Quantum Gravity
Mathematical Models and Experimental Bounds
Bertfried Fauser Jürgen Tolksdorf Eberhard Zeidler Editors
Birkhäuser Verlag Basel . Boston . Berlin Editors:
Bertfried Fauser Vladimir Rabinovich Jürgen Tolksdorf Instituto Politecnico Nacional Eberhard Zeidler ESIME Zacatenco Avenida IPN Max-Planck-Institut für Mathematik in den Mexico, D. F. 07738 Naturwissenschaften Mexico Inselstrasse 22–26 e-mail: [email protected] D-04103 Leipzig Germany e-mail: [email protected] Sergei M. Grudsky [email protected] Nikolai Vasilevski [email protected] Departamento de Matematicas CINVESTAV Apartado Postal 14-740 07000 Mexico, D.F. Mexico e-mail: [email protected] [email protected]
2000 Mathematical Subject Classification: primary 81-02; 83-02; secondary: 83C45; 81T75; 83D05; 83E30; 83F05
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9 8 7 6 5 4 3 2 1 www.birkhauser.ch CONTENTS
Preface ...... xi Bertfried Fauser, J¨urgen Tolksdorf and Eberhard Zeidler Quantum Gravity – A Short Overview...... 1 Claus Kiefer 1. Why do we need quantum gravity? 1 2. Quantum general relativity 5 2.1. Covariant approaches 5 2.2. Canonical approaches 5 2.2.1. Quantum geometrodynamics 6 2.2.2. Connection and loop variables 7 3. String theory 8 4. Loops versus strings – a few points 9 5. Quantum cosmology 10 6. Some central questions about quantum gravity 11 References 12 The Search for Quantum Gravity...... 15 Claus L¨ammerzahl 1. Introduction 15 2. The basic principles of standard physics 16 3. Experimental tests 17 3.1. Tests of the universality of free fall 17 3.2. Tests of the universality of the gravitational redshift 18 3.3. Tests of local Lorentz invariance 19 3.3.1. Constancy of c 20 3.3.2. Universality of c 21 3.3.3. Isotropy of c 21 3.3.4. Independence of c from the velocity of the laboratory 21 3.3.5. Time dilation 21 3.3.6. Isotropy in the matter sector 22 3.4. Implications for the equations of motion 22 3.4.1. Implication for point particles and light rays 22 1 3.4.2. Implication for spin– 2 particles 23 3.4.3. Implications for the Maxwell field 23 3.4.4. Summary 24 3.5. Implications for the gravitational field 24 3.6. Tests of predictions – determination of PPN parameters 25 3.6.1. Solar system effects 25 3.6.2. Strong gravity and gravitational waves 27 vi Contents
4. Unsolved problems: first hints for new physics? 27 5. On the magnitude of quantum gravity effects 29 6. How to search for quantum gravity effects 30 7. Outlook 31 Acknowledgements 32 References 32 Time Paradox in Quantum Gravity ...... 41 Alfredo Mac´ıas and Hernando Quevedo 1. Introduction 41 2. Time in canonical quantization 43 3. Time in general relativity 45 4. Canonical quantization in minisuperspace 49 5. Canonical quantization in midisuperspace 51 6. The problem of time 52 7. Conclusions 56 Acknowledgements 57 References 57 Differential Geometry in Non-Commutative Worlds ...... 61 Louis H. Kauffman 1. Introduction to non-commutative worlds 61 2. Differential geometry and gauge theory in a non-commutative world 66 3. Consequences of the metric 69 Acknowledgements 74 References 74 Algebraic Approach to Quantum Gravity III: Non-Commutative Riemannian Geometry ...... 77 Shahn Majid 1. Introduction 77 2. Reprise of quantum differential calculus 79 2.1. Symplectic connections: a new field in physics 81 2.2. Differential anomalies and the orgin of time 82 3. Classical weak Riemannian geometry 85 3.1. Cotorsion and weak metric compatibility 86 3.2. Framings and coframings 87 4. Quantum bundles and Riemannian structures 89 5. Quantum gravity on finite sets 94 6. Outlook: Monoidal functors 97 References 98 Quantum Gravity as a Quantum Field Theory of Simplicial Geometry ...... 101 Daniele Oriti 1. Introduction: Ingredients and motivations for the group field theory 101 1.1. Why path integrals? The continuum sum-over-histories approach 102 Contents vii
1.2. Why topology change? Continuum 3rd quantization of gravity 103 1.3. Why going discrete? Matrix models and simplicial quantum gravity 105 1.4. Why groups and representations? Loop quantum gravity/spin foams 107 2. Group field theory: What is it? The basic GFT formalism 109 2.1. A discrete superspace 109 2.2. The field and its symmetries 111 2.3. The space of states or a third quantized simplicial space 112 2.4. Quantum histories or a third quantized simplicial spacetime 112 2.5. The third quantized simplicial gravity action 113 2.6. The partition function and its perturbative expansion 114 2.7. GFT definition of the canonical inner product 115 2.8. Summary: GFT as a general framework for quantum gravity 116 3. An example: 3d Riemannian quantum gravity 117 4. Assorted questions for the present, but especially for the future 120 Acknowledgements 124 References 125 An Essay on the Spectral Action and its Relation to Quantum Gravity ...... 127 Mario Paschke 1. Introduction 127 2. Classical spectral triples 130 3. On the meaning of noncommutativity 134 4. NC description of the standard model: the physical intuition behind it 136 4.1. The intuitive idea: an picture of quantum spacetime at low energies 136 4.2. The postulates 138 4.3. How such a noncommutative spacetime would appear to us 139 5. Remarks and open questions 140 5.1. Remarks 140 5.2. Open problems, perspectives, more speculations 141 5.3. Comparision: intuitive picture/other approaches to Quantum Gravity143 6. Towards a quantum equivalence principle 145 6.1. Globally hyperbolic spectral triples 145 6.2. Generally covariant quantum theories over spectral geometries 147 References 149 Towards a Background Independent Formulation of Perturbative Quantum Grav- ity ...... 151 Romeo Brunetti and Klaus Fredenhagen 1. Problems of perturbative Quantum Gravity 151 2. Locally covariant quantum field theory 152 3. Locally covariant fields 155 4. Quantization of the background 158 viii Contents
References 158 Mapping-Class Groups of 3-Manifolds ...... 161 Domenico Giulini 1. Some facts about Hamiltonian general relativity 161 1.1. Introduction 161 1.2. Topologically closed Cauchy surfaces 163 1.3. Topologically open Cauchy surfaces 166 2. 3-Manifolds 169 3. Mapping class groups 172 3.1. A small digression on spinoriality 174 3.2. General Diffeomorphisms 175 4. A simple yet non-trivial example 183 4.1. The RP3 geon 183 4.2. The connected sum RP3 RP3 185 5. Further remarks on the general structure of GF(Σ) 190 6. Summary and outlook 192 Appendix: Elements of residual finiteness 193 References 197 Kinematical Uniqueness of Loop Quantum Gravity ...... 203 Christian Fleischhack 1. Introduction 203 2. Ashtekar variables 204 3. Loop variables 205 3.1. Parallel transports 205 3.2. Fluxes 205 4. Configuration space 206 4.1. Semianalytic structures 206 4.2. Cylindrical functions 207 4.3. Generalized connections 207 4.4. Projective limit 207 4.5. Ashtekar-Lewandowski measure 208 4.6. Gauge transforms and diffeomorphisms 208 5. Poisson brackets 208 5.1. Weyl operators 209 5.2. Flux derivations 209 5.3. Higher codimensions 209 6. Holonomy-flux ∗ -algebra 210 6.1. Definition 210 6.2. Symmetric state 210 6.3. Uniqueness proof 211 7. Weyl algebra 212 7.1. Definition 212 7.2. Irreducibility 213 Contents ix
7.3. Diffeomorphism invariant representation 213 7.4. Uniqueness proof 213 8. Conclusions 215 8.1. Theorem – self-adjoint case 215 8.2. Theorem – unitary case 215 8.3. Comparison 216 8.4. Discussion 216 Acknowledgements 217 References 218 Topological Quantum Field Theory as Topological Quantum Gravity...... 221 Kishore Marathe 1. Introduction 221 2. Quantum Observables 223 3. Link Invariants 224 4. WRT invariants 226 5. Chern-Simons and String Theory 227 6. Conifold Transition 228 7. WRT invariants and topological string amplitudes 229 8. Strings and gravity 232 9. Conclusion 233 Acknowledgements 234 References 234 Strings, Higher Curvature Corrections, and Black Holes...... 237 Thomas Mohaupt 1. Introduction 237 2. The black hole attractor mechanism 240 3. Beyond the area law 244 4. From black holes to topological strings 247 5. Variational principles for black holes 250 6. Fundamental strings and ‘small’ black holes 253 7. Dyonic strings and ‘large’ black holes 256 8. Discussion 258 Acknowledgements 259 References 260 The Principle of the Fermionic Projector: An Approach for Quantum Gravity? ...... 263 Felix Finster 1. A variational principle in discrete space-time 264 2. Discussion of the underlying physical principles 266 3. Naive correspondence to a continuum theory 268 4. The continuum limit 270 5. Obtained results 271 xContents
6. Outlook: The classical gravitational field 272 7. Outlook: The field quantization 274 References 280 Gravitational Waves and Energy Momentum Quanta ...... 283 Tekin Dereli and Robin W. Tucker 1. Introduction 283 2. Conserved quantities and electromagnetism 285 3. Conserved quantities and gravitation 286 4. The Bel-Robinson tensor 287 5. Wave solutions 289 6. Conclusions 291 References 292 Asymptotic Safety in Quantum Einstein Gravity: Nonperturbative Renormalizability and Fractal Spacetime Structure ...... 293 Oliver Lauscher and Martin Reuter 1. Introduction 293 2. Asymptotic safety 294 3. RG flow of the effective average action 296 4. Scale dependent metrics and the resolution function (k) 300 5. Microscopic structure of the QEG spacetimes 304 6. The spectral dimension 307 7. Summary 310 References 311 Noncommutative QFT and Renormalization ...... 315 Harald Grosse and Raimar Wulkenhaar 1. Introduction 315 2. Noncommutative Quantum Field Theory 316 3. Renormalization of φ4 -theory on the 4D Moyal plane 318 4. Matrix-model techniques 323 References 324 Index ...... 327 Preface
This Edited Volume is based on a workshop on “Mathematical and Physical As- pects of Quantum Gravity” held at the Heinrich-Fabri Institute in Blaubeuren (Germany) from July 28th to August 1st, 2005. This workshop was the succes- sor of a similar workshop held at the same place in September 2003 on the issue of “Mathematical and Physical Aspects of Quantum Field Theories”. Both work- shops were intended to bring together mathematicians and physicists to discuss profound questions within the non-empty intersection of mathematics and physics. The basic idea of this series of workshops is to cover a broad range of different approaches (both mathematical and physical) to a specific subject in mathemati- cal physics. The series of workshops is intended, in particular, to discuss the basic conceptual ideas behind different mathematical and physical approaches to the subject matter concerned. The workshop on which this volume is based was devoted to what is com- monly regarded as the biggest challenge in mathematical physics: the “quantiza- tion of gravity”. The gravitational interaction is known to be very different from the known interactions like, for instance, the electroweak or strong interaction of elementary particles. First of all, to our knowledge, any kind of energy has a gravitational coupling. Second, since Einstein it is widely accepted that gravity is intimately related to the structure of space-time. Both facts have far reaching consequences for any attempt to develop a quantum theory of gravity. For in- stance, the first fact questions our understanding of “quantization” as it has been developed in elementary particle physics. In fact, this understanding is very much related to the “quantum of energy” encountered in the concept of photons in the quantum theory of electromagnetism. However, in Einstein’s theory of gravity the gravitational field does not carry (local) energy. While general relativity is a local theory, the notion of gravitational energy is still a “global” issue which, however, is not yet well-defined even within the context of classical gravity. The second fact seems to clearly indicate that a quantum theory of gravity will radically chance our ideas about the structure of space-time. It is thus supposed that a quantum version of gravity is deeply related to our two basic concepts: “quantum” and “space-time” which have been developed so successfully over the last one-hundred years. The idea of the second workshop was to provide a forum to discuss differ- ent approaches to a possible theory of quantum gravity. Besides the two major accepted roads provided by String Theory and Loop Quantum Gravity, also other ideas were discussed, like those, for instance, based on A. Connes’ non-commutative xii Preface geometry. Also, possible experimental evidence of a quantum structure of gravity was discussed. However, it was not intended to cover the latest technical results but instead to summarize some of the basic features of the existing ans¨atze to for- mulate a quantum theory of gravity. The present volume provides an appropriate cross-section of the discussion. The refereed articles are written with the intention to bring together experts working in different fields in mathematics and physics and who are interested in the subject of quantum gravity. The volume provides the reader with some overview about most of the accepted approaches to develop a quantum gravity theory. The articles are purposely written in a less technical style than usual and are mainly intended to discuss the major questions related to the subject of the workshop. Since this volume covers rather different perspectives, the editors thought it might be helpful to start the volume by providing a brief summary of each of the various articles. Obviously, such a summary will necessarily reflect the editors’ understanding of the subject matter.
Thevolumestartswithanoverview,presentedbyClaus Kiefer,onthemain roads towards a quantum theory of relativity. The chapter is nontechnical and comprehensibly written. The author starts his article with a brief motivation why there is a need to consider a quantum theory of gravity. Next, he discusses several aspects of the different approaches presented in this volume. For instance, he con- trasts background independent approaches with background dependent theories. The chapter closes with a brief summary of some of the main results obtained so far to achieve a quantum version of gravity. In the second article Claus L¨ammerzahl reports on the experimental status of quantum gravity effects. On the one hand, gravity is assumed to be “universal”. On the other hand, quantum theory is regarded as being “fundamental”. As a con- sequence, one should expect that a quantum theory of gravity will yield corrections to any physical process. Recent experiments, however, confirm with high precision the theory of relativity and quantum theory. Nonetheless, the author indicates how astrophysical as well as laboratory and satelite experiments may be improved in accuracy so that possible quantum gravity effects could be observed not too far in the future. L¨ammerzahl describes at which scales and parameter ranges it could be more promising to push forward experimental efforts. The article closes with a discussion on recent proposals to increase experimental accuracy. A wealth of information is presented in a very readable style.
In their contribution the authors Alfredo Macias and Hernando Quevedo review in a very precise and compelling way the role of time in the process of (canonical) quantization. They discuss different approaches to solve the so-called “time paradox”. The different conclusions drawn from their analysis imply that, after 70 years of attempts to quantize gravity, the fundamental “problem of time” is still an unresolved and fascinating issue. Preface xiii
In the search for quantum gravity the approach proposed by Louis Kauffman assumes non-commutative variables. In his contribution the author considers a non-commutative description of the world from an operational point of view. He introduces in a fascinating and straightforward approach a differential calculus that permits rephrasing some of the most basic notions known from classical differential geometry without the use of smooth manifolds. This includes, especially, the notion of Riemannian curvature, which is less simple to algebraically rephrase than the notion of Yang-Mills curvature of ordinary gauge theories. The discussion presented in this article should be contrasted with the contributions by Majid and Paschke as well as with the ideas presented by Grosse and Wulkenhaar concerning a non- commutative quantum field theory. In contrast to the approach by Kauffman, the contribution to this volume by Shahn Majid uses the framework of Hopf algebras and (bi)covariant calculi on the former to address the problem of quantum gravity within an elaborated algebraic framework. The author starts by presenting the basic mathematical material in order to afterwards discuss a whole series of examples. These examples provide the reader with an introduction to a possible quantum theory of gravity on finite sets. The outlook of the article addresses further generalizations toward a purely functorial setting and a statement about the author’s viewpoint on why this is needed to put quantum theory and gravity into a single framework. The author concludes his contribution with a number of remarks concerning some links to several other contributions of this volume. Group field theory is an elaborated extension of Penrose’s spin networks and spin foams. Daniele Oriti critically describes the challenges and achievements of group field theory. It seems possible that this way of generalized loop quan- tum gravity provides a richer framework that permits to handle problems like the Hamiltonian constraint. The author puts emphasis on the viewpoint that group field theory should be regarded as being a theoretical framework of its own. Alain Connes’ non-commutative geometry may provide an alternative ap- proach to a quantum theory of gravity. In his contribution Mario Paschke gives an overview on the present status of this approach. The author focusses his at- tention on the role of the so-called “spectral action”. In particular, he critically discusses the need to extend non-commutative geometry to Lorentzian signature and to study globally hyperbolic spectral triples. This viewpoint may provide a Lorentzian covariant and hence a more physically convincing approach to a non- commutative generalization of gravity. In this respect the article is closely related to the ideas concerning a covariant description of a perturbative quantum field theory as it is proposed by Brunetti and Fredenhagen in the next contribution. Romeo Brunetti and Klaus Fredenhagen propose a certain background inde- pendent axiomatic formulation of perturbative quantum gravity. This formulation is based on a functorial mapping from the category of globally hyperbolic mani- folds to the category of ∗ -algebras. As explained in some details, the axioms for xiv Preface this functor are physically well motivated and permit to consider a quantum field as a fundamental local observable. A major question of interest is the study of representations of the diffeomor- phism group for any diffeomorphism invariant theory, like general relativity. In the case of globally hyperbolic space-times, these representations are related to the topology of spatially closed orientable 3-manifolds. One expects that a quan- tum theory of gravity would yield a super-selection structure that is induced by the topology of the classical (limiting) space. In his contribution to this volume, Domenico Giulini provides a well written introduction to this fascinating topic. He puts emphasis on a geometrical understanding of the mapping-class group of 3-manifolds and includes many illuminating pictures for illustration. Next, Christian Fleischhack studies uniqueness theorems in loop quantum gravity analogous to the famous Stone – von Neumann theorem of ordinary quan- tum mechanics that guarantees the unitary equivalence of all the irreducible repre- sentations of the Heisenberg algebra. Due to the tremendously more complicated configuration space of loop quantum gravity, it is of utmost importance to know whether different quantization schemes may give rise to different physical pre- dictions. Fleischhack’s discussion of two uniqueness theorems proves that, under certain technical assumptions, an almost unique quantization procedure can be obtained for Ashtekar’s formulation of a quantum gravity. String theory is known to naturally include a spin-two field. When quantized, this field is commonly interpreted as graviton analogous to the photon in quantum electrodynamics. A basic object in any string theoretical formulation of a quan- tum theory of gravity plays the partition function defined in terms of appropriate functional integrals. A perturbative evaluation of the partition functions yields topological invariants of the background manifolds under consideration. Kishore Marathe discusses several aspects of the interplay between topological quantum field theory and quantum gravity. For instance, he discusses the Jones polynomial and other related knot invariants of low-dimensional smooth manifolds. A rather different route to quantum gravity is proposed by Felix Finster.His “principle of the fermionic projector” summarizes the idea to start the formulation of a quantum theory of gravity from a set of points, a certain set of projectors re- lated to these points and a discrete variational principle. The author summarizes the basic ideas how gravity and the gauge theory may be formulated within the framework presented in his contribution. Contrary to the common belief, the au- thor considers locality and causality as fundamental notions only in the continuum limit of “quantum space-time”. Black holes are models of actual astrophysical effects involving strong gravi- tational fields. Hence, black holes are optimally suited as a theoretical laboratory for quantum gravity. In his article, Thomas Mohaupt deals with a string theo- retical approach to black hole physics. The usual black hole quantum theory is heuristic and needs to be supported by a microscopic (statistical) theory. Formal Preface xv arguments from string theory permit possible scenarios to construct densities of states that give rise to a statistical definition of entropy. Furthermore, Mohaupt’s article demonstrates how effects even next to the leading order can be given a satisfactory explanation by the identification of different statistical ensembles.
Quantum mechanics originated in the attempt to understand experimental results which were in sharp contrast with Maxwell’s electrodynamics. In this re- spect, one of the most crucial experimental effects was what is called today the “photon electric effect”. Together with the black body radiation, the photon elec- tric effect may be considered as the birth of the idea of the “quantum of (electro- magnetic) energy”. This idea, in turn, is known to have been fundamental for the development of the quantum theory of Maxwell’s electrodynamics. The authors Tekin Dereli and Robin W. Tucker start out from the question of whether there is a similar effect related to the “quantum of gravitational energy”. Analogous to electrodynamics one may look for a Hamiltonian that incorporates the energy of the classical gravitational field. In Einstein’s theory of gravity this is known to be a non-trivial task. For this the authors introduce a different Lagrangian density which includes additional degrees of freedom and from which they derive an en- ergy momentum tensor of the gravitational field. Moreover, the authors discuss specific gravitational (plane) wave like solutions of their generalized gravitational field equations which may be regarded as being similar to the electromagnetic plane waves in ordinary electromagnetism.
General relativity is known to be a perturbatively non-renormalizable theory. Such a theory needs the introduction of infinitely many free parameters (“counter terms”), which seems to spoil any predictive power of the corresponding quantum theory. Using the renormalization group Oliver Lauscher and Martin Reuter dis- cuss the existence of non-Gaussian fixed points of the renormalization group flow such that the number of counter terms can be restricted to a finite number. Such a scenario can be obtained from numerical techniques called “asymptotic safety”. Employing techniques from random walks one can show that a scale dependent effective theory which probes the nature of space-time on that particular scale can be obtained. The renormalization group trajectories permit discussing space-time properties on changing scales. While for large scales a smooth four dimensional manifold occurs, at small scales a fractal space-time of dimension two is obtained. Similar results are obtained using the idea of numerical dynamical triangulation which has been introduced by other groups.
The idea to change the structure of space-time at small distances in order to cure divergence problems in ordinary quantum field theory was introduced by Heisenberg and Schr¨odinger and was firstly published by Snyder. Recent devel- opments concerning D-branes in string theory also support such a scenario. Yet another approach to a not point-like structure of space-time has been introduced by Dopplicher, Fredenhagen and Roberts in the 1990’s. It is based on the idea to also obtain an uncertainty principle for the configuration space similar to the xvi Preface known phase space uncertainty of ordinary quantum mechanics. Harald Grosse and Raimar Wulkenhaar close this volume by a summary of their pioneering work on the construction of a specific model of a renormalizable quantum field theory on such a so-called “ θ− deformed” space-time. They also describe the relation of their model to multi-scale matrix models.
Acknowledgements It is a great pleasure for the editors to thank all of the participants of the workshop for their contributions, which have made the workshop so successful. We would like to express our gratitude to the staff of the Max Planck Institute, especially to Regine L¨ubke, who managed the administrative work excellently. The editors would like to thank the German Science Foundation (DFG) and the Max Planck Institute for Mathematics in the Sciences in Leipzig (Germany) for their generous financial support. Furthermore, they would also like to thank Thomas Hempfling from Birkh¨auser for the excellent cooperation.
Bertfried Fauser, J¨urgen Tolksdorf and Eberhard Zeidler Leipzig, October 1, 2006 Quantum Gravity B. Fauser, J. Tolksdorf and E. Zeidler, Eds., 1–13 c 2006 Birkh¨auser Verlag Basel/Switzerland
Quantum Gravity — A Short Overview
Claus Kiefer
Abstract. The main problems in constructing a quantum theory of gravity as well as some recent developments are briefly discussed on a non-technical level. Mathematics Subject Classification (2000). Primary 83-02; Secondary 83C45; 83C47; 83F05; 83E05; 83E30. Keywords. Quantum gravity, string theory, quantum geometrodynamics, loop quantum gravity, black holes, quantum cosmology, experimental tests.
1. Why do we need quantum gravity? The task to formulate a consistent quantum theory of gravity has occupied physi- cists since the first attempts by L´eon Rosenfeld in 1930. Despite much work it is fair to say that this goal has not yet been reached. In this short contribution I shall attempt to give a concise summary of the situation in 2005 from my point of view. The questions to be addressed are: Why is this problem of interest? What are the main difficulties? And where do we stand? A comprehensive technical treatment can be found in my monograph [1] as well as in the Proceedings volumes [2] and [3]. A more detailed review with focus on recent developments is presented in [4]. The reader is referred to these sources for details and references. Why should one be interested in developing a quantum theory of the gravi- tational field? The main reasons are conceptual. The famous singularity theorems show that the classical theory of general relativity is incomplete: Under very gen- eral conditions, singularities are unavoidable. Such singularities can be rather mild, that is, of a purely topological nature, but they can also consist of diverging cur- vatures and energy densities. In fact, the latter situation seems to be realized in two important physical cases: The Big Bang and black holes. The presence of the cosmic microwave background (CMB) radiation indicates that a Big Bang has happened in the past. Curvature singularities seem to lurk inside the event horizon of black holes. One thus needs a more comprehensive theory to understand these
I thank Bertfried Fauser and J¨urgen Tolksdorf for inviting me to this stimulating workshop. 2 Claus Kiefer situations. The general expectation is that a quantum theory of gravity is needed, in analogy to quantum mechanics in which the classical instability of atoms has disappeared. The origin of our universe cannot be described without such a theory, so cosmology remains incomplete without quantum gravity. Because of its geometric nature, gravity interacts with all forms of energy. Since all non-gravitational degrees of freedom are successfully described by quan- tum fields, it would seem natural that gravity itself is described by a quantum theory. It is hard to understand how one should construct a unified theory of all interactions in a hybrid classical–quantum manner. In fact, all attempts to do so have up to now failed. A theory of gravity is also a theory of spacetime. Quantum gravity should thus make definite statements about the behaviour of spacetime at the smallest scales. For this reason it has been speculated long ago that the inclusion of gravity can avoid the divergences that plague ordinary quantum field theories. These di- vergences arise from the highest momenta and thus from the smallest scales. This speculation is well motivated. Non-gravitational field theories are given on a fixed background spacetime, that is, on a non-dynamical structure. Quantum gravity ad- dresses the background itself. One thus has to seek for a background-independent formulation. If the usual divergences have really to do with the smallest scales of the background spacetime, they should disappear together with the background. A particular aspect of background independence is the ‘problem of time’, which should arise in any approach to quantum gravity, [1, 2, 6]. On the one hand, time is external in ordinary quantum theory; the parameter t in the Schr¨odinger equation, ∂ψ i = Hψ , (1) ∂t is identical to Newton’s absolute time — it is not turned into an operator and is presumed to be prescribed from the outside. This is true also in special relativity where the absolute time t is replaced by Minkowski spacetime, which is again an absolute structure. On the other hand, time (as part of spacetime) is dynamical in general relativity. The implementation of gravity into the quantum framework should thus entail the disappearance of absolute time. A major obstacle in the search for quantum gravity is the lack of experimental data. This is connected with the smallness of the scales that are connected with such a theory: Combining the gravitational constant, G, the speed of light, c,and the quantum of action, , into units of length, time, and mass, respectively, one arrives at the famous Planck units, G ≈ × −33 lP = 3 1.62 10 cm , (2) c G ≈ × −44 tP = 5 5.40 10 s , (3) c c m = ≈ 2.17 × 10−5 g ≈ 1.22 × 1019 GeV . (4) P G Quantum Gravity — A Short Overview 3
To probe, for example, the Planck length with contemporary accelerators one would have to built a machine of the size of our Milky Way. Direct observations are thus expected to come mainly from the astrophysical side — the early universe and black holes. In addition there may of course occur low-energy effects such as a small violation of the equivalence principle [5, 3]. The search for such effects is often called ‘quantum gravity phenomenology’. The irrelevance of quantum gravity in usual astrophysical investigations can be traced back to the huge discrepancy between the Planck mass and the proton mass: It is the small constant 2 2 Gmpr mpr −39 αg = = ≈ 5.91 × 10 , (5) c mP where mpr denotes the proton mass, that enters astrophysical quantities of interest such as stellar masses and stellar lifetimes. On the road towards quantum gravity it is important to study all levels of interaction between quantum systems and the gravitational field. The first level, where experiments exist, concerns the level of quantum mechanical systems inter- acting with Newtonian gravity [7]. The next level is quantum field theory on a given (usually curved) background spacetime. Here one has a specific prediction: Black holes radiate with a temperature proportional to (‘Hawking radiation’),