Two-dimensional Quantum Gravity 8 9 Juri Rolf 9 1 Niels Bohr Institute t c University of Copenhagen O 5 1 v 7 2 0 0 1 8 9 / h t - p e h Thesis submitted for the Ph.D. degree in physi s : v at the Fa ulty of S ien e, University of Copenhagen. i X May 1998 r a Contents A knowledgement iii Introdu tion 1 1 Two-dimensional quantum gravity 2 1.1 S aling in quantum gravity . . . . . . . . . . . . . . . . . . . 3 1.1.1 Partition fun tion . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Hartle-Hawking wavefun tionals . . . . . . . . . . . . 6 1.1.3 The two-point fun tion . . . . . . . . . . . . . . . . . 7 1.2 Liouville theory: A brief reminder . . . . . . . . . . . . . . . 11 1.2.1 Fun tional measures . . . . . . . . . . . . . . . . . . . 11 1.2.2 Fa torization of the di(cid:11)eomorphisms . . . . . . . . . . 14 1.2.3 Gravitational dressing of s aling exponents . . . . . . 16 1.3 Fra tal dimensions . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3.1 Extrinsi Hausdor(cid:11) dimension . . . . . . . . . . . . . 21 1.3.2 Spe tral dimension . . . . . . . . . . . . . . . . . . . . 23 1.3.3 Intrinsi Hausdor(cid:11) dimension . . . . . . . . . . . . . . 28 2d 2 Dis retization of quantum gravity 32 2.1 Dynami al triangulation . . . . . . . . . . . . . . . . . . . . . 33 2.1.1 Dis retization of geometry . . . . . . . . . . . . . . . . 33 2.1.2 De(cid:12)nition of the model . . . . . . . . . . . . . . . . . 36 2.1.3 The two-point fun tion . . . . . . . . . . . . . . . . . 39 2.1.4 Bran hed polymers . . . . . . . . . . . . . . . . . . . . 43 2.2 Matrix models . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.2.1 Dynami al triangulation by matrix models . . . . . . 46 2.2.2 The loop equations . . . . . . . . . . . . . . . . . . . . 49 γ 2.2.3 S aling limit and omputation of . . . . . . . . . . . 52 2.3 The fra tal stru ture of pure gravity . . . . . . . . . . . . . . 54 2.3.1 The geodesi two-loop fun tion . . . . . . . . . . . . . 54 i ii CONTENTS 2.3.2 S aling of the two-point fun tion . . . . . . . . . . . . 55 2.4 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 The failure of quantum Regge al ulus 58 3.1 Formulation of quantum Regge al ulus . . . . . . . . . . . . 59 3.2 Regge integration measures . . . . . . . . . . . . . . . . . . . 60 3.2.1 DeWitt-like measures . . . . . . . . . . . . . . . . . . 61 3.2.2 The DeWitt-like measure in other dimensions than two 63 3.2.3 The DeWitt-like measures for spe ial geometries . . . 64 3.2.4 Commonly used measures . . . . . . . . . . . . . . . . 65 3.2.5 Numeri alsimulationsintwo-dimensionalquantumRegge al ulus . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3 The appearan e of spikes . . . . . . . . . . . . . . . . . . . . 69 3.3.1 General proof of appearan e . . . . . . . . . . . . . . . 70 3.3.2 Spikes in spe ial geometries . . . . . . . . . . . . . . . 72 3.4 Con luding remarks . . . . . . . . . . . . . . . . . . . . . . . 75 3.4.1 The appearan e of spikes in higher dimensions . . . . 75 3.4.2 The role of di(cid:11)eomorphisms in two dimensions . . . . 76 3.4.3 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . 79 Con lusion 82 Bibliography 83 A knowledgement I want to thank all the people who have dire tly or indire tly ontributed to this work and made my stay in Copenhagen possible, in parti ular: (cid:15) my supervisor Jan Ambj(cid:28)rn for the interesting proje t, for many dis- ussions, for a good ollaboration and for partial (cid:12)nan ial support of my visit in Santa Fe at the workshop New Dire tions in Simpli ial Quantum Gravity, (cid:15) Jakob L. Nielsen for many dis ussions and for a good ollaboration, (cid:15) my ollaborators Dimitrij Boulatov, George Savvidy and Yoshiyuki Watabiki for the good work done together, (cid:15) Martin Harris and Jakob L. Nielsen for proofreading the thesis, (cid:15) my parents and my wife for their support and en ouragement, (cid:15) Konstantinos N. Anagnostopoulos, Martin Harris, Lars Jensen, Jakob L.Nielsen, Kaspar Olsen, J(cid:28)rgenRasmussen and Morten Weis for some dis ussions and for reating the ni e atmosphere in our oÆ e, (cid:15) the Niels Bohr Institute for warm hospitality, (cid:15) the Studienstiftung des deuts hen Volkes for (cid:12)nan ial support. iii Introdu tion This Ph.D. thesis pursues two goals: The study of the geometri al stru - ture of two-dimensionalquantum gravity and in parti ular its fra tal nature. To address these questions we review the ontinuum formalism of quantum gravity with spe ial fo us on the s aling properties of the theory. We dis- uss several on epts of fra tal dimensions whi h hara terize the extrinsi and intrinsi geometry of quantum gravity. This work is partly based on work done in ollaboration with Jan Ambj(cid:28)rn, Dimitrij Boulatov, Jakob L. Nielsen and Yoshiyuki Watabiki [1℄. The other goal is the dis ussion of the dis retization of quantum gravity and to address the so alled quantum failure of Regge al ulus. We review dynami al triangulations and show that it agrees with the ontinuum theory in two dimensions. Then we dis uss Regge al ulus and prove that a ontin- uum limit annot be taken in a sensible way and that it does not reprodu e ontinuum results. This work is partly based on work done in ollaboration with Jan Ambj(cid:28)rn, Jakob L. Nielsen and George Savvidy [2℄. In hapter 1 we introdu e the main ingredients for the formulation of two-dimensional quantum gravity as an Eu lidean fun tional integral over geometries. It ontains a brief reminder of Liouvilletheory and the te hni al issues in the ontinuum formalism. We use these te hniques to dis uss the extrinsi and intrinsi Hausdor(cid:11) dimension and the spe tral dimension of two-dimensional quantum gravity. Chapter 2 is a review of dynami al triangulation in two dimensions. We begin with an introdu tion of the main ideas of how to dis retize two- dimensional quantum geometries. The s aling properties are illustrated by means of the two-point fun tion of pure gravity and of bran hed polymers. In hapter 3wedis ussquantumRegge al uluswhi hhasbeen suggested as an alternative method to dis retize quantum geometries. We prove by a simples alingargumentthatasensible ontinuumlimitofthistheory annot be taken and that it disagrees with ontinuum results. 1 Chapter 1 Two-dimensional quantum gravity Eu lideanquantumgravity is anattemptto quantizegeneralrelativitybased on Feynman's fun tional integral and on the Einstein-Hilbert a tion prin i- d ple. One integrates over allRiemannianmetri son a -dimensionalmanifold M . It is based on the hope that one an re over the Lorentzian signature af- ter performingthe integrationanalogouslyto the Wi k rotationin Eu lidean quantum (cid:12)eld theory. For a general dis ussion of further problems and for motivation of a theory of quantum gravity we refer to [3, 4, 5℄. General relativity is a reparametrization invariant theory whi h an be formulated with no referen e to oordinates at all. This di(cid:11)eomorphism invarian e is a entral issue in the quantum theory. Its importan e is most apparent in two dimensions, sin e the Einstein-Hilbert a tion is trivial and onsists only of a topologi al term and the osmologi al onstant oupled to the volume of the spa etime. All the non-trivial dynami s of the two- dimensional theory of quantum gravity thus ome from gauge (cid:12)xing the di(cid:11)eomorphismswhilekeepingthegeometryexa tly(cid:12)xed. Thisisthefamous representation of the fun tional integral over geometries as a Liouville (cid:12)eld theory by Polyakov [6℄. Based on this formulation the s aling exponents an be obtained [7, 8, 9℄. Any theory of quantum gravity must aim at answering questions about the geometri al stru ture of quantum spa etime. The interplay between matter and geometry is well known from general relativity. The quantum average over allgeometries hanges the dynami sof this intera tion. It turns out that the quantum spa etime has a fra tal nature and even its dimension is a dynami al quantity. The hara terization of the fra tal nature of the quantum spa etime in two-dimensionalquantum gravity is one of the entral themes of this work. In se tion 1.1 the main on epts in the ontinuum formalism are intro- 2 1.1. SCALING IN QUANTUM GRAVITY 3 du ed. In the following se tion we review te hni al details about the fa tor- izationof the di(cid:11)eomorphismsfrom the fun tional integral. In se tion 1.3 we introdu e fra tal dimensions to hara terize the fra tal nature of quantum spa etime. That se tion, and in parti ular se tion 1.3.2 are partly based on work presented in [1, 10℄. Reviews about the ontinuum approa h to two-dimensional quantum gravity an be found in [11℄ and the referen es therein. For an introdu - tion to onformal (cid:12)eld theory see [12, 13, 14℄. 1.1 S aling in quantum gravity In this se tion we introdu e the basi on epts in two-dimensional quantum gravity. We begin with the partition fun tion and the Hartle-Hawkingwave- fun tionals. The most natural obje t to address questions about the s aling behaviour of the theory is the geodesi two-point fun tion. We dis uss its s aling behaviour in detail and introdu e the on ept of the intrinsi fra tal dimension whi h illustrates the fra tal nature of the intrinsi geometry of the two-dimensional quantum spa e-time. 1.1.1 Partition fun tion M Let beatwo-dimensional, losed, ompa t, onne ted, orientablemanifold h ofgenus . Then the partitionfun tionfor two-dimensionalquantumgravity an formally be written as the fun tional integral Z(G,Λ) = [g ] e−S (g;G,Λ) X e−S (g,X). µν EH g matter D D (1.1) h 0Z Z X≥ Here 1 S (g;G,Λ) = Λ d2ξ√g− d2ξ√g (ξ) EH ZM 4πG ZM R (1.2) is the lassi al reparametrization invariant Einstein-Hilbert a tion [15, 16℄ G Λ with the gravitational oupling onstant , the osmologi al onstant and the urvature s alar R. A ording to the Gauss{Bonnet theorem, the last χ term in (1.2) is a topologi al invariant, alled the Euler hara teristi of M : 1 χ(h) = d2ξ√g (ξ) = 2−2h, 4π R (1.3) ZM 4 CHAPTER 1. TWO-DIMENSIONAL QUANTUM GRAVITY V M g g µν while the (cid:12)rst term is the volume of equipped with the metri : V = d2ξ√g. g (1.4) ZM Therefore, (1.1) an be rewritten as χ(h) Z(G,Λ) = e G Z(Λ), (1.5) h 0 X≥ Z(Λ) where is de(cid:12)ned as Z(Λ) = [g ] e−S(g,Λ) X e−S (g,X), µν g matter D D (1.6) Z Z S(g,Λ) = ΛV g with . S (g,X) matter in (1.1) denotes any reparametrization invariant a tion for X D onformalmatter(cid:12)elds with entral harge oupledto gravity. A typi al D X1,... ,XD example is the oupling of free s alar matter (cid:12)elds to gravity. In this ase 1 S (g,X) = d2ξ√g gµν∂ Xa∂ Xa, µ ν matter 8π ZM (1.7) whi h is di(cid:11)eomorphisminvariant and invariant under Weyl res alings of the metri : S (eφg,X) = S (g,X). matter matter (1.8) M D Note that (1.7) an also be interpreted as an embedding of in a - dimensional Eu lidean spa e, thus leading to an interpretation of (1.1) as D bosoni string theory in dimensions [6℄. [g ] µν The fun tional integration D in (1.1) is an integration over all ge- [g ] g ometries, that means all di(cid:11)eoRmorphism lasses µν of metri s µν on the M manifold . This is often denoted formally as g µν D , (1.9) ZVol(Di(cid:11)) M where Vol(Di(cid:11)) is the volume of the group of di(cid:11)eomorphisms on . Sin e this group is not ompa t the quotient (1.9) does not make sense beyond a g [g ] µν µν formal level. We will derive expressions for the measures D , D and X g D in se tion 1.2. In (1.1) the sum goes over all topologies of two-dimensional manifolds, h that means over all genera . It is presently unknown how to de(cid:12)ne su h a