reproduces the time dependence ( 42 ) of the moduli ; see [ 87 , 135 ] for details .
the poisson brackets among the traces of the holonomies ( 38 ) are also easy to compute . if we let 47\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\begin{array}{*{20}c}{r_1^ \pm = { 1 \over 2}{\rm{tr}}\;{\rho ^ \pm}[{\gamma _ 1 } ] = \cosh { { r_1^ \pm } \over 2 } , } \\{r_2^ \pm = { 1 \over 2}{\rm{tr}}\;{\rho ^ \pm}[{\gamma _ 2 } ] = \cosh { { r_2^ \pm } \over 2 } , } \\{r_{12}^ \pm = { 1 \over 2}{\rm{tr}}\;{\rho ^ \pm}[{\gamma _ 1 } \cdot { \gamma _ 2 } ] = \cosh { { ( r_1^ \pm + r_2^
\pm ) } \over 2 } , } \\\end{array}$$\end{document } it is not hard to check that 48\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\{r_1^ \pm , r_2^ \pm \ } = \mp { 1 \over { 4\ell}}(r_{12}^ \pm - r_1^ \pm r_2^ \pm)\;\;\;{\rm{and}}\;{\rm{cyclical}}\;{\rm{permutations,}}$$\end{document } reproducing the poisson algebra of nelson , regge , and zertuche .
this group is generated by two dehn twists , which act on 1(t ) by 49\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\begin{array}{*{20}c}{s:{\gamma _ 1 } \rightarrow \gamma _ 2^{- 1 } , } & { { \gamma _ 2 } \rightarrow { \gamma _ 1 } , } \\
{ t:{\gamma _ 1 } \rightarrow { \gamma _ 1 } \cdot { \gamma _ 2 } , } & { { \gamma _ 2 } \rightarrow { \gamma _ 2}. } \\\end{array}$$\end{document } these transformations act on the parameters \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$r_a^ \pm$\end{document } and the adm moduli and momenta as 50\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\begin{array}{*{20}c}{s : r_1^ \pm \rightarrow r_2^ \pm } & { r_2^ \pm \rightarrow - r_1^ \pm } & { \tau \rightarrow - { 1 \over \tau } } & { p \rightarrow { { \bar \tau}^2}p } \\ { t : r_1^ \pm \rightarrow r_1^ \pm + r_2^ \pm } & { r_2^ \pm \rightarrow r_2^ \pm } & { \tau \rightarrow \tau + 1 } & { p \rightarrow p. } \\ \end{array}$$\end{document } these transformations are consistent with the relationships between the adm and holonomy variables , and that they preserve all poisson brackets . for a torus universe with zero or positive cosmological constant ,
i refer the reader to   for details . for the torus universe of the preceding section 2.7
the area of a slice of constant t is essentially the hamiltonian ( 44 ) ; it increases from 0 at t = 0 to a maximum at t = and then shrinks to zero at t = /2 at the  big bang  and  big crunch  the modulus ( 42 ) is purely real , 2 = 0 .
this means that even apart from the  crunch  in volume , the geometry is singular : a real value of  represents a torus that has collapsed to a line . for   0 , the final big crunch disappears , and the torus universe expands forever from an initial big bang .
it is not hard to check that as time increases , the modulus ( 42 ) moves along a semicircle in the upper half of the complex plane , with a center on the real axis .
such a curve is a geodesic in the natural weil - petersson ( or poincar ) metric on the torus moduli space [ 159 , 135 ] . because of the invariance under the mapping class group ( 50 )
, however , the true physical motion in the moduli space of the torus  the space of physical configurations with the large diffeomorphisms modded out  is much more complicated ; there are arbitrarily long geodesics , and the flow is , in fact , ergodic . for spacetimes    with  being a surface of genus g > 1 , no explicit metrics analogous to equation ( 40 ) are known , except for the special case of solutions with constant moduli .
the problem is in part that no simple form such as equation ( 41 ) for the  standard  constant curvature metrics exists , and in part that the adm hamiltonian becomes a complicated , nonlocal function of the moduli . for the case of an asymptotically flat genus g space ,
some interesting progress has been made by krasnov ; i do not know whether these methods can be extended to the spatially closed case .
one can write down the holonomies of the geometric structure for a higher genus surface , of course  though even there , it is nontrivial to ensure that they represent spacetimes with spacelike genus g slices  but to a physicist , these holonomies in themselves give fairly little insight into the dynamics . in principle , the adm and chern - simons approaches might be viewed as complimentary : as moncrief has pointed out , one could evaluate the holonomies in terms of adm variables in a nice time - slicing , set these equal to constants , and thereby solve the adm equations of motion . in practice , though , this approach seems intractable except for the genus one case . for 
= 0 , it may be possible to extract a useful physical picture from the geometrical results of , which relate holonomies to the structure of the initial singularity and the asymptotic future geometry , but the implications have not yet been explored in any depth .
the singular behavior of the torus universe carries over to higher genus : spacetimes with  < 0 expand from a big bang and recollapse in a big crunch , while those with   0 expand forever [ 200 , 20 ] .
moreover , the degeneration of the spatial geometry at the initial singularity carries over to the higher genus case [ 200 , 55 ] . by introducing a global  cosmological time  and exploiting recent results in two- and three - dimensional geometry , benedetti and guadagnini
have shown that when  = 0 , a set of parameters describing the initial singularity and a second set describing the geometry in the asymptotic future together completely determine the spacetime .
it seems likely that these two sets are canonically conjugate , and a better understanding of the symplectic structure could be useful for quantum gravity .
the reader may well have decided that for an author reviewing quantum gravity , i have spent an inordinate amount of time on the classical theory .
there is a good reason for this , though : each of the approaches described in the preceding sections leads very naturally to an approach to quantization , which is now  with a few twists  fairly straightforward .
indeed , the main reason that 2 + 1 dimensions offer such an attractive setting for quantum gravity is that the classical solutions can be completely described by a finite set of parameters .
such a description effectively reduces quantum gravity to quantum mechanics , allowing us to evade the complications of quantum field theory .
this is not to imply that all approaches to quantum gravity simplify  the wheeler - dewitt equation , for example , apparently does not  but it allows us to explore at least a few approaches in depth .
perhaps the simplest approach to quantum gravity in 2 + 1 dimensions [ 66 , 160 ] begins with the reduced phase space action ( 34 ) , which describes a finite - dimensional system of physical degrees of freedom , albeit one with a complicated , time - dependent hamiltonian . we know , at least in principle , how to quantize such a system :
we simply replace the poisson brackets ( 36 ) with commutators , 51\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$[{\hat{m}_\alpha},{\hat{p}^\beta } ] = i\hbar \delta _ \alpha ^\beta,$$\end{document } represent the momenta as derivatives , p = i/m , and choose our wave functions to be square integrable functions (m ,
t ) that evolve according to the schrdinger equation 52\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$i\hbar { { \partial \psi ( { m_\alpha},t ) } \over { \partial t } } = \hat h \psi ( { m_\alpha},t),$$\end{document } where the hamiltonian  is obtained from equation ( 35 ) in a suitable operator ordering .
invariance under the mapping class group of section 2.6 can be incorporated by demanding that (m , t ) transform under a representation of \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\mathcal d}(m)$\end{document}. a similar requirement may help determine the operator ordering in the hamiltonian operator [ 68 , 70 ] , although some ambiguities will remain . for spatial surfaces of genus
g > 1 , the complexity of the constraint ( 32 ) seems to make this approach to quantization impractical . a perturbative expression for
 may still exist , though , as discussed in [ 217 , 218 ] , and the gauss map has been proposed as a useful tool . for genus one ,
the classical hamiltonian ( 44 ) becomes , up to operator ordering ambiguities , 53\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\hat{h } = { \left({{t^2 } + { 4 \over { { \ell ^2 } } } } \right)^{- 1/2}}\delta _ 0^{1/2},\quad { \delta _ 0 } = - { \tau _ 2}^2\left({{{{\partial ^2 } } \over { \partial { \tau _ 1}^2 } } + { { { \partial ^2 } } \over { \partial { \tau _ 2}^2 } } } \right),$$\end{document } where 0 is the ordinary scalar laplacian for the constant negative curvature poincar metric on moduli space , and one chooses the positive square root in order to have a hamiltonian that is bounded below . this laplacian is invariant under the modular transformations ( 50 ) , and its invariant eigenfunctions , the weight zero maass forms , have been studied extensively by mathematicians ( see , for example , ) .
the behavior of the corresponding wave functions has been explored by puzio , who argues that they are well - behaved and nonsingular at the boundaries of moduli space .
such behavior is relevant to the question of how quantum gravity handles singularities : the degeneration of the torus geometry at the big bang , described in section 2.8 , corresponds to an approach to the boundary of moduli space , and puzio s results suggest that the classical singularity may be better - behaved in the quantum theory .
a related form of quantization comes from reexpressing the moduli space for the torus as a quotient space sl(2 , )sl(2 , )/so(2 ) [ 193 , 273 ] . here
, the symmetric space sl(2 , )/so(2 ) describes the transverse traceless deformations of the spatial metric , while sl(2 ,  ) is the modular group .
as waldron has observed , this makes it possible to reinterpret the quantum mechanical problem as that of a fictitious free particle , with mass proportional \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\sqrt \lambda$\end{document } , moving in a quotient space of the ( flat ) three - dimensional milne universe . with a suitable choice of coordinates ,
though , the problem again reduces to that of understanding the hamiltonian ( 53 ) and the corresponding maass forms .
while the choice ( 53 ) of operator ordering is not unique , the number of alternatives is smaller than one might expect .
the key restriction is diffeomorphism invariance : the eigenfunctions of the hamiltonian should transform under a one - dimensional unitary representation of the mapping class group ( 50 ) .
the representation theory of this group is well - understood [ 120 , 184 ] ; one finds that the possible hamiltonians are all of the form ( 53 ) , but with 0 replaced by 54\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${\delta _ n } = - { \tau _ 2}^2\left({{{{\partial ^2 } } \over { \partial { \tau _ 1}^2 } } + { { { \partial ^2 } } \over { \partial { \tau _ 2}^2 } } } \right ) + 2 in{\tau _
2}{\partial \over { \partial { \tau _ 1 } } } + n(n + 1),\quad 2 n\;\in \;{\mathbb z},$$\end{document } the maass laplacian acting on automorphic forms of weight n. it has been suggested in   that the choice n = 1/2 is most natural from the point of view of chern - simons quantization .
note that when written in terms of the momentum p , the operators n differ from each other by terms of order  , as one would expect for operator ordering ambiguities
. nevertheless , the choice of ordering may have dramatic effects on the physics , since the spectra of the various maass laplacians are quite different .
this ordering ambiguity may be viewed as arising from the structure of the classical phase space .
the torus moduli space is not a manifold , but rather has orbifold singularities , and quantization on an orbifold is generally not unique . since the space of solutions of the einstein equations in 3 + 1 dimensions has a similar orbifold structure , we might expect a similar ambiguity in realistic ( 3 + 1)-dimensional quantum gravity .
it appears to be self - consistent , and like ordinary quantum mechanics , it is guaranteed to have the correct classical limit on the reduced phase space of section 2.4 .
the principal drawback is that the method relies on a classical choice of time coordinate , which occurs as part of the gauge - fixing needed to solve the constraints .
in particular , the analysis of section 2.4 required that we choose the york time - slicing from the start ; a different choice might lead to a different quantum theory , as it is known to do in quantum field theory . in other words , it is not clear that this approach to quantum gravity preserves general covariance .
the problem may be rephrased as a statement about the kinds of questions we can ask in this quantum theory .
the model naturally allows us to compute the transition amplitude between the spatial geometry of a time slice of constant mean curvature  tr k = t1 and the geometry of a later slice of constant mean curvature  tr k = t2 .
indeed , such amplitudes are given explicitly in , where it is shown that they are peaked around the classical trajectory .
but it is far less clear how to ask for transition amplitudes between other spatial slices , on which tr k is not constant .
such questions would seem to require a different classical time - slicing , and thus a different  and perhaps inequivalent  quantum theory .
we will eventually find a possible way out of this difficulty in section 3.4 . as a first step
, we next turn to an alternative approach to quantization , one that starts from the first order formalism . as we saw in section 2.3 , ( 2 + 1)-dimensional general relativity in first order form
the quantization of chern - simons theory is well understood [ 278 , 243 , 244 , 116 , 34 ] . for noncompact groups such as those that appear in gravity , much less
is known , though there has been some promising work [ 280 , 155 , 43 , 150 ] .
nevertheless , interesting progress can be made , especially in the simple case of a manifold with topology   t. in contrast to the reduced phase space quantization of the preceding section 3.1 , our understanding of the quantum chern - simons gravity depends strongly on the sign of the cosmological constant . for  < 0 , the relevant gauge group is so(2 , 2 ) or its cover sl(2 ,  )  sl(2 ,  ) .
this is the most poorly understood case ; an explicit quantization of the algebra holonomies exists for genus one ( see below ) and genus two , but more general results do not yet exist . for 
> 0 , the relevant gauge group is so(3 , 1 ) or its cover sl(2 ,  ) , a complex gauge group whose chern - simons theory is somewhat better understood [ 280 , 155 , 43 ] . as noted in section 2.3 , the poisson brackets for this theory are related to the quantum double of the lorentz group , and buffenoir et al . have used this structure to write down an explicit quantization .
as far as i know , the relationship between this work , which is based on a hamiltonian formalism and combinatorial quantization , and that of witten and hayashi [ 280 , 155 ] , which is based on geometric quantization , has not yet been explored . for 
= 0 , the relevant gauge group is iso(2 , 1 ) , the ( 2 + 1)-dimensional poincar group , or its universal cover .
here there is again a connection to the quantum double of the lorentz group , which has been used in [ 37 , 36 , 201 ] to explore the quantum theory , although largely in the context of point particles . in this case , one has the nice feature that the phase space has a natural cotangent bundle structure , allowing us to immediately identify the holonomies of the spin connection  as generalized positions , and their derivatives as generalized momenta .
this provides a direct link to the loop variables of ashtekar , rovelli , and smolin [ 26 , 29 ] , 55\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${t^0}[\gamma ] = { 1 \over 2}{\rm{tr}}\;{\rho _ 0}[\gamma , x],\quad \;{t^1}[\gamma ] = \int\nolimits_\gamma { { \rm{tr } } } \;\{{\rho _ 0}[\gamma , x(s)]{e^a}(\gamma ( s)){{\mathcal j}_a}\},$$\end{document } where 56\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${\rho _ 0}[\gamma , x ] = p\;\exp \left({\int\nolimits_\gamma { { \omega^a } } { { \mathcal j}_a } } \right)$$\end{document } is the sl(2 ,  ) holonomy of the spin connection and t [  ] can be expressed as a derivative of t[ ] along a path in the space of flat connections .
note that the generator \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\mathcal j}$\end{document } may , in principle , be in any representation of sl(2 ,  ) , and that the trace in equation ( 55 ) may depend on the choice of representation .
i will return to the resulting quantum theory , loop quantization , in section 3.5 . as in reduced phase space quantization , matters simplify considerably for the torus universe   t. let us again focus on the case  < 0 . a complete  in fact , overcomplete  set of observables is given by the traces ( 47 ) of the holonomies , and our goal is to quantize the algebra ( 48 ) .
to do so , we proceed as follows : 
 we replace the classical poisson brackets { , } by commutators [ , ] , with the rule [ x , y ] = xy  yx = i{x , y}.on the right hand side of equation ( 48 ) , we replace the product by the symmetrized product , \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$xy \to { 1 \over 2}(xy + yx)$\end{document}. we replace the classical poisson brackets { , } by commutators [ , ] , with the rule [ x , y ] = xy  yx = i{x , y}. on the right hand side of equation ( 48 ) , we replace the product by the symmetrized product , \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$xy \to { 1 \over 2}(xy + yx)$\end{document}. the resulting algebra is defined by the relations 57\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\hat{r}_1^ \pm \hat{r}_2^ \pm { e^{\pm i\theta } } - \hat{r}_2^ \pm \hat{r}_1^ \pm { e^{\mp i\theta } } = \pm 2i\;\sin \;\theta \;\hat{r}_{12}^ \pm \quad { \rm{and}}\;{\rm{cyclical}}\;{\rm{permutations}}$$\end{document } with tan  = /(8 ) . the algebra ( 57 ) is not a lie algebra , but it is related to the lie algebra of the quantum group uq(sl(2 ) ) with q = exp 4i .
classically , the observables \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$r_1^ \pm , r_2^ \pm$\end{document } , and \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$r_{12}^ \pm$\end{document } are not independent ; in the quantum theory , the corresponding statement is that the quantities 58\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${\hat{f}^ \pm } = 1 - { \tan ^2}\theta - { e^{\pm 2i\theta}}\left({{{(\hat{r}_1^ \pm)}^2 } + { { ( \hat{r}_{12}^ \pm)}^2 } } \right ) - { e^{\mp 2i\theta}}{(\hat{r}_2^ \pm)^2 } + 2{e^{\pm i\theta}}\cos \theta \hat{r}_1^ \pm \hat{r}_2^
\pm \hat{r}_{12}^ \pm$$\end{document } commute with the holonomies , and can be consistently set to zero . in terms of the parameters \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$r_a^ \pm$\end{document } of equation ( 38 ) , the algebra can be represented by [ 87 , 86 ] 59\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\begin{array}{*{20}c}{\hat{r}_1^ \pm = \sec \;\theta \;\cosh { { \hat{r}_1^ \pm } \over 2 } , } & { \hat{r}_2^ \pm = \sec \;\theta \;\cosh { { \hat{r}_2^ \pm } \over 2 } , } & { \hat{r}_{12}^ \pm = \sec \;\theta \;\cosh { { ( \hat{r}_1^ \pm + \hat{r}_2^ \pm ) } \over 2 } , } \\
\end{array}$$\end{document } with 60\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$[\hat{r}_1^ \pm , \hat{r}_2^ \pm ] = \pm 8i\theta , \quad [ \hat{r}_a^ + , \hat{r}_b^ - ] = 0.$$\end{document } for  small , these commutators differ from the naive quantization of the classical brackets ( 45 ) by terms of order . an alternative quantization , also differing by terms of order  , works directly with the holonomy matrices ( 38 ) , imposing a quantum - group - like quantization condition   61\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${\rho ^ + } [ { \gamma _ 1}]{\rho ^ + } [ { \gamma _ 2 } ] = q{\rho ^ + } [ { \gamma _ 2}]{\rho ^ + } [ { \gamma _ 1}],\quad \quad q = { e^{- i\hbar/4\ell}},$$\end{document } with a similar condition for . we must also implement the action of the modular group ( 50 ) on the operators \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\hat r_a^ \pm$\end{document}. one can find an action preserving the algebraic relations ( 57 ) , corresponding to a particular factor ordering of the classical modular group .
the nelson - picken quantization ( 61 ) admits a similar modular group action . for a full quantum theory , of course
, one needs not only an abstract operator algebra , but a hilbert space upon which the operators act . for the   t universe , equation ( 60 )
suggests that a natural choice is to take wave functions to be square integrable functions of the \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$r_2^ \pm$\end{document}. there is a potential difficulty here , however : the modular group does not act properly discontinuously on this configuration space .
this means that the quotient of this space by the modular group is badly behaved ; in fact , there are no nonconstant modular invariant functions of the \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$r_2^ \pm$\end{document } [ 182 , 143 , 221 ] .
the technique of chern - simons quantization relies on special features of general relativity in 2 + 1 dimensions , and does not readily generalize to higher dimensions .
it is , however , closely related to a much more general approach , covariant canonical quantization [ 31 , 28 , 98 , 175 , 272 ] , or  quantization of the space of classical solutions  .
our starting point is the observation that the phase space of a well - behaved classical theory is isomorphic to the space of classical solutions . indeed ,
if \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\mathcal c}$\end{document } is an arbitrary cauchy surface , then a point in the phase space determines initial data on \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\mathcal c}$\end{document } , which can be evolved to give a unique solution , while , conversely , a classical solution restricted to \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\mathcal c}$\end{document } determines a point in the phase space .
moreover , the space of solutions has natural symplectic structure [ 175 , 272 ] , which can be shown to be equivalent to the standard symplectic structure on phase space . for the case of ( 2 + 1)-dimensional gravity ,
this equivalence is demonstrated in section 6.1 of . for ( 2 + 1)-dimensional gravity ,
the space of classical solutions is the space of geometric strictures of section 2.2 . if we restrict our attention to spacetimes with the topology    with s closed and   0 , the holonomies of a geometric structure determine a unique maximal domain of dependence , exactly the right setting for covariant canonical quantization .
but as we saw in section 2.3 , the holonomies of a geometric structure are precisely the holonomies of the chern - simons formalism , and the symplectic structures are the same as well . thus in this
setting , chern - simons quantization is covariant canonical quantization . if  > 0 or point particles are present , the holonomies do not quite determine a unique geometric structure , and the chern - simons theory is not quite equivalent to general relativity .
in that case , additional discrete variables might be necessary ; see , for example ,   for the case of a torus universe with  > 0 . as we shall see in section 3.4 , the construction of dynamical observables and time - dependent states in covariant canonical quantum theory requires an explicit isomorphism between the phase space and the space of classical solutions . for the torus universe , such an isomorphism is known . for higher genus spaces , however  and certainly for realistic ( 3 + 1)-dimensional gravity  it is not .
often , however , we can determine such an isomorphism perturbatively in the neighborhood of a known classical solution .
this raises the interesting question , so far answered only in simple models , of whether classical perturbation theory can be used to define a perturbative covariant canonical quantum theory .
when one attempts to interpret the quantum theories coming from the chern - simons formalism or covariant canonical quantization , one finds an immediate and rather profound difficulty .
the gauge - invariant observables  the traces of the holonomies  are automatically nonlocal and time - independent , and one obtains a 
frozen time formalism ,  or what kucha has called  quantum gravity without time  . in one sense
, this is a good thing : one knows from general arguments that the diffeomorphism - invariant observables in any quantum theory of gravity must have these features . on the other hand
, it is not at all easy to see how to extract local geometry and dynamics from such a picture : if our only observables are nonlocal and time - independent , how can we recover a classical limit with local excitations that evolve in time ?
a geometric structure determines a spacetime , and must contain within it all of the dynamics of that spacetime . on the other hand , the basic data that fix the geometric structure  the transition functions , or , often , the holonomies  have no obvious dynamics . in principle , the classical answer is simple : 
 use , say , the holonomies to determine a spacetime geometry.select a favorite time-slicing.read off the spatial metric and its time derivatives from the spacetime metric of step 1 in this slicing . 
 this procedure can be understood as a concrete realization of the isomorphism described in section 3.3 between the phase space and the space of classical solutions , with the cauchy surface \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\mathcal c}$\end{document } fixed by the choice of time - slicing .
read off the spatial metric and its time derivatives from the spacetime metric of step 1 in this slicing .
for the simple case of the torus universe , these steps can be transcribed almost directly to the quantum theories .
equations ( 42 , 43 , 44 ) become definitions of operators , 62\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\begin{array}{*{20}c}{{{\hat\tau}_t } = \left({\hat{r}_1^ - { e^{it/\ell } } + \hat{r}_1^ + { e^{- it/\ell } } } \right){{\left({\hat{r}_2^ - { e^{it/\ell } } + \hat{r}_2^ + { e^{- it/\ell } } } \right)}^{- 1 } } , } \\ { { { \hat{p}}_t } = - { { i\ell } \over { 2\sin { { 2 t } \over \ell}}}{{\left({\hat{r}_2^ + { e^{it/\ell } } + \hat{r}_2^ - { e^{- it/\ell } } } \right)}^2 } , } \\ { { { \hat{h}}_t } = { { { \ell ^2 } } \over 4}\sin { { 2 t } \over \ell}(\hat{r}_1^ - \hat{r}_2^ + - \hat{r}_1^ + \hat{r}_2^ - ) , } \\ \end{array}$$\end{document } where the operator ordering has been chosen to respect the modular transformations ( 50 ) .
the parameter t is now merely a label for a one - parameter family of diffeomorphism - invariant observables .
these observables obtain their physical significance from the classical limit : \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${{\hat \tau}_t}$\end{document } , for example , is the operator whose expectation value gives the mean value of the modulus on a time slice of constant mean curvature \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$t = - { 2 \over \ell}\cot { { 2 t } \over \ell}$\end{document}. such observables are examples of what rovelli has called  evolving constants of motion  [ 236 , 237 ] . from this point of view , we should think of chern - simons / covariant canonical quantization as a sort of heisenberg picture , with time - independent states and  time-dependent operators . to obtain the corresponding schrodinger picture
, we proceed as in ordinary quantum mechanics : we diagonalize \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${{\hat \tau}_t}$\end{document } , obtaining a transition matrix \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$k(\tau , \bar \tau ; t\vert r_2^ + , r_2^ - ) = \left\langle { \tau , \bar \tau ; t\vert r_2^ + , r_2^ - } \right\rangle$\end{document } that allows us to transform between representations [ 68 , 88 ] . the resulting 
time-dependent wave functions obey a schrdinger equation of the form ( 52 , 53 ) , but with the laplacian in  replaced by the weight 1/2 maass laplacian 1/2 of equation ( 54 ) .
in , it has been shown that these wave functions are peaked around the correct classical trajectories .
( different operator orderings in equation ( 62 ) give different weight laplacians . ) as a useful byproduct , this analysis allows us to solve the problem of the poorly - behaved action of the modular group discussed at the end of section 3.2 [ 88 , 89 ] .
if we start with a reduced phase space wave function \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\tilde \psi ( \tau , \bar \tau ; t)$\end{document } and use the transition matrix k to determine a chern - simons wave function \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\psi ( r_2^ + , r_2^ -)$\end{document } , we find , indeed , that \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\psi ( r_2^ + , r_2^ -)$\end{document } is not modular invariant .
instead , though , the entire hilbert space of chern - simons wave functions splits into  fundamental regions ,  orthogonal subspaces that transform into each other under the action of the modular group .
any one of these fundamental regions is equivalent to any other , and each is equivalent to the hilbert space arising from reduced phase space quantization .
moreover , matrix elements of any modular invariant function vanish unless they are taken between states in the same fundamental region .
modular invariance thus takes a slightly unexpected form , but can still be imposed by restricting the theory to a single fundamental region of the hilbert space .
we can also begin to address the problem raised at the end of section 3.1 , the limited and slicing - dependent range of questions one can ask in reduced phase space quantization .
the operators ( 62 ) introduced here on the covariant canonical hilbert space were obtained from a particular classical time - slicing , and answer questions about spatial geometry in that slicing . in principle , however , we can choose any other slicing , with a new time coordinate \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\bar t}$\end{document } , and determine the corresponding operators \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${{\hat \tau}_t},{{\hat p}_{\bar t}}$\end{document } , and \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${{\hat h}_{\bar t}}$\end{document}. the operator ordering of such operators will , of course , be ambiguous , though one might hope that the action of the modular group might again restrict the choices .
but such an ambiguity need not be seen as a problem with the theory ; rather , it is merely a statement that many different quantum operators can have the same classical limit , and that ultimately experiment must decide which operator we are really observing .
there is , to be sure , a danger that the  schrdinger pictures  coming from different time - slicings may not be consistent .
suppose , for example , that we choose two slicings that agree on an initial and a final slice 1 and 2 , but disagree in between .
if we start with an initial wave function on 1 , we must check that the hamiltonians coming from the different slicings evolve us to the same final wave function on 2 . for field theories , even in flat spacetime , this will not always happen . for ( 2 + 1)-dimensional gravity ,
on the other hand , there is evidence that one can always find operator orderings of the hamiltonians that ensure consistent evolution .
if this ultimately turns out not to be the case , however , it may simply mean that we should treat the covariant canonical picture as fundamental , and discard the schrodinger pictures of time - dependent states .
we now resume the discussion of alternative approaches to quantum gravity in 2 + 1 dimensions . in 3 + 1 dimensions ,
one of the most attractive programs of quantization is  loop quantum gravity ,  or  quantum geometry 
= 0 , this approach is closely related to the first order formalism of section 2.3 , but takes as its fundamental observables the loop variables t[ ] and t[ ] of equation ( 55 ) .
more precisely , loop quantum gravity starts with a hamiltonian formulation of the first order formalism , with constraints , written in analogy to the ( 3 + 1)-dimensional case , that take the form 63\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${d_i}{\tilde{e}^{ia } } = 0,\quad { \tilde{e}^i}_ar_{ij}^a = 0,\quad { \epsilon _ { abc}}{\tilde{e}^{ib}}{\tilde{e}^{jc}}r_{ij}^a = 0.$$\end{document } here , the indices i , j , k are spatial indices on a surface of constant time , \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${{\tilde e}^{ia } } = { \in ^{ij}}{e_j}^a$\end{document } , di is the so(2 , 1 ) gauge - covariant derivative for the connection  , and the \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$r_{ij}^a$\end{document } are the spatial components of the curvature two - form ( 16 ) . when the spatial metric \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${g_{ij } } = { e_i}^a{e_{ja}}$\end{document } is nondegenerate , it is straightforward to show that these constraints are equivalent to the standard constraints of first order gravity , and the classical theories are identical . when gij , is noninvertible , on the other hand , the constraints are not equivalent . instead , the constraints ( 63 ) yield a phase space with infinitely many degrees of freedom , arising from the geometries formed from an arbitrary collection of independent patches of ordinary spacetime separated by regions with degenerate metrics [ 44 , 45 ] .
let us restrict ourselves to invertible spatial metrics , and attempt to quantize the algebra of loop variables \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${{\hat t}^0}[{\rm{\gamma ] } } $ \end{document } and \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${{\hat t}^1}[{\rm{\gamma ] } } $ \end{document}. for the torus universe , it is not hard to show that such a quantization simply reproduces the theory we already obtained in the chern - simons formulation ( see , for example , section 7.2 of ) .
there is another way to look at the operator algebra of the operators \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${{\hat t}^0}[{\rm{\gamma ] } } $ \end{document } and \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${{\hat t}^1}[{\rm{\gamma ] } } $ \end{document } , however , which leads to a new approach , the loop representation . up to now , we have been thinking of the operators \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\hat t}$\end{document } as a set of functions of the triad and spin connection , indexed by loops . our wave functions are thus functionals of the  configuration space  variable  , or , more precisely , functions on the moduli space of flat so(2 , 1 ) or sl(2 ,  ) connections on . but we could equally well view the \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\hat t}$\end{document } operators as functions of loops  or , in 2 + 1 dimensions , homotopy classes [  ] of loops  indexed by e and . wave functions would then be functions of loops or sets of loops .
this change of viewpoint is rather like the decision in ordinary quantum mechanics to view a wave function e as a function on momentum space , indexed by q , rather than a function on position space , indexed by p. the loop representation is complicated by the existence of mandelstam identities   among holonomies of loops , but for the case of the torus universe , a complete , explicit description of the states is again possible [ 26 , 29 ] . the simplest construction begins with a vacuum state |0 annihilated by every operator \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${{\hat t}^1}[{\rm{\gamma ] } } $ \end{document } , and treats the \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${{\hat t}^0}[{\rm{\gamma ] } } $ \end{document } as  creation operators .  since any homotopy class [  ] of loops on the torus is completely characterized by a pair of winding numbers ( m , n ) , one can write these states as |m , n. the action 64\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\begin{array}{*{20}c}{{{\widehat{t}}^0}[m , n]\vert p , q\rangle = \quad { 1 \over 2}(\vert m + p , n + q\rangle + \vert m - p , n - q\rangle ) } \\ { { { \widehat{t}}^1}[m , n]\vert p , q\rangle = \quad - { i \over 8}(mq - np)(\vert m + p , n + q\rangle - \vert m - p , n - q\rangle ) } \\ \end{array}$$\end{document } then gives a representation of the loop algebra .
observe now that the loop variables t[ ] depend only on the  configuration space 
variable . we can thus relate the loop representation to the chern - simons representation by simultaneously diagonalizing these operators , obtaining wave functions that are functions of the so(2 , 1 ) holonomies alone . for the torus universe , this 
loop transform  can be obtained explicitly [ 26 , 29 , 192 ] , and written as a simple integral transform .
the properties of this transform depend on the holonomies , that is , the eigenvalues of \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${{\hat t}^0}[{\rm{\gamma ] } } $ \end{document}. for simplicity , let us take the generator \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\mathcal j}$\end{document } in equation ( 56 ) to be in the two - dimensional representation of sl(2 ,  ) . in the  timelike sector , 
in which the traces of the two holonomies are both less than two , the loop transform is a simple fourier transformation , and chern - simons and loop quantization are equivalent .
unfortunately , though , this is not the physically relevant sector : it does not correspond to a geometric structure with spacelike t slices . for a physically interesting geometry
, one must go to the  spacelike sector ,  in which the traces of the holonomies are both greater than two . in this sector , the transform is not very well - behaved : in fact , a dense set of chern - simons states transforms to zero .
the loop representation thus appears to be rather drastically different from the chern - simons formulation .
the problems in the physical sector can be traced back to the fact that sl(2 ,  ) is a noncompact group .
one is to start with a different dense set of chern - simons states that transform faithfully , determine the inner product and the action of the \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\hat t}$\end{document } operators on the resulting loop states , and then form the cauchy completion to define the hilbert space in the loop representation .
this is a consistent procedure , but many of the resulting states in the cauchy completion are no longer functions of loops in any clear sense ; they correspond instead to  extended loops  , whose geometrical interpretation is not entirely clear .
a second possibility is to change the integration measure in the loop transform to make various integrals converge better .
such a choice introduces order  corrections to the action of the \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${{\hat t}^1}$\end{document } operators , and one must be careful that the algebra remains consistent .
this is possible , but at some cost  the inner products between loop states become considerably more complex , as does the action of the mapping class group  and it is not obvious that there is a canonical choice of the new measure and algebra . a third possibility is suggested by recent work on spin networks for noncompact groups [ 129 , 130 ] .
this new technology essentially allows one to consider holonomies ( 56 ) that lie in infinite - dimensional unitary representations of the lorentz group , with a finite inner product defined by appropriate gauge - fixing .
the quantities t and t can be represented as hermitian operators on this space of holonomies ( or on a larger space of spin networks ) . at this writing ,
implications of this approach for the loop transform in 2 + 1 dimensions have not yet been investigated .
finally , i should briefly mention the role of spin networks in ( 2 + 1)-dimensional quantum geometry . in the ( 3 + 1)-dimensional theory ,
loop states have been largely superseded by spin network states , states characterized by a graph  with edges labeled by representations and vertices labeled by intertwiners .
such states can be defined in 2 + 1 dimensions as well , and there has been some interesting recent work on their role as 
, however , the full constraints imply that such states have their support on flat connections , and only holonomies around noncontractible curves describe nontrivial physics .
an interesting step toward projecting out the physical states has recently been taken in , in the context of euclidean quantum gravity ; the ultimate effect is to reduce spin network states to loop states of the sort we have considered above
. a better understanding of the relationship to the gauge - fixing procedure of [ 129 , 131 ] would be valuable .
a long - standing approach to quantum gravity in 3 + 1 dimensions has been to look for discrete approximations to the path integral [ 179 , 232 ] : quantized regge calculus , for example , or sums over random triangulations .
the basic idea is that although the full  sum over geometries  may be impossible to evaluate , a sum over appropriately discretized geometries might give a good approximation , perhaps becoming extremely good near a phase transition at which lattice spacing can go to zero . when applied to 2 + 1 dimensions , such methods have the added feature of sometimes being exact : since geometries satisfying the constraints have constant or zero curvature , a discrete  approximation  may give a complete description .
regge calculus in 2 + 1 dimensions begins with a triangulated three - manifold , consisting of a collection of flat simplices joined along one - dimensional edges .
the curvature of such a manifold is concentrated entirely at the edges . for a simplicial three - manifold with riemannian signature , composed of simplices with edges of length le ,
regge s form of the einstein - hilbert action is 65\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${i_{{\rm{regge } } } } = 2\sum\limits_{{\rm{edges}}:\;e } { { \delta _ e}{\ell _ e}},$$\end{document } where e is the conical deficit angle at the edge labeled by the index e. a similar expression exists for lorentzian signature , although the definition of the deficit angle is a bit more complicated .
the first hint that ( 2 + 1)-dimensional gravity might be exceptional came from the observation by ponzano and regge   that the regge action in 2 + 1 dimensions can be re - expressed in terms of wigner - racah 6j - symbols .
consider first a single tetrahedron with edge lengths \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\ell _ i } = { 1 \over 2}({j_i } + { 1 \over 2})$\end{document } , where the ji are integers or half - integers . ponzano and regge noticed , and roberts later proved rigorously , that in the limit of large j , 66\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\exp \left({\pi i\sum\limits_{i = 1}^6 { ji } } \right)\left\{{\begin{array}{*{20}c}{{j_1}\;{j_2}\;{j_3 } } \\ { { j_4}\;{j_5}\;{j_6 } } \\
\end{array } } \right\ } \sim { 1 \over { \sqrt { 6\pi v}}}\left\{{\exp \left [ { i\left({{i_{{\rm{regge } } } } + { \pi \over 4 } } \right ) } \right ] + \exp \left [ { - i\left({{i_{{\rm{regge } } } } + { \pi \over 4 } } \right ) } \right ] } \right\},$$\end{document } where \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\left\ { { \begin{array}{*{20}{c } } 
   { { j_1}\;{j_2}\;{j_3 } } \\ 
   { { j_4}\;{j_5}\;{j_6 } } 
 \end{array } } \right\}$\end{document } is a 6j - symbol , iregge is the regge action ( 65 ) for the tetrahedron , and v is its volume . for a manifold made of a collection of such tetrahedra
, the full regge action will occur in a product of such 6j - symbols .
this suggests that the ( 2 + 1)-dimensional path integral , which is essentially a sum over geometries of terms of the form exp(iiregge ) , might be expressible as a sum of such products .
ponzano and regge s specific proposal , developed by hasslacher and perry   and modified by ooguri   to account for boundaries , was the following : consider a three - manifold m with boundary m , with a given triangulation  of m .
label interior edges of tetrahedra by integers or half - integers and exterior ( boundary ) edges by ji , and for a given tetrahedron t , let ji(t ) denote the spins that color its ( interior and exterior ) edges . then 67\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\begin{array}{*{20}{c } } 
   { { z_\delta } [ \ { { j_i}\ } ] = \mathop { \lim } \limits_{l \to \infty } \sum\limits_{{x_e}l } { \left ( { \prod\limits_{{\text{ext}}.{\text{edges}}:i } { { { ( - 1)}^{2{j_i}}}\sqrt { 2{j_i } + 1 } } } \right.\prod\limits_{{\text{int}}.v{\text{ertices } } } { \lambda { { ( l)}^ { - 1 } } } } } \\ 
   { \left . { \quad \quad \quad \quad   \times \prod\limits_{{\text{int}}.{\text{edges}}:\ell } { ( 2{x_\ell } + 1)\prod\limits_{{\text{tetra}}:t } { ( - 1)\sigma _ { i = 1}^6{j_i}(t)\left\ { { \begin{array}{*{20}{c } } 
   { { j_1}(t)\;{j_2}(t)\;{j_3}(t ) } \\ 
   { { j_4}(t)\;{j_5}(t)\;{j_6}(t ) } 
 \end{array } } \right\ } } } } \right ) , } 
 \end{array}$$\end{document } where  int  and  ext  mean  interior  and  exterior  and 68\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\lambda ( l ) = { \sum\limits_{j \leq l } { ( 2j + 1 ) } ^2}$$\end{document } is a regularization factor that controls divergences in the sum over interior lengths . with this weighting ,
identities among 6j - symbols may be used to show that the amplitude is invariant under refinement  that is , subdivision of a tetrahedron into four smaller tetrahedra  suggesting that we are dealing with a 
this is , of course , what one would hope for , based on the classical characteristics of ( 2 + 1)-dimensional gravity .
topological  feature of the ponzano - regge model was made more precise by turaev and viro , who discovered an improved regularization , based on the technology of quantum groups .
the  spins  j in equation ( 67 ) can be viewed as labeling representations of su(2 ) . if these are replaced by representations of the quantum group uq(sl(2 ) )
(  quantum su(2 )  ) , with \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$q = \exp \left({{{2\pi i } \over { k + 2 } } } \right)$\end{document } , k   , the number of such representations is finite , and the sum over interior edge lengths is automatically cut off . with appropriate substitutions ( e.g. ,  quantum 
6j - symbols ) , the ponzano - regge amplitude ( 67 ) becomes well - defined without any regularization .
the construction of physical states as appropriate functions of boundary edge lengths is described in section 11.2 of .
the resulting amplitudes can be computed for simple topologies [ 162 , 161 ] , and have several key features : 
 for large but finite k , the turaev - viro quantum group regularization introduces a cosmological constant to the regge action [ 204 , 203 ] , 69\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\lambda { \left({{{4\pi } \over k } } \right)^2}.$$\end{document } correspondingly , the quantum 6j - symbols are related to spherical tetrahedra rather than flat tetrahedra .in the large k limit , the turaev - viro hilbert space is isomorphic to the space of gauge - invariant functions of flat su(2 ) connections [ 219 , 220 , 242 ] .
this establishes a direct link to chern - simons quantization : just as ( 2 + 1)-dimensional lorentzian gravity can be written as an iso(2 , 1 ) chern - simons theory with a configuration space of flat so(2 , 1 ) connections , three - dimensional euclidean gravity can be written as an isu(2 ) chern - simons theory with a configuration space of flat su(2 ) connections.for a closed three - manifold m , the turaev - viro amplitude  now interpreted as a partition function  is equal to the absolute square of the partition function of an su(2 ) chern - simons theory with coupling constant k [ 228 , 260 , 233 ] , 70\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${z_{{\rm{tv } } } } = \vert { z_{{\rm{cs}}}}{\vert ^2}.$$\end{document } this again establishes an equivalence with euclidean gravity in first - order form : the first - order euclidean action with  > 0 can be written as a difference of su(2 ) chern - simons actions , so 71\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${z_{{\text{grav } } } } = \int { [ d{a^ + } ] [ d{a^ - } ] } \;{e^{i(i[{a^ + } ] - i[{a^ - } ] ) } } = { \text { } } { \left| { \int { [ d{a^ + } ] { e^{ii[{a^ + } ] } } } } \right|^2},$$\end{document } in agreement with equation ( 70).a candidate for a discrete version of the wheeler - dewitt equation in three dimensions has been found , for which the ponzano - regge wave functions are solutions . 

although it has not been universally appreciated , the existence of a divergence in the sum ( 67 )  regulated either by an explicit cut - off or by quantum group tricks  is rather mysterious , given the absence of local excitations and the general well - behavedness of gravity in three dimensions .
this mystery may have recently been solved by freidel and louapre , who show that a residual piece of the diffeomorphism symmetry has not been factored out of the ponzano - regge action .
because of this symmetry , the sum ( 67 ) overcounts physical configurations , and the regulator (l ) is simply the remaining gauge volume .
freidel and louapre further show that the symmetry can instead be gauge - fixed , leading to a sum over a restricted and considerably simplified class of  collapsed  triangulations . for large but finite k , the turaev - viro quantum group regularization introduces a cosmological constant to the regge action [ 204 , 203 ] , 69\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\lambda { \left({{{4\pi } \over k } } \right)^2}.$$\end{document } correspondingly , the quantum 6j - symbols are related to spherical tetrahedra rather than flat tetrahedra . in the large k limit ,
the turaev - viro hilbert space is isomorphic to the space of gauge - invariant functions of flat su(2 ) connections [ 219 , 220 , 242 ] .
this establishes a direct link to chern - simons quantization : just as ( 2 + 1)-dimensional lorentzian gravity can be written as an iso(2 , 1 ) chern - simons theory with a configuration space of flat so(2 , 1 ) connections , three - dimensional euclidean gravity can be written as an isu(2 ) chern - simons theory with a configuration space of flat su(2 ) connections . for a closed three - manifold m , the turaev - viro amplitude
 now interpreted as a partition function  is equal to the absolute square of the partition function of an su(2 ) chern - simons theory with coupling constant k [ 228 , 260 , 233 ] , 70\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${z_{{\rm{tv } } } } = \vert { z_{{\rm{cs}}}}{\vert ^2}.$$\end{document } this again establishes an equivalence with euclidean gravity in first - order form : the first - order euclidean action with  > 0 can be written as a difference of su(2 ) chern - simons actions , so 71\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${z_{{\text{grav } } } } = \int { [ d{a^ + } ] [ d{a^ - } ] } \;{e^{i(i[{a^ + } ] - i[{a^ - } ] ) } } = { \text { } } { \left| { \int { [ d{a^ + } ] { e^{ii[{a^ + } ] } } } } \right|^2},$$\end{document } in agreement with equation ( 70 ) .
a candidate for a discrete version of the wheeler - dewitt equation in three dimensions has been found , for which the ponzano - regge wave functions are solutions .
while the mathematics of ponzano - regge and turaev - viro models has been studied extensively , so far only a bit of attention has been given to the  traditional  issues of quantum gravity .
a few numerical investigations of the ponzano - regge path integral have been undertaken , but the evidence of a continuum limit is thus far inconclusive . the model has been used to study conditional probabilities and the emergence of quasiclassical behavior in quantum gravity , but the cut - off dependence of these results makes their physical significance unclear . in an interesting recent paper , colosi et al .
have investigated the dynamics of a single tetrahedron , showing that a quantum description of the evolution can be given in terms of a boundary amplitude .
a number of observables , whose expectation values generally give topological information about the spacetime or about knots within spacetime , have been discussed in [ 23 , 54 , 261 , 139 ] . with a few exceptions , though , work in this area has remained largely mathematical in nature ; fairly little is understood about the physics of these observables , although some are probably related to length spectra   and perhaps volumes [ 127 , 64 ] , and others are almost certainly connected to scattering amplitudes for test particles .
the ponzano - regge and turaev - viro models are examples of  spin foam  models [ 35 , 222 ] , that is , a model based on simplicial complexes with faces , edges , and vertices labeled by group representations and intertwiners .
it has been known for several years how to generalize the ponzano - regge action for a single tetrahedron [ 48 , 101 , 191 ] , and recently considerable progress has been made in constructing lorentzian spin foam models [ 222 , 125 , 102 ] .
probably the most elegant derivation of a lorentzian spin foam description starts with the first - order action ( 14 ) , with  = 0 , for a triangulated manifold [ 125 , 128 ] .
one can rewrite the action in terms of a set of discrete variables : a lie algebra element e corresponding to the integral of e along the edge  of a tetrahedron in the triangulation , and a holonomy g of the connection  around the edge . the path integral then becomes an integral over these variables . as in the continuum path integral of section 3.10 ,
this translates back to the geometric statement that the constraints require the connection w to be flat , and thus to have trivial holonomy around a contractible curve surrounding an edge . for the euclidean ponzano - regge action , g  su(2 ) , and the key trick is now to use the plancherel formula to express each (g ) as a sum over the characters of finite - dimensional representations of su(2 ) .
fairly straightforward arguments then permit an exact evaluation of the remaining integrals over the g , reproducing the 6j symbols in the ponzano - regge action . to obtain a lorentzian version
the corresponding plancherel formula involves a sum over both the ( continuous ) principle series of representations of so(2 , 1 ) and the discrete series .
similar methods may be used for supergravity . the resulting rather complicated expression for the partition function
the appearance of both continuous and discrete labels has a nice physical interpretation : continuous representations describe spacelike edges , and seem to imply a continuous length spectrum , while discrete representations label timelike edges , and suggest discrete time .
these results should probably not yet be considered conclusive , since they require operators that do not commute with all of the constraints , but they are certainly suggestive . while spin foam models ordinarily assume a fixed spacetime topology , recent work has suggested a method for summing over all topologies as well , thus allowing quantum fluctuations of spacetime topology . these results will be discussed in section 3.11 .
methods from 2 + 1 dimensions have also been generalized to higher dimensions , leading to new insights into the construction of spin foams .
spin foam models are based on a fixed triangulation of spacetime , with edge lengths serving as the basic gravitational variables .
an alternative scheme is  dynamical triangulation ,  in which edge lengths are fixed and the path integral is represented as a sum over triangulations .
( for reviews of this approach in arbitrary dimensions , see [ 8 , 179 ] . )
dynamical triangulation has been proven to be quite useful in two - dimensional gravity , and some important steps have been taken in higher dimensions , especially with the recent progress in understanding lorentzian triangulations .
the starting point is now a simplicial complex , diffeomorphic to a manifold m , composed of an arbitrary collection of equilateral tetrahedra , with sides of length a. metric information is no longer contained in the choice of edge lengths , but rather depends on the combinatorial pattern .
such a model is not exact in 2 + 1 dimensions , but one might hope that as a becomes small and the number of tetrahedra becomes large it may be possible to approximate an arbitrary geometry . in particular , it is plausible ( although not rigorously proven ) that a suitable model lies in the same universality class as genuine ( 2 + 1)-dimensional gravity , in which case the continuum limit should be exact .
the einstein - hilbert action for such a theory takes the standard regge form ( 65 ) , which for spherical spatial topology reduces to a sum 72\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$i = - { k_0}{n_0 } + { k_3}{n_3},$$\end{document } where n0 and n3 are the numbers of vertices and tetrahedra in the triangulation , k0 = a/(4 g ) , and k3 is related to the cosmological constant . as the number of tetrahedra becomes large , the number of distinct triangulations ( the  entropy  ) increases exponentially , while the n3 term in equation ( 72 ) provides an exponential suppression .
path integral \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\sum \exp ( - i)$\end{document } should thus converge for k3 greater than a critical value \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$k_3^{\rm{c}}({k_0})$\end{document}. as k3 approaches \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$k_3^{\rm{c}}({k_0})$\end{document } from above , expectation values of n3 will diverge , and one may hope for a finite - volume continuum limit as a  0 . for ordinary 
the system appears to exhibit two phases  a  crumpled  phase , in which the hausdorff dimension is extremely large , and a  branched polymer  phase  neither of which look much like a classical spacetime .
lorentzian  model , introduced by ambjrn and loll [ 16 , 9 , 12 , 10 , 180 , 13 ] , however , has much nicer properties , including a continuum limit that appears numerically to match a finite - sized , spherical 
the lorentzian model begins with a slicing of spacetime into constant time surfaces , each of which is given an equilateral triangulation .
the region between two neighboring slices is then filled in by tetrahedra , which can come only in the three varieties shown in figure 3 .
this setup automatically restricts spacetime to have the topology    , and by declaring each slice to be spacelike and each edge joining adjacent slices to be timelike , one has a well - defined  wick rotation  to a riemannian signature metric with regge action ( 72 ) .
note that for convergence , this method requires a positive value of k3 , and thus a positive cosmological constant . 

the path integral for such a system can be evaluated numerically , using monte carlo methods and a set of  moves  that systematically change an initial triangulation [ 12 , 10 ] .
the system splits into uncorrelated two - dimensional spaces , each well - described by two - dimensional gravity . at weak coupling , however , a 
regime appears that resembles the picture obtained from other approaches to ( 2 + 1)-dimensional gravity . in particular
, one may evaluate the expectation value a(t) of the spatial area at fixed time and the correlation a(t)a(t
+ 1) of successive areas ; the results match the classical de sitter behavior for a spacetime   s quite well . the more  local  behavior
 the hausdorff dimension of a constant time slice , for example  is not yet well - understood .
neither is the role of moduli for spatial topologies more complicated than s , although initial steps have been taken for the torus universe .
the lorentzian dynamical triangulation model can also be translated into a two - matrix model , the so - called abab model .
the feynman diagrams of the matrix model correspond to dual graphs of a triangulation , and matrix model amplitudes become particular sums of transfer matrix elements in the gravitational theory [ 11 , 14 , 15 ] . in principle , this connection can be used to solve the gravitational model analytically . while this goal has not yet been achieved ( though see ) ,
for example , the matrix model connection can be used to show that newton s constant and the cosmological constant are additively renormalized , and to analyze the apparent nonrenormalizability of ordinary field theoretical approach . in principle the discrete approaches described in section 2.5  in particular , the lattice descriptions of t hooft and waelbroeck  should be straightforward to quantize . in practice ,
there has been fairly little work in this area , and most of the literature that does exist involves point particles rather than closed universes. t hooft has emphasized that the hamiltonian in his approach is an angle , and that time should therefore be discrete , in agreement with the lorentzian spin foam analysis of section 3.6. t hooft has also found that for a particular representation of the commutation relations for a point particle in ( 2 + 1)-dimensional gravity , space may also be discrete , although it remains unclear whether these results can be generalized beyond this one special example .
have examined waelbroeck s lattice hamiltonian approach for the quantized torus universe , investigating the implication of the choice of an internal time variable , and waelbroeck has studied the role of the mapping class group . the approaches to quantization of sections 3.1 , 3.2 , 3.3 , 3.4 , and 3.5 share an important feature : all are  reduced phase space 
quantizations , quantum theories based on the true physical degrees of freedom of the classical theory .
that is , the classical constraints have been solved before quantizing , eliminating classically redundant  gauge  degrees of freedom . in dirac s
approach to quantization [ 112 , 113 , 114 ] , in contrast , one quantizes the entire space of degrees of freedom of classical theory , and only then imposes the constraints .
states are initially determined from the full classical phase space ; in the adm formulation of quantum gravity , for instance , they are functionals [gij , ] of the full spatial metric .
the constraints then act as operators on this auxiliary hilbert space ; the physical hilbert space consists of those states that are annihilated by the constraints , with a suitable new inner product , acted on by physical operators that commute with the constraints . for gravity ,
in particular , the hamiltonian constraint acting on states leads to a functional differential equation , the wheeler - dewitt equation [ 110 , 276 ] . in the first order formalism
, it is straightforward to show that dirac quantization is equivalent to the chern - simons quantum theory we have already seen .
details can be found in chapter 8 of , but the basic argument is fairly clear : at least for  = 0 , the first order constraints coming from equations ( 15 , 16 ) are at most linear in the momenta , and are thus uncomplicated to solve . in the second order formalism , matters become considerably more complicated .
we begin with a wave function [gij ] , upon which we wish to impose the constraints ( 30 ) , with momenta acting as functional derivatives , 73\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${\pi ^{ij } } = - i{\delta \over { \delta { g_{ij}}}}.$$\end{document } the first difficulty is that we are no longer allowed to choose a nice time - slicing such as york time ; that would be a form of gauge - fixing , and is not permitted in dirac quantization .
we can still decompose the spatial metric and momentum as in equation ( 31 ) , but only up to a spatial diffeomorphism , which depends on an undetermined vector field y appearing in the momentum  .
the momentum constraint fixes in terms of the scale factor  , but it does so nonlocally . as a consequence ,
the hamiltonian constraint becomes a nonlocal functional differential equation , and very little is understood about its solutions , even for the simplest case of the torus universe .
further complications come from the fact that the inner product on the space of solutions of the wheeler - dewitt equation must be gauge - fixed [ 282 , 144 ] ; again , little is understood about the resulting hilbert space . in view of the difficulty in finding exact solutions to the wheeler - dewitt equation
, it is natural to look for perturbative methods , for example an expansion in powers of newton s constant g. one can solve the momentum constraints order by order by insisting that each term depend only on ( nonlocal ) spatially diffeomorphism - invariant quantities .
such an expansion has been studied by banks , fischler , and susskind for the physically trivial topology   s , following much earlier work by leutwyler . even in this simple case ,
other attempts have been made [ 47 , 186 ] to write a discrete version of the wheeler - dewitt equation in the ponzano - regge formalism of section 3.6 .
this approach has the advantage that the spatial diffeomorphisms have already been largely eliminated , removing the main source of nonlocality discussed above .
the wheeler - dewitt - like equation in   has been shown to agree with the the ponzano - regge model .
up to now , i have mainly concentrated on approaches to quantum gravity that fall under the broad heading of canonical quantization .
an alternative approach  already implicit in the discussion of discrete methods  starts with the feynman path integral , or 
 in an important sense , path integral methods are less precise than those of canonical quantization : the infinite - dimensional  integral  over histories can rarely be rigorously defined , we do not really know what classes of paths to sum over , and ordering ambiguities in the operator formalism reemerge as ambiguities in the integration measure . on the other hand ,
path integrals allow us to ask questions  for example , about amplitudes for spatial topology change  that are difficult or impossible to formulated in a canonical theory .
the simplest path integral approach to ( 2 + 1)-dimensional quantum gravity is the phase space path integral , in which the action is written in the adm form ( 29 , 30 ) , and the spatial metric gij and momentum  are treated as independent integration variables . the lapse n and shift appear as lagrange multipliers , and the integrals over these quantities yield delta functionals for the constraints  and i .
one might therefore expect the result to be equivalent to the canonical quantization of section 3.1 , in which the constraints are set to zero and solved for the physical degrees of freedom .
this is indeed true , as shown in [ 77 , 245 ] for spatially closed universes and   for geometries with point particles .
the main subtlety comes from the appearance of many different determinants , arising from gauge - fixing and from the delta functionals , which must be shown to cancel .
the phase space path integral for the first order formulation similarly reproduces the corresponding canonically quantized theory .
it is perhaps more interesting to look at the covariant metric path integral , in which one starts with the ordinary einstein - hilbert action and gauge - fixes the full ( 2 + 1)-dimensional diffeomorphism group .
this approach does not require a topology    , and could potentially describe topology - changing amplitudes .
section 9.2 of   describes a partial gauge - fixing , which takes advantage of the fact that every metric on a three - manifold is conformal to one of constant scalar curvature .
but while this leads to some simplification , we are still left with an infinite - dimensional integral about which very little can yet be said . by far the most useful results in the path integral
approach to ( 2 + 1)-dimensional quantum gravity have come from the covariant first - order action ( 14 ) .
the path integral for this action was first fully analyzed in two seminal papers by witten [ 277 , 279 ] , who showed that it reduced to a ratio of determinants that has an elegant topological interpretation as the analytic or ray - singer torsion .
the partition function for a closed three - manifold with  = 0 takes the form 74\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$${z_m } = { { { { \left\vert { \det \delta _
\varpi ^{(1 ) } } \right\vert}^{1/2 } } } \over { \left\vert { \det \delta _
\varpi ^{(2 ) } } \right\vert}},$$\end{document } where \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\delta _ { \bar \omega}^{(n ) } = { d_{\bar \omega}}*{d_{\bar \omega } } * + * { d_{\bar \omega}}*{d_{\bar \omega}}$\end{document } is the gauge - covariant laplacian acting on n - forms and \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}${\bar \omega}$\end{document } is a flat so(2 , 1 ) connection .
when m admits more than one such flat connection , equation ( 74 ) must be integrated over the moduli space of such connections .
although it was originally derived for closed manifolds , equation ( 74 ) can be extended to manifolds with boundary in a straightforward manner .
the path integral then gives a transition function that depends on specified boundary data  most simply , the induced spin connection  , with some additional restrictions on the normal component of  and the triad e [ 283 , 84 ] . for a manifold with the topology    , the results agree with those of covariant canonical quantization : the transition amplitude between two surfaces with prescribed spin connections is nonzero only if the holonomies agree . but the path integral can also give transition amplitudes between states on surfaces i and f with different topologies . if we demand that the initial and final surfaces be nondegenerate and spacelike , their topologies are severely restricted : amano and higuchi have shown that i and f must have equal euler numbers . for such manifolds , concrete computations can exploit the topological invariance of the ray - singer torsion .
carlip and cosgrove , for example , explicitly compute amplitudes for a transition between a genus three surface and a pair of genus two surfaces .
lorentzian path integrals allow us to compute interesting topology - changing amplitudes , in which the universe tunnels from one spatial topology to another .
they do not , however , directly address a principle issue of quantum cosmology , the problem of describing the birth of a universe from  nothing  . here ,
most of the literature has focused on the hawking s euclidean path integral   and the hartle - hawking  no boundary  proposal , which describes the universe in terms of a path integral over riemannian metrics on manifolds with a single , connected boundary . as in 3 + 1 dimensions , most of the work in 2 + 1 dimensions has concentrated on the saddle point approximation .
so far , the main benefit of the lower - dimensional model has been the possibility of treating topology more systematically , revealing interesting effects that are only now being explored in 3 + 1 dimensions . in the hartle - hawking approach to quantum cosmology
, the initial wave function of the universe is described by a path integral for a compact manifold m with a single spatial boundary  , as in figure 4 . 

figure 4a manifold m with a single boundary  describes the birth of a universe in the hartle - hawking approach to quantum cosmology .
a manifold m with a single boundary  describes the birth of a universe in the hartle - hawking approach to quantum cosmology . in 2 + 1 dimensions , the selection rules of  
imply that such a process can be described in lorentzian signature only if ( ) = 0 , that is , only for  a torus .
moreover , the known examples of such metrics always yield a degenerate metric on . if one allows riemannian signature , on the other hand , such a path integral makes sense for any spatial topology , and if one further requires that  be totally geodesic  that is , that the extrinsic curvature of  vanish  one can smoothly join on a lorentzian metric at  .
wave function 75\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\psi [ h,\varphi { \vert _ \sigma};\sigma ] = \sum\limits_{m:\partial m = \sigma } { \int { [ dg][d\varphi ] } \;\exp \;(- { i_{\rm{e}}}[g,\varphi ] ) } , $ $ \end{document } where the value of the path integral is determined by a specified induced metric h and matter configuration on the boundary .
the summation represents a sum over topologies of m ; in the absence of any basis for picking out a preferred topology , all manifolds with a given boundary  are assumed to contribute .
the wave function  is to be interpreted as an amplitude for finding a universe , with a prescribed spatial topology  , characterized by an  initial 
this approach finesses the question of initial conditions for the universe by simply omitting an initial boundary , and it postpones the question of the nature of time in quantum gravity : information about time is hidden in the boundary geometry h , but the path integral can be formulated without making a choice of time explicit . the path integral ( 75 ) can not , in general , be evaluated exactly , even in 2 + 1 dimensions .
indeed , there are general reasons to expect the expression to be ill - defined : a conformal excitation g  eg contributes to ie with the wrong sign , and the action is unbounded below .
in the ( 2 + 1)-dimensional lorentzian dynamical triangulation models of section 3.7 , however , it is known that these wrong sign contributions are unimportant ; they are overwhelmed by the much larger number of well - behaved geometries in the path integral .
this has led to a suggestion [ 100 , 99 ] that the conformal contribution is canceled by a faddeev - popov determinant ( see also ) , and some preliminary supporting computations have been made in a proper time gauge . assuming that the  conformal factor problem  is solved , a saddle point evaluation of the path integral is arguably a good approximation . for simplicity ,
an easy computation shows that the leading contribution to equation ( 75 ) is a sum of terms of the form 76\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$\exp \;(-
{ \bar{i } _ e } ) = { \delta _ m}\exp \left({{\rm{sign}}(\lambda){{{\rm{vol}}_{\bar g}(m ) } \over { 4\pi g\hbar \vert \lambda { \vert ^{1/2 } } } } } \right),$$\end{document } where  is an einstein metric on m , vol(m ) is the volume of m with the metric rescaled to constant curvature 1 , and the prefactor m is related , as in section 3.10 , to the ray - singer torsion of m. for  > 0 , three - manifolds that admit einstein metrics are all elliptic  that is , they have constant positive curvature , and can be described as quotients of the three - sphere by discrete groups of isometries .
the largest value of vol(m ) comes from the three - sphere itself , and one might expect it to dominate the sum over topologies . as shown in , though , the number of topologically distinct lens spaces with volumes less than vol(s ) grows fast enough that these spaces dominate , leading to a divergent partition function for closed three - manifolds .
the implications for the hartle - hawking wave function have not been examined explicitly , but it seems likely that a divergence will appear there as well . for 
< 0 , three - manifolds that admit einstein metrics are hyperbolic , and the single largest contribution to equation ( 76 ) comes from the smallest such manifold .
this contribution has been worked out in detail , for a genus 2 boundary , in . here , too
, however , manifolds with larger volumes  although individually exponentially suppressed  are numerous enough to lead to a divergence in the partition function . in this case
, the hartle - hawking wave function has been examined as well , and it has been shown that the wave function acquires infinite peaks at certain specific spatial geometries : again , topologically complicated manifolds whose individual contributions are small occur in large enough numbers to dominate the path integral , and  entropy
the benefit of restricting to 2 + 1 dimensions here is a bit different from the advantages seen earlier .
we are now helped not so much by the simplicity of the geometry ( although this helps in the computation of the prefactors m ) , but by the fact that three - manifold topology is much better understood than four - manifold topology .
it is only quite recently that similar results for sums over topologies have been found in four dimensions [ 79 , 80 , 229 , 19 ] . as noted in section 3.6 ,
recent work on spin foams has also suggested a new nonperturbative approach to evaluating the sum over topologies .
building on work by boulatov , freidel and loupre have recently considered a variant of the ponzano - regge model , and have shown that although the sum over topologies diverges , it is borel summable .
this result involves a clever representation of a spacetime triangulation as a feynman graph in a field theory on a group manifold , allowing the sum over topologies to be reexpressed as a sum of field theory feynman diagrams .
the model considered in   is not exactly the ponzano - regge model , and it is not clear that it is really  ordinary  quantum gravity .
moreover , study of the physical meaning of the borel resummed partition function has barely begun . nonetheless , these results suggest that a full treatment of the sum over topologies in ( 2 + 1)-dimensional quantum gravity may not be hopelessly out of reach .
there are also indications that string theory might have something to say about the sum over topologies .
in particular , the ads / cft correspondence may impose boundary conditions that limit the topologies allowed in the sum . whether such results can be extended to spatially closed manifolds
the world is not ( 2 + 1)-dimensional , and ( 2 + 1)-dimensional quantum gravity is certainly not a realistic model of our universe .
nonetheless , the ( 2 + 1)-dimensional model reflects many of the fundamental conceptual issues of real world quantum gravity , and work in this field has provided some valuable insights . 


existence and nonuniqueness 
 perhaps the most important lesson of ( 2 + 1)-dimensional quantum gravity is that general relativity can , in fact , be quantized .
while additional ingredients  strings , for instance  may have their own attractions , they are evidently not necessary for the existence of quantum gravity .
more than an  existence theorem  , though , the ( 2 + 1)-dimensional models also provide a  nonuniqueness theorem  : many approaches to the quantum theory are possible , and they are not all equivalent .
this is perhaps a bit of a disappointment , since many in this field had hoped that once we found a self - consistent quantum theory of gravity , the consistency conditions might be stringent enough to make that theory unique .
though , we should not be so surprised : quantum gravity is presumably more fundamental than classical general relativity , and it is not so strange to learn that more than one quantum theory can have the same classical limit . 

( 2 + 1)-dimensional gravity as a test bed 
 general relativity in 2 + 1 dimensions has provided a valuable test bed for a number of specific proposals for quantum gravity . some of these are  classics   the wheeler - dewitt equation , for instance , and reduced phase space quantization  while others , like spin foams , lorentzian dynamical triangulations , and covariant canonical quantization , are less well established.we have discovered some rather unexpected features , such as the difficulties caused by spatial diffeomorphism invariance and the consequent nonlocality in wheeler - dewitt quantization , and the necessity of understanding the representations of the group of large diffeomorphisms in almost all approaches . for particular quantization programs , ( 2 + 1)-dimensional models
have also offered more specific guidance : special properties of the loop operators ( 55 ) , methods for treating noncompact groups in spin foam models , and properties of the sums over topologies described in section 3.11 have all been generalized to 3 + 1 dimensions . 

lorentzian dynamical triangulations 
 a particular application of ( 2 + 1)-dimensional gravity as a test bed is important enough to deserve special mention .
the program of  lorentzian dynamical triangulations  described in section 3.7 is a genuinely new approach to quantum gravity .
euclidean dynamical triangulations  , one might normally be quite skeptical of such a method .
but the success in reproducing semiclassical states in 2 + 1 dimensions , although still fairly limited , provides a strong argument that the approach should be taken seriously . 

one of the deepest conceptual difficulties in quantum gravity has been the problem of reconstructing local , dynamical spacetime from the nonlocal diffeomorphism - invariant observables required by quantum gravity .
the notorious  problem of time  is a special case of this more general problem of observables . as we saw in section 3.4 , ( 2 + 1)-dimensional quantum gravity points toward a solution , allowing the construction of families of  local  and  time - dependent  observables that nevertheless commute with all constraints.the idea that  frozen time  quantum gravity is a heisenberg picture corresponding to a fixed - time - slicing schrdinger picture is a central insight of ( 2 + 1)-dimensional gravity . in practice ,
though , we have also seen that the transformation between these pictures relies on our having a detailed description of the space of classical solutions of the field equations .
we can not expect such a fortunate circumstance to carry over to full ( 3 + 1)-dimensional quantum gravity ; it is an open question , currently under investigation , whether one can use a perturbative analysis of classical solutions to find suitable approximate observables . 

singularities 
 it has long been hoped that quantum gravity might smooth out the singularities of classical general relativity . although the ( 2 + 1)-dimensional model has not yet provided a definitive test of this idea , some progress has been made .
puzio , for example , has shown that a wave packet initially concentrated away from the singular points in moduli space will remain non - singular . on the other hand ,
minassian has recently demonstrated   that quantum fluctuations do not  push singularities off to infinity  ( as suggested in ) , and that several classically singular ( 2 + 1)-dimensional quantum spacetimes also have singular  quantum b - boundaries  . 

another long - standing expectation has been that quantum gravity will lead to discrete , quantized lengths , with a minimum length on the order of the planck length .
partial results in quantum geometry and spin foam approaches to ( 2 + 1)-dimensional quantum gravity suggest that this may be true , but also that the problem is a bit subtle [ 197 , 251 , 241 ] .
the most recent result in this area   relates the spectrum of lengths to representations of the ( 2 + 1)-dimensional lorentz group , which can be discrete or continuous .
argue that spacelike intervals are continuous , while timelike intervals are discrete , with a spectrum of the form \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\sqrt { n(n - 1 ) } { \ell _ { \rm{p}}}$\end{document}. the analysis is a bit tricky , since the length 
a first step towards defining truly invariant operators describing distances between point particles supports this picture , but the results are not yet conclusive . 

quantum gravity contains two fundamental dimensionful constants , the planck length p and the speed of light c. this has suggested to some that special relativity might itself be altered so that both p and c are constants .
this requires a nonlinear deformation of the poincar algebra , and leads to a set of theories collectively called  doubly special relativity 
it has recently been pointed out that ( 2 + 1)-dimensional gravity automatically displays such a deformation [ 197 , 18 , 126 ] .
a few attempts have been made to connect this picture to noncommutative spacetime , mainly in the context of point particles [ 197 , 275 , 38 ] , but it seems too early to evaluate them . 

the answer in 2 + 1 dimensions is unequivocally no : canonical quantization gives a perfectly consistent description of a universe with a fixed spatial topology . on the other hand ,
the path integrals of section 3.10 seem to allow the computation of amplitudes for tunneling from one topology to another .
problems with these topology - changing amplitudes remain , particularly in the regulation of divergent integrals over zero - modes .
if these can be resolved , however , we will have to conclude that we have found genuinely and deeply inequivalent quantum theories of gravity.sums over topologiesin conventional descriptions of the hartle - hawking wave function , and in other euclidean path integral descriptions of quantum cosmology , it is usually assumed that a few simple contributions dominate the sum over topologies .
the results of ( 2 + 1)-dimensional quantum gravity indicate that such claims should be treated with skepticism ; as discussed in section 3.11 , the sum over topologies is generally dominated by an infinite number of complicated topologies , each individually exponentially suppressed .
this is a new and unexpected result , whose implications for realistic ( 3 + 1)-dimensional gravity are just starting to be explored . 
 existence and nonuniqueness 
  perhaps the most important lesson of ( 2 + 1)-dimensional quantum gravity
while additional ingredients  strings , for instance  may have their own attractions , they are evidently not necessary for the existence of quantum gravity .
more than an  existence theorem  , though , the ( 2 + 1)-dimensional models also provide a 
nonuniqueness theorem  : many approaches to the quantum theory are possible , and they are not all equivalent .
this is perhaps a bit of a disappointment , since many in this field had hoped that once we found a self - consistent quantum theory of gravity , the consistency conditions might be stringent enough to make that theory unique .
in retrospect , though , we should not be so surprised : quantum gravity is presumably more fundamental than classical general relativity , and it is not so strange to learn that more than one quantum theory can have the same classical limit . 

( 2 + 1)-dimensional gravity as a test bed 
  general relativity in 2 + 1 dimensions has provided a valuable test bed for a number of specific proposals for quantum gravity . some of these are  classics   the wheeler - dewitt equation , for instance , and reduced phase space quantization  while others , like spin foams , lorentzian dynamical triangulations , and covariant canonical quantization , are less well established . we have discovered some rather unexpected features , such as the difficulties caused by spatial diffeomorphism invariance and the consequent nonlocality in wheeler - dewitt quantization , and the necessity of understanding the representations of the group of large diffeomorphisms in almost all approaches
. for particular quantization programs , ( 2 + 1)-dimensional models have also offered more specific guidance : special properties of the loop operators ( 55 ) , methods for treating noncompact groups in spin foam models , and properties of the sums over topologies described in section 3.11 have all been generalized to 3 + 1 dimensions . 

lorentzian dynamical triangulations 
  a particular application of ( 2 + 1)-dimensional gravity as a test bed is important enough to deserve special mention .
the program of  lorentzian dynamical triangulations  described in section 3.7 is a genuinely new approach to quantum gravity .
euclidean dynamical triangulations  , one might normally be quite skeptical of such a method .
but the success in reproducing semiclassical states in 2 + 1 dimensions , although still fairly limited , provides a strong argument that the approach should be taken seriously . 

 
  one of the deepest conceptual difficulties in quantum gravity has been the problem of reconstructing local , dynamical spacetime from the nonlocal diffeomorphism - invariant observables required by quantum gravity .
the notorious  problem of time  is a special case of this more general problem of observables . as we saw in section 3.4 , ( 2 + 1)-dimensional quantum gravity points toward a solution , allowing the construction of families of  local  and  time - dependent  observables that nevertheless commute with all constraints .
quantum gravity is a heisenberg picture corresponding to a fixed - time - slicing schrdinger picture is a central insight of ( 2 + 1)-dimensional gravity . in practice , though , we have also seen that the transformation between these pictures relies on our having a detailed description of the space of classical solutions of the field equations .
we can not expect such a fortunate circumstance to carry over to full ( 3 + 1)-dimensional quantum gravity ; it is an open question , currently under investigation , whether one can use a perturbative analysis of classical solutions to find suitable approximate observables .
it has long been hoped that quantum gravity might smooth out the singularities of classical general relativity .
although the ( 2 + 1)-dimensional model has not yet provided a definitive test of this idea , some progress has been made .
puzio , for example , has shown that a wave packet initially concentrated away from the singular points in moduli space will remain non - singular . on the other hand ,
minassian has recently demonstrated   that quantum fluctuations do not  push singularities off to infinity  ( as suggested in ) , and that several classically singular ( 2 + 1)-dimensional quantum spacetimes also have singular  quantum b - boundaries  . 

another long - standing expectation has been that quantum gravity will lead to discrete , quantized lengths , with a minimum length on the order of the planck length .
partial results in quantum geometry and spin foam approaches to ( 2 + 1)-dimensional quantum gravity suggest that this may be true , but also that the problem is a bit subtle [ 197 , 251 , 241 ] .
the most recent result in this area   relates the spectrum of lengths to representations of the ( 2 + 1)-dimensional lorentz group , which can be discrete or continuous .
argue that spacelike intervals are continuous , while timelike intervals are discrete , with a spectrum of the form \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$\sqrt { n(n - 1 ) } { \ell _ { \rm{p}}}$\end{document}. the analysis is a bit tricky , since the length 
a first step towards defining truly invariant operators describing distances between point particles supports this picture , but the results are not yet conclusive . 

quantum gravity contains two fundamental dimensionful constants , the planck length p and the speed of light c. this has suggested to some that special relativity might itself be altered so that both p and c are constants .
this requires a nonlinear deformation of the poincar algebra , and leads to a set of theories collectively called  doubly special relativity  [ 17 , 185 , 171 ] .
it has recently been pointed out that ( 2 + 1)-dimensional gravity automatically displays such a deformation [ 197 , 18 , 126 ] .
a few attempts have been made to connect this picture to noncommutative spacetime , mainly in the context of point particles [ 197 , 275 , 38 ] , but it seems too early to evaluate them . does consistent quantum gravity require spatial topology change ?
the answer in 2 + 1 dimensions is unequivocally no : canonical quantization gives a perfectly consistent description of a universe with a fixed spatial topology . on the other hand ,
the path integrals of section 3.10 seem to allow the computation of amplitudes for tunneling from one topology to another .
problems with these topology - changing amplitudes remain , particularly in the regulation of divergent integrals over zero - modes .
if these can be resolved , however , we will have to conclude that we have found genuinely and deeply inequivalent quantum theories of gravity .
sums over topologiesin conventional descriptions of the hartle - hawking wave function , and in other euclidean path integral descriptions of quantum cosmology , it is usually assumed that a few simple contributions dominate the sum over topologies .
the results of ( 2 + 1)-dimensional quantum gravity indicate that such claims should be treated with skepticism ; as discussed in section 3.11 , the sum over topologies is generally dominated by an infinite number of complicated topologies , each individually exponentially suppressed .
this is a new and unexpected result , whose implications for realistic ( 3 + 1)-dimensional gravity are just starting to be explored .
we know immensely more about ( 2 + 1)-dimensional quantum gravity than we did twenty years ago .
, it is only quite recently that the general tools developed over the past few years have been brought to bear on particular physical problems  the resolution of singularities , for example , and the question of whether space is discrete at the planck scale .
a sketchy and rather personal list of open questions would include the following : 

 singularities 
 a key question in quantum gravity is whether quantized spacetime  resolves  the singularities of classical general relativity .
this is a difficult question  already classically , it is highly nontrivial to even define a singularity , and the quantum extensions of the classical definitions are far from obvious .
this is an area in which ( 2 + 1)-dimensional gravity provides a natural arena , but results so far are highly preliminary [ 227 , 202 ] . 

sums over topologies 
 another long - standing question in quantum gravity is whether spacetime topology can ( or must ) undergo quantum fluctuations .
as we saw in section 3.11 , some real progress has been made in 2 + 1 dimensions .
often , though , the results require saddle point approximations , and pick out particular classes of saddle points .
the nonperturbative summation techniques discussed at the end of section 3.11 promise much deeper results , and may point toward a measure on the space of topologies analogous to the measure on the space of geometries induced by the dewitt metric . 
 quantized geometry 
 we saw above that there is some evidence for quantization of timelike intervals in ( 2 + 1)-dimensional gravity .
a systematic exploration of this issue might teach us a good deal about differences among approaches to quantization .
in particular , it would be very interesting to see whether any corresponding result appears in reduced phase space quantization , wheeler - dewitt quantization , or path integral approaches . to address this problem properly
, one must introduce genuine observables for quantities such as length and area , either by adding point particles   or by looking at shortest geodesics around noncontractible cycles .
note that for the torus universe , the moduli can be considered as ratios of lengths , and there is no sign that these need be discrete .
this does not contradict the claims of , since the lengths in question are spacelike , but it does suggest an interesting dilemma in euclidean quantum gravity , where spacelike as well as timelike intervals might naturally be quantized . 

euclidean vs. lorentzian gravity 
 in the chern - simons formalism of section 2.3 ,  euclidean  and  lorentzian  quantum gravity seem to be dramatically inequivalent : they have different gauge groups , different holonomies , and very different behaviors under the actions of large diffeomorphisms . in the adm approach of section 2.4 , on the other hand
this suggests that further study might finally tell us whether euclideanization is merely a technical trick , analogous to wick rotation in ordinary quantum field theory , or whether it gives a genuinely different theory ; and , if the latter , just how different the euclidean and lorentzian theories are . in canonical quantization
, a key step would be to relate chern - simons and adm amplitudes in the euclidean theory , perhaps using the methods of section 3.4 . in spin foam and path integral approaches
, it might be possible to explicitly compare amplitudes . 
 which approaches are equivalent ? 

a more general problem is to understand which of the approaches described here are equivalent .
in particular , it is not obvious how much of the difference among various methods of quantization can be attributed to operator ordering ambiguities , and how much reflects a deeper inequivalence , as reflected ( for instance ) in different length spectra or different possibilities for topology change .
an answer might help us understand just how nonunique quantum gravity in higher dimensions will be . 

higher genus 
 most of the detailed , explicit results in ( 2 + 1)-dimensional quantum gravity hold only for the torus universe   t. as noted in section 2.7 , this topology has some exceptional features , and might not be completely representative . in particular , the relationship between the adm and chern - simons quantizations in section 3.4 relied on a particularly simple operator ordering ; it is not obvious that such an ordering can be found for the higher genus case .
an extension to arbitrary genus might be too difficult , but a full treatment of the genus two topology , using the relation to hyperelliptic curves or the sigma model description of , may be possible .
it could also be worthwhile to further explore the case of spatially nonorientable manifolds   to see whether any important new features arise . 

some limited progress has been made : for example , there is some evidence that ( 2 + 1)-dimensional gravity is renormalizable in the 1/n expansion when coupled to scalar fields [ 174 , 205 ] .
this is apparently no longer the case when gravity is coupled to fermions and a u(1 ) chern - simons gauge theory , although anselmi has argued that if coupling constants are tuned to exact values , renormalizability can be restored , and in fact the theory can be made finite .
certain matter couplings in supergravity have been studied [ 104 , 196 ] , and work on circularly symmetric  midi - superspace models  has led to some surprising results , including unexpected bounds on the hamiltonian [ 27 , 32 , 137 , 53 , 265 , 224 ] .
but the general problem of coupling matter remains very difficult , not least because  except in the special case of  topological matter  [ 140 , 85 ]  we lose the ability to represent diffeomorphisms as iso(2 , 1 ) gauge transformations.difficult as it is , however , an understanding of matter couplings may be the key to many of the conceptual issues of quantum gravity .
one can explore the properties of a singularity , for example , by investigating the reaction of nearby matter , and one can look for quantization of time by examining the behavior of physical clocks .
moreover , some of the deep questions of quantum gravity can be answered only in the presence of matter .
this idea is an old one [ 109 , 165 , 166 ] , and it gets some support from the boundedness of the hamiltonian in midi - superspace models , but it is only in the context of a full quantum field theory that a final answer can be given . 
 the cosmological constant 
 undoubtedly , the biggest embarrassment in quantum gravity today is the apparent prediction , at least in effective field theory , that the cosmological constant should be some 120 orders of magnitude larger than the observed limit .
several attempts have been made to address this problem in the context of ( 2 + 1)-dimensional quantum gravity .
first , witten has suggested a novel mechanism by which supersymmetry in 2 + 1 dimensions might cancel radiative corrections to  without requiring the equality of superpartner masses , essentially because even if the vacuum is supersymmetric , the asymptotics forbid the existence of unbroken supercharges for massive states [ 281 , 52 ] .
this argument requires special features of 2 + 1 dimensions , though , and it is not at all clear that it can be generalized to 3 + 1 dimensions ( although some attempts have been made in the context of  deconstruction  ) .second , the discovery that the sum over topologies can lead to a divergent partition function has been extended to 3 + 1 dimensions , at least for  < 0 , and it has been argued that this behavior might signal a phase transition that could prohibit a conventional cosmology with a negative cosmological constant [ 79 , 80 ]
. the crucial case of a positive cosmological constant is not yet understood , however , and if a phase change does indeed occur , its nature is still highly obscure
. it may be that the nonperturbative summation over topologies discussed at the end of section 3.11 could cast light on this question.one might also hope that a careful analysis of the coupling of matter in 2 + 1 dimensions could reveal useful details concerning the vacuum energy contribution to  , perhaps in a setting that goes beyond the usual effective field theory approach .
for example , there is evidence that the matter hamiltonian is bounded above in ( 2 + 1)-dimensional gravity ; perhaps this could cut off radiative contributions to the cosmological constant at an interesting scale . 
 again ,
( 2 + 1)-dimensional gravity as a test bed 
 as new approaches to quantum gravity are developed , the ( 2 + 1)-dimensional model will undoubtedly remain important as a simplified test bed .
for example , a bit of work has been done on the null surface formulation of classical gravity in 2 + 1 dimensions ; a quantum treatment might be possible , and could tell us more about the utility of this approach in 3 + 1 dimensions . similarly , ( 2 + 1)-dimensional gravity has recently been examined as an arena in which to test for a new partially discrete , constraint - free formulation of quantum gravity .
a key question in quantum gravity is whether quantized spacetime  resolves  the singularities of classical general relativity .
this is a difficult question  already classically , it is highly nontrivial to even define a singularity , and the quantum extensions of the classical definitions are far from obvious .
this is an area in which ( 2 + 1)-dimensional gravity provides a natural arena , but results so far are highly preliminary [ 227 , 202 ] . 

sums over topologies 
  another long - standing question in quantum gravity is whether spacetime topology can ( or must ) undergo quantum fluctuations . as we saw in section 3.11 , some real progress has been made in 2 + 1 dimensions .
often , though , the results require saddle point approximations , and pick out particular classes of saddle points .
the nonperturbative summation techniques discussed at the end of section 3.11 promise much deeper results , and may point toward a measure on the space of topologies analogous to the measure on the space of geometries induced by the dewitt metric .
we saw above that there is some evidence for quantization of timelike intervals in ( 2 + 1)-dimensional gravity .
a systematic exploration of this issue might teach us a good deal about differences among approaches to quantization .
in particular , it would be very interesting to see whether any corresponding result appears in reduced phase space quantization , wheeler - dewitt quantization , or path integral approaches .
to address this problem properly , one must introduce genuine observables for quantities such as length and area , either by adding point particles   or by looking at shortest geodesics around noncontractible cycles .
note that for the torus universe , the moduli can be considered as ratios of lengths , and there is no sign that these need be discrete .
this does not contradict the claims of , since the lengths in question are spacelike , but it does suggest an interesting dilemma in euclidean quantum gravity , where spacelike as well as timelike intervals might naturally be quantized . 

euclidean vs. lorentzian gravity 
  in the chern - simons formalism of section 2.3 ,  euclidean  and  lorentzian  quantum gravity seem to be dramatically inequivalent : they have different gauge groups , different holonomies , and very different behaviors under the actions of large diffeomorphisms . in the adm approach of section 2.4 , on the other hand
this suggests that further study might finally tell us whether euclideanization is merely a technical trick , analogous to wick rotation in ordinary quantum field theory , or whether it gives a genuinely different theory ; and , if the latter , just how different the euclidean and lorentzian theories are . in canonical quantization
, a key step would be to relate chern - simons and adm amplitudes in the euclidean theory , perhaps using the methods of section 3.4 . in spin foam and path integral approaches
which approaches are equivalent ? 
  a more general problem is to understand which of the approaches described here are equivalent .
in particular , it is not obvious how much of the difference among various methods of quantization can be attributed to operator ordering ambiguities , and how much reflects a deeper inequivalence , as reflected ( for instance ) in different length spectra or different possibilities for topology change .
an answer might help us understand just how nonunique quantum gravity in higher dimensions will be .
most of the detailed , explicit results in ( 2 + 1)-dimensional quantum gravity hold only for the torus universe   t. as noted in section 2.7 , this topology has some exceptional features , and might not be completely representative . in particular , the relationship between the adm and chern - simons quantizations in section 3.4 relied on a particularly simple operator ordering ; it is not obvious that such an ordering can be found for the higher genus case .
an extension to arbitrary genus might be too difficult , but a full treatment of the genus two topology , using the relation to hyperelliptic curves or the sigma model description of , may be possible .
it could also be worthwhile to further explore the case of spatially nonorientable manifolds   to see whether any important new features arise .
some limited progress has been made : for example , there is some evidence that ( 2 + 1)-dimensional gravity is renormalizable in the 1/n expansion when coupled to scalar fields [ 174 , 205 ] .
this is apparently no longer the case when gravity is coupled to fermions and a u(1 ) chern - simons gauge theory , although anselmi has argued that if coupling constants are tuned to exact values , renormalizability can be restored , and in fact the theory can be made finite .
certain matter couplings in supergravity have been studied [ 104 , 196 ] , and work on circularly symmetric  midi - superspace models  has led to some surprising results , including unexpected bounds on the hamiltonian [ 27 , 32 , 137 , 53 , 265 , 224 ] .
but the general problem of coupling matter remains very difficult , not least because  except in the special case of  topological matter 
[ 140 , 85 ]  we lose the ability to represent diffeomorphisms as iso(2 , 1 ) gauge transformations .
difficult as it is , however , an understanding of matter couplings may be the key to many of the conceptual issues of quantum gravity .
one can explore the properties of a singularity , for example , by investigating the reaction of nearby matter , and one can look for quantization of time by examining the behavior of physical clocks .
moreover , some of the deep questions of quantum gravity can be answered only in the presence of matter .
this idea is an old one [ 109 , 165 , 166 ] , and it gets some support from the boundedness of the hamiltonian in midi - superspace models , but it is only in the context of a full quantum field theory that a final answer can be given . 
 the cosmological constant 
  undoubtedly , the biggest embarrassment in quantum gravity today is the apparent prediction , at least in effective field theory , that the cosmological constant should be some 120 orders of magnitude larger than the observed limit .
several attempts have been made to address this problem in the context of ( 2 + 1)-dimensional quantum gravity .
first , witten has suggested a novel mechanism by which supersymmetry in 2 + 1 dimensions might cancel radiative corrections to  without requiring the equality of superpartner masses , essentially because even if the vacuum is supersymmetric , the asymptotics forbid the existence of unbroken supercharges for massive states [ 281 , 52 ] .
this argument requires special features of 2 + 1 dimensions , though , and it is not at all clear that it can be generalized to 3 + 1 dimensions ( although some attempts have been made in the context of  deconstruction  ) .
second , the discovery that the sum over topologies can lead to a divergent partition function has been extended to 3 + 1 dimensions , at least for  < 0 , and it has been argued that this behavior might signal a phase transition that could prohibit a conventional cosmology with a negative cosmological constant [ 79 , 80 ]
. the crucial case of a positive cosmological constant is not yet understood , however , and if a phase change does indeed occur , its nature is still highly obscure
. it may be that the nonperturbative summation over topologies discussed at the end of section 3.11 could cast light on this question .
one might also hope that a careful analysis of the coupling of matter in 2 + 1 dimensions could reveal useful details concerning the vacuum energy contribution to  , perhaps in a setting that goes beyond the usual effective field theory approach .
for example , there is evidence that the matter hamiltonian is bounded above in ( 2 + 1)-dimensional gravity ; perhaps this could cut off radiative contributions to the cosmological constant at an interesting scale . 

again , ( 2 + 1)-dimensional gravity as a test bed 
  as new approaches to quantum gravity are developed , the ( 2 + 1)-dimensional model will undoubtedly remain important as a simplified test bed .
for example , a bit of work has been done on the null surface formulation of classical gravity in 2 + 1 dimensions ; a quantum treatment might be possible , and could tell us more about the utility of this approach in 3 + 1 dimensions . similarly , ( 2 + 1)-dimensional gravity has recently been examined as an arena in which to test for a new partially discrete , constraint - free formulation of quantum gravity .
 in three spacetime dimensions , general relativity drastically simplifies , becoming a  
 topological  theory with no propagating local degrees of freedom 
 . nevertheless , many of the difficult conceptual problems of quantizing gravity are still present . in this review , 
 i summarize the rather large body of work that has gone towards quantizing ( 2 + 1)-dimensional vacuum gravity in the setting of a spatially closed universe . 
Introduction
Classical Gravity in 2 + 1 Dimensions
Quantum Gravity in 2 + 1 Dimensions
What Have We Learned?
What Can We Still Learn?