This page intentionally left blank APPROACHES TO QUANTUM GRAVITY Toward a New Understanding of Space, Time and Matter
The theory of quantum gravity promises a revolutionary new understanding of gravity and spacetime, valid from microscopic to cosmological distances. Research in this field involves an exciting blend of rigorous mathematics and bold speculations, foundational questions and technical issues. Containing contributions from leading researchers in this field, this book presents the fundamental issues involved in the construction of a quantum theory of gravity and building up a quantum picture of space and time. It introduces the most current approaches to this problem, and reviews their main achievements. Each part ends in questions and answers, in which the contributors explore the merits and problems of the various approaches. This book provides a complete overview of this field from the frontiers of theoretical physics research for graduate students and researchers.
DANIELE ORITI is a Researcher at the Max Planck Institute for Gravitational Physics, Potsdam, Germany, working on non-perturbative quantum gravity. He has previously worked at the Perimeter Institute for Theoretical Physics, Canada; the Institute for Theoretical Physics at Utrecht University, The Netherlands; and the Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK. He is well known for his results on spin foam models, and is among the leading researchers in the group field theory approach to quantum gravity.
APPROACHES TO QUANTUM GRAVITY Toward a New Understanding of Space, Time and Matter
Edited by
DANIELE ORITI Max Planck Institute for Gravitational Physics, Potsdam, Germany CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521860451 © Cambridge University Press 2009
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Contents
List of contributors page x Preface xv
Part I Fundamental ideas and general formalisms 1 1 Unfinished revolution 3 C. Rovelli 2 The fundamental nature of space and time 13 G. ’t Hooft 3 Does locality fail at intermediate length scales? 26 R. D. Sorkin 4 Prolegomena to any future Quantum Gravity 44 J. Stachel 5 Spacetime symmetries in histories canonical gravity 68 N. Savvidou 6 Categorical geometry and the mathematical foundations of Quantum Gravity 84 L. Crane 7 Emergent relativity 99 O. Dreyer 8 Asymptotic safety 111 R. Percacci 9 New directions in background independent Quantum Gravity 129 F. Markopoulou Questions and answers 150
Part II String/M-theory 167 10 Gauge/gravity duality 169 G. Horowitz and J. Polchinski
vii viii Contents 11 String theory, holography and Quantum Gravity 187 T. Banks 12 String field theory 210 W. Taylor Questions and answers 229
Part III Loop quantum gravity and spin foam models 233 13 Loop quantum gravity 235 T. Thiemann 14 Covariant loop quantum gravity? 253 E. Livine 15 The spin foam representation of loop quantum gravity 272 A. Perez 16 Three-dimensional spin foam Quantum Gravity 290 L. Freidel 17 The group field theory approach to Quantum Gravity 310 D. Oriti Questions and answers 332
Part IV Discrete Quantum Gravity 339 18 Quantum Gravity: the art of building spacetime 341 J. Ambjørn, J. Jurkiewicz and R. Loll 19 Quantum Regge calculus 360 R. Williams 20 Consistent discretizations as a road to Quantum Gravity 378 R. Gambini and J. Pullin 21 The causal set approach to Quantum Gravity 393 J. Henson Questions and answers 414
Part V Effective models and Quantum Gravity phenomenology 425 22 Quantum Gravity phenomenology 427 G. Amelino-Camelia 23 Quantum Gravity and precision tests 450 C. Burgess 24 Algebraic approach to Quantum Gravity II: noncommutative spacetime 466 S. Majid Contents ix 25 Doubly special relativity 493 J. Kowalski-Glikman 26 From quantum reference frames to deformed special relativity 509 F. Girelli 27 Lorentz invariance violation and its role in Quantum Gravity phenomenology 528 J. Collins, A. Perez and D. Sudarsky 28 Generic predictions of quantum theories of gravity 548 L. Smolin Questions and answers 571 Index 580 Contributors
J. Ambjørn The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark and Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, NL-3584 CE Utrecht, The Netherlands G. Amelino-Camelia Dipartimento di Fisica, Universitá di Roma “La Sapienza”, P.le A. Moro 2, 00185 Rome, Italy T. Banks Department of Physics, University of California, Santa Cruz, CA 95064, USA and NHETC, Rutgers University, Piscataway, NJ 08854, USA C. Burgess Department of Physics & Astronomy, McMaster University, 1280 Main St. W, Hamilton, Ontario, Canada, L8S 4M1 and Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo N2L 2Y5, Ontario, Canada J. Collins Physics Department, Pennsylvania State University, University Park, PA 16802, USA L. Crane Mathematics Department, Kansas State University, 138 Cardwell Hall Manhattan, KS 66506-2602, USA
x List of contributors xi O. Dreyer Theoretical Physics, Blackett Laboratory, Imperial College London, London, SW7 2AZ, UK L. Freidel Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo N2L 2Y5, Ontario, Canada R. Gambini Instituto de Física, Facultad de Ciencias, Iguá 4225, Montevideo, Uruguay F. Girelli SISSA, via Beirut 4, Trieste, 34014, Italy, and INFN, sezione di Trieste, Italy J. Henson Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, NL-3584 CE Utrecht, The Netherlands G. Horowitz Physics Department, University of California, Santa Barbara, CA 93106, USA J. Jurkiewicz Institute of Physics, Jagellonian University, Reymonta 4, PL 30-059 Krakow, Poland J. Kowalski-Glikman Institute for Theoretical Physics, University of Wroclaw 50-204 Wroclaw, pl. M. Borna 9, Poland E. Livine Ecole Normale Supérieure de Lyon, 46 Allée d’Italie, 69364 Lyon Cedex 07, France R. Loll Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, NL-3584 CE Utrecht, The Netherlands S. Majid School of Mathematical Sciences, Queen Mary, University of London 327 Mile End Rd, London E1 4NS, UK and Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo ON N2L 2Y5, Canada F. Markopoulou Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo ON N2L 2Y5, Canada xii List of contributors
D. Oriti Max Planck Institute for Gravitational Physics, Am Mühlenberg 1, D 14476 Golm, Germany R. Percacci SISSA, via Beirut 4, Trieste, 34014, Italy, and INFN, sezione di Trieste, Italy A. Perez Centre de Physique Théorique, Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix-Marseille I, Aix-Marseille II, et du Sud Toulon-Var, laboratoire afilié à la FRUMAM (FR 2291), Campus de Luminy, 13288 Marseille, France J. Polchinski Department of Physics, University of California, Santa Barbara CA 93106, USA J. Pullin Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803 USA C. Rovelli Centre de Physique Théorique, Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix-Marseille I, Aix-Marseille II, et du Sud Toulon-Var, laboratoire afilié à la FRUMAM (FR 2291), Campus de Luminy, 13288 Marseille, France N. Savvidou Theoretical Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK L. Smolin Perimeter Institute for Theoretical Physics, Waterloo N2J 2W9, Ontario, Canada and Department of Physics, University of Waterloo, Waterloo N2L 3G1, Ontario, Canada R. D. Sorkin Perimeter Institute for Theoretical Physics, Waterloo N2J 2W9, Ontario, Canada J. Stachel CAS Physics, Boston University, 745 Commonwealth Avenue, MA 02215, USA D. Sudarsky Instituto de Ciencias Nucleares, Universidad Autónoma de México, A. P. 70-543, México D.F. 04510, México List of contributors xiii W. Taylor Massachusetts Institute of Technology, Lab for Nuclear Science and Center for Theoretical Physics, 77 Massachusetts Ave., Cambridge, MA 02139-4307, USA T. Thiemann Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, D-14476 Golm, Germany and Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo N2L 2Y5, Ontario, Canada G. ’t Hooft Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, NL-3584 CE Utrecht, The Netherlands R. Williams Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Preface
Quantum Gravity is a dream, a theoretical need and a scientific goal. It is a theory which still does not exist in complete form, but that many people claim to have had glimpses of, and it is an area of research which, at present, comprises the collective efforts of hundreds of theoretical and mathematical physicists. This yet-to-be-found theory promises to be a more comprehensive and com- plete description of the gravitational interaction, a description that goes beyond Einstein’s General Relativity in being possibly valid at all scales of distances and energy; at the same time it promises to provide a new and deeper understanding of the nature of space, time and matter. As such, research in Quantum Gravity is a curious and exciting blend of rig- orous mathematics and bold speculations, concrete models and general schemata, foundational questions and technical issues, together with, since recently, tentative phenomenological scenarios. In the past three decades we have witnessed an amazing growth of the field of Quantum Gravity, of the number of people actively working in it, and consequently of the results achieved. This is due to the fact that some approaches to the prob- lem started succeeding in solving outstanding technical challenges, in suggesting ways around conceptual issues, and in providing new physical insights and scenar- ios. A clear example is the explosion of research in string theory, one of the main candidates to a quantum theory of gravity, and much more. Another is the develop- ment of Loop Quantum Gravity, an approach that attracted much attention recently, due to its successes in dealing with many long standing problems of the canonical approach to Quantum Gravity. New techniques have been then imported to the field from other areas of theoretical physics, e.g. Lattice Gauge Theory, and influenced in several ways the birth or growth of even more directions in Quantum Gravity research, including for example discrete approaches. At the same time, Quantum Gravity has been a very fertile ground and a powerful motivation for developing
xv xvi Preface new mathematics as well as alternative ways of thinking about spacetime and mat- ter, which in turn have triggered the exploration of other promising avenues toward a Quantum Gravity theory. I think it is fair to say that we are still far from having constructed a satis- factory theory of Quantum Gravity, and that any single approach currently being considered is too incomplete or poorly understood, whatever its strengths and suc- cesses may be, to claim to have achieved its goal, or to have proven to be the only reasonable way to proceed. On the other hand every single one of the various approaches being pursued has achieved important results and insights regarding the Quantum Gravity problem. Moreover, technical or conceptual issues that are unsolved in one approach have been successfully tackled in another, and often the successes of one approach have clearly come from looking at how similar difficulties had been solved in another. It is even possible that, in order to achieve our common goal, formulate a com- plete theory of Quantum Gravity and unravel the fundamental nature of space and time, we will have to regard (at least some of) these approaches as different aspects of the same theory, or to develop a more complete and more general approach that combines the virtues of several of them. However strong faith one may have in any of these approaches, and however justified this may be in light of recent results, it should be expected, purely on historical grounds, that none of the approaches currently pursued will be understood in the future in the same way as we do now, even if it proves to be the right way to proceed. Therefore, it is useful to look for new ideas and a different perspective on each of them, aided by the the insights provided by the others. In no area of research a “dogmatic approach” is less produc- tive, I feel, than in Quantum Gravity, where the fundamental and complex nature of the problem, its many facets and long history, combined with a dramatically (but hopefully temporarily) limited guidance from Nature, suggest a very open-minded attitude and a very critical and constant re-evaluation of one’s own strategies. I believe, therefore, that a broad and well-informed perspective on the vari- ous present approaches to Quantum Gravity is a necessary tool for advancing successfully in this area. This collective volume, benefiting from the contributions of some of the best Quantum Gravity practitioners, all working at the frontiers of current research, is meant to represent a good starting point and an up-to-date support reference, for both students and active researchers in this fascinating field, for developing such a broader perspective. It presents an overview of some of the many ideas on the table, an introduction to several current approaches to the construction of a Quantum The- ory of Gravity, and brief reviews of their main achievements, as well as of the many outstanding issues. It does so also with the aim of offering a comparative perspec- tive on the subject, and on the different roads that Quantum Gravity researchers Preface xvii are following in their searches. The focus is on non-perturbative aspects of Quan- tum Gravity and on the fundamental structure of space and time. The variety of approaches presented is intended to ensure that a variety of ideas and mathematical techniques will be introduced to the reader. More specifically, the first part of the book (Part I) introduces the problem of Quantum Gravity, and raises some of the fundamental questions that research in Quantum Gravity is trying to address. These concern for example the role of local- ity and of causality at the most fundamental level, the possibility of the notion of spacetime itself being emergent, the possible need to question and revise our way of understanding both General Relativity and Quantum Mechanics, before the two can be combined and made compatible in a future theory of Quantum Gravity. It provides as well suggestions for new directions (using the newly available tools of category theory, or quantum information theory, etc.) to explore both the construc- tion of a quantum theory of gravity, as well as our very thinking about space and time and matter. The core of the book (Parts II–IV) is devoted to a presentation of several approaches that are currently being pursued, have recently achieved important results, and represent promising directions. Among these the most developed and most practiced are string/M-theory, by far the one which involves at present the largest amount of scholars, and loop quantum gravity (including its covariant version, i.e. spin foam models). Alongside them, we have various (and rather dif- ferent in both spirit and techniques used) discrete approaches, represented here by simplicial quantum gravity, in particular the recent direction of causal dynam- ical triangulations, quantum Regge calculus, and the “consistent discretization scheme”, and by the causal set approach. All these approaches are presented at an advanced but not over-technical level, so that the reader is offered an introduction to the basic ideas characterizing any given approach as well as an overview of the results it has already achieved and a perspective on its possible development. This overview will make manifest the variety of techniques and ideas currently being used in the field, ranging from con- tinuum/analytic to discrete/combinatorial mathematical methods, from canonical to covariant formalisms, from the most conservative to the most radical conceptual settings. The final part of the book (Part V) is devoted instead to effective models of Quantum Gravity. By this we mean models that are not intended to be of a fun- damental nature, but are likely to provide on the one hand key insights on what sort of features the more fundamental formulation of the theory may possess, and on the other powerful tools for studying possible phenomenological consequences of any Quantum Gravity theory, the future hopefully complete version as well xviii Preface as the current tentative formulations of it. The subject of Quantum Gravity phe- nomenology is a new and extremely promising area of current research, and gives ground to the hope that in the near future Quantum Gravity research may receive experimental inputs that will complement and direct mathematical insights and constructions. The aim is to convey to the reader the recent insight that a Quantum Gravity theory need not be forever detached by the experimental realm, and that many possibilities for a Quantum Gravity phenomenology are instead currently open to investigation. At the end of each part, there is a “Questions & Answers” session. In each of them, the various contributors ask and put forward to each other questions, com- ments and criticisms to each other, which are relevant to the specific topic covered in that part. The purpose of these Q&A sessions is fourfold: (a) to clarify further subtle or particularly relevant features of the formalisms or perspectives presented; (b) to put to the forefront critical aspects of the various approaches, including potential difficulties or controversial issues; (c) to give the reader a glimpse of the real-life, ongoing debates among scholars working in Quantum Gravity, of their different perspectives and of (some of) their points of disagreement; (d) in a sense, to give a better picture of how science and research (in particular, Quantum Gravity research) really work and of what they really are. Of course, just as the book as a whole cannot pretend to represent a com- plete account of what is currently going on in Quantum Gravity research, these Q&A sessions cannot really be a comprehensive list of relevant open issues nor a faithful portrait of the (sometimes rather heated) debate among Quantum Gravity researchers. What this volume makes manifest is the above-mentioned impressive develop- ment that occurred in the field of Quantum Gravity as a whole, over the past, say, 20–30 years. This is quickly recognized, for example, by comparing the range and content of the following contributed papers to the content of similar collective vol- umes, like Quantum Gravity 2: a second Oxford symposium, C. Isham, ed., Oxford University Press (1982), Quantum structure of space and time, M. Duff, C. Isham, eds., Cambridge University Press (1982), Quantum Theory of Gravity, essays in honor of the 60th Birthday of Bryce C DeWitt,S.D.Christensen,ed.,Taylorand Francis (1984), or even the more recent Conceptual problems of Quantum Grav- ity, A. Ashtekar, J. Stachel, eds., Birkhauser (1991), all presenting overviews of the status of the subject at their time. Together with the persistence of the Quan- tum Gravity problem itself, and of the great attention devoted, currently just as then, to foundational issues alongside the more technical ones, it will be impos- sible not to notice the greater variety of current approaches, the extent to which researchers have explored beyond the traditional ones, and, most important, the Preface xix enormous amount of progress and achievements in each of them. Moreover, the very existence of research in Quantum Gravity phenomenology was un-imaginable at the time. Quantum Gravity remains, as it was in that period, a rather esoteric subject, within the landscape of theoretical physics at large, but an active and fascinating one, and one of fundamental significance. The present volume is indeed a collective report from the frontiers of theoretical physics research, reporting on the latest and most exciting developments but also trying to convey to the reader the sense of intellectual adventure that working at such frontiers implies.
It is my pleasure to thank all those that have made the completion of this project possible. First of all, I gratefully thank all the researchers who have contributed to this volume, reporting on their work and on the work of their colleagues in such an excellent manner. This is a collective volume, and thus, if it has any value, it is solely due to all of them. Second, I am grateful to all the staff at the Cambridge University Press, and in particular to Simon Capelin, for supporting this project since its conception, and for guiding me through its development. Last, I would like to thank, for very useful comments, suggestions and advice, several colleagues and friends: John Baez, Fay Dowker, Sean Hartnoll, Chris Isham, Prem Kumar, Pietro Massignan, and especially Ted Jacobson.
Daniele Oriti
Part I Fundamental ideas and general formalisms
1 Unfinished revolution
C. ROVELLI
One hundred and forty-four years elapsed between the publication of Copernicus’s De Revolutionibus, which opened the great scientific revolution of the seventeenth century, and the publication of Newton’s Principia, the final synthesis that brought that revolution to a spectacularly successful end. During those 144 years, the basic grammar for understanding the physical world changed and the old picture of reality was reshaped in depth. At the beginning of the twentieth century, General Relativity (GR) and Quantum Mechanics (QM) once again began reshaping our basic understanding of space and time and, respectively, matter, energy and causality – arguably to a no lesser extent. But we have not been able to combine these new insights into a novel coherent synthesis, yet. The twentieth-century scientific revolution opened by GR and QM is therefore still wide open. We are in the middle of an unfinished scientific rev- olution. Quantum Gravity is the tentative name we give to the “synthesis to be found”. In fact, our present understanding of the physical world at the fundamental level is in a state of great confusion. The present knowledge of the elementary dynam- ical laws of physics is given by the application of QM to fields, namely Quantum Field Theory (QFT), by the particle-physics Standard Model (SM), and by GR. This set of fundamental theories has obtained an empirical success nearly unique in the history of science: so far there isn’t any clear evidence of observed phenom- ena that clearly escape or contradict this set of theories – or a minor modification of the same, such as a neutrino mass or a cosmological constant.1 But, the theories in this set are based on badly self-contradictory assumptions. In GR the gravitational field is assumed to be a classical deterministic dynamical field, identified with the (pseudo) Riemannian metric of spacetime: but with QM we have understood that all dynamical fields have quantum properties. The other way around, conventional
1 Dark matter (not dark energy) might perhaps be contrary evidence.
Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, ed. Daniele Oriti. Published by Cambridge University Press. c Cambridge University Press 2009. 4 C. Rovelli QFT relies heavily on global Poincaré invariance and on the existence of a non-dynamical background spacetime metric: but with GR we have understood that there is no such non-dynamical background spacetime metric in nature. In spite of their empirical success, GR and QM offer a schizophrenic and con- fused understanding of the physical world. The conceptual foundations of classical GR are contradicted by QM and the conceptual foundation of conventional QFT are contradicted by GR. Fundamental physics is today in a peculiar phase of deep conceptual confusion. Some deny that such a major internal contradiction in our picture of nature exists. On the one hand, some refuse to take QM seriously. They insist that QM makes no sense, after all, and therefore the fundamental world must be essentially classical. This doesn’t put us in a better shape, as far as our understanding of the world is concerned. Others, on the other hand, and in particular some hard-core particle physicists, do not accept the lesson of GR. They read GR as a field theory that can be consistently formulated in full on a fixed metric background, and treated within conventional QFT methods. They motivate this refusal by insisting than GR’s insight should not be taken too seriously, because GR is just a low-energy limit of a more fundamen- tal theory. In doing so, they confuse the details of the Einstein’s equations (which might well be modified at high energy), with the new understanding of space and time brought by GR. This is coded in the background independence of the funda- mental theory and expresses Einstein’s discovery that spacetime is not a fixed back- ground, as was assumed in special relativistic physics, but rather a dynamical field. Nowadays this fact is finally being recognized even by those who have long refused to admit that GR forces a revolution in the way to think about space and time, such as some of the leading voices in string theory. In a recent interview [1], for instance, Nobel laureate David Gross says: “ [...] this revolution will likely change the way we think about space and time, maybe even eliminate them com- pletely as a basis for our description of reality”. This is of course something that has been known since the 1930s [2] by anybody who has taken seriously the prob- lem of the implications of GR and QM. The problem of the conceptual novelty of GR, which the string approach has tried to throw out of the door, comes back by the window. These and others remind me of Tycho Brahe, who tried hard to conciliate Coper- nicus’s advances with the “irrefutable evidence” that the Earth is immovable at the center of the universe. To let the background spacetime go is perhaps as difficult as letting go the unmovable background Earth. The world may not be the way it appears in the tiny garden of our daily experience. Today, many scientists do not hesitate to take seriously speculations such as extra dimensions, new symmetries or multiple universes, for which there isn’t a Unfinished revolution 5 wit of empirical evidence; but refuse to take seriously the conceptual implications of the physics of the twentieth century with the enormous body of empirical evi- dence supporting them. Extra dimensions, new symmetries, multiple universes and the like, still make perfectly sense in a pre-GR, pre-QM, Newtonian world, while to take GR and QM seriously together requires a genuine reshaping of our world view. After a century of empirical successes that have equals only in Newton’s and Maxwell’s theories, it is time to take seriously GR and QM, with their full con- ceptual implications; to find a way of thinking the world in which what we have learned with QM and what we have learned with GR make sense together – finally bringing the twentieth-century scientific revolution to its end. This is the problem of Quantum Gravity.
1.1 Quantum spacetime Roughly speaking, we learn from GR that spacetime is a dynamical field and we learn from QM that all dynamical field are quantized. A quantum field has a gran- ular structure, and a probabilistic dynamics, that allows quantum superposition of different states. Therefore at small scales we might expect a “quantum spacetime” formed by “quanta of space” evolving probabilistically, and allowing “quantum superposition of spaces”. The problem of Quantum Gravity is to give a precise mathematical and physical meaning to this vague notion of “quantum spacetime”. Some general indications about the nature of quantum spacetime, and on the problems this notion raises, can be obtained from elementary considerations. The size of quantum mechanical effects is determined by Planck’s constant .The strength of the gravitational force is determined by Newton’s constant G, and the relativistic domain is determined by the speed of light c. By combining these three 3 −33 fundamental constants we obtain the Planck length lP = G/c ∼ 10 cm. Quantum-gravitational effects are likely to be negligible at distances much larger than lP, because at these scales we can neglect quantities of the order of G, or 1/c. Therefore we expect the classical GR description of spacetime as a pseudo- Riemannian space to hold at scales larger than lP, but to break down approaching this scale, where the full structure of quantum spacetime becomes relevant. Quan- tum Gravity is therefore the study of the structure of spacetime at the Planck scale.
1.1.1 Space
Many simple arguments indicate that lP may play the role of a minimal length, in the same sense in which c is the maximal velocity and the minimal exchanged action. 6 C. Rovelli For instance, the Heisenberg principle requires that the position of an object of mass m can be determined only with uncertainty x satisfying mvx > , where v is the uncertainty in the velocity; special relativity requires v
1.1.2 Time Time is affected even more radically by the quantization of gravity. In conventional QM, time is treated as an external parameter and transition probabilities change in time. In GR there is no external time parameter. Coordinate time is a gauge variable which is not observable, and the physical variable measured by a clock is a nontrivial function of the gravitational field. Fundamental equations of Quantum Gravity might therefore not be written as evolution equations in an observable time variable. And in fact, in the quantum-gravity equation par excellence, the Wheeler– deWitt equation, there is no time variable t at all. Much has been written on the fact that the equations of nonperturbative Quantum Gravity do not contain the time variable t. This presentation of the “problem of time in Quantum Gravity”, however, is a bit misleading, since it mixes a problem of classical GR with a specific Quantum Gravity issue. Indeed, classical GR as well can be entirely formulated in the Hamilton–Jacobi formalism, where no time variable appears either. In classical GR, indeed, the notion of time differs strongly from the one used in the special-relativistic context. Before special relativity, one assumed that there is a universal physical variable t, measured by clocks, such that all physical phenom- ena can be described in terms of evolution equations in the independent variable t. In special relativity, this notion of time is weakened. Clocks do not measure a uni- versal time variable, but only the proper time elapsed along inertial trajectories. If we fix a Lorentz frame, nevertheless, we can still describe all physical phenomena in terms of evolution equations in the independent variable x0, even though this description hides the covariance of the system. Unfinished revolution 7 In general relativity, when we describe the dynamics of the gravitational field (not to be confused with the dynamics of matter in a given gravitational field), there is no external time variable that can play the role of observable independent evolution variable. The field equations are written in terms of an evolution param- eter, which is the time coordinate x0; but this coordinate does not correspond to anything directly observable. The proper time τ along spacetime trajectories can- not be used as an independent variable either, as τ is a complicated non-local function of the gravitational field itself. Therefore, properly speaking, GR does not admit a description as a system evolving in terms of an observable time vari- able. This does not mean that GR lacks predictivity. Simply put, what GR predicts are relations between (partial) observables, which in general cannot be repre- sented as the evolution of dependent variables on a preferred independent time variable. This weakening of the notion of time in classical GR is rarely emphasized: after all, in classical GR we may disregard the full dynamical structure of the theory and consider only individual solutions of its equations of motion. A single solution of the GR equations of motion determines “a spacetime”, where a notion of proper time is associated to each timelike worldline. But in the quantum context a single solution of the dynamical equation is like a single “trajectory” of a quantum particle: in quantum theory there are no physical individual trajectories: there are only transition probabilities between observable eigenvalues. Therefore in Quantum Gravity it is likely to be impossible to describe the world in terms of a spacetime, in the same sense in which the motion of a quantum electron cannot be described in terms of a single trajectory. To make sense of the world at the Planck scale, and to find a consistent con- ceptual framework for GR and QM, we might have to give up the notion of time altogether, and learn ways to describe the world in atemporal terms. Time might be a useful concept only within an approximate description of the physical reality.
1.1.3 Conceptual issues The key difficulty of Quantum Gravity may therefore be to find a way to understand the physical world in the absence of the familiar stage of space and time. What might be needed is to free ourselves from the prejudices associated with the habit of thinking of the world as “inhabiting space” and “evolving in time”. Technically, this means that the quantum states of the gravitational field cannot be interpreted like the n-particle states of conventional QFT as living on a given spacetime. Rather, these quantum states must themselves determine and define a spacetime – in the manner in which the classical solutions of GR do. 8 C. Rovelli Conceptually, the key question is whether or not it is logically possible to under- stand the world in the absence of fundamental notions of time and time evolution, and whether or not this is consistent with our experience of the world. The difficulties of Quantum Gravity are indeed largely conceptual. Progress in Quantum Gravity cannot be just technical. The search for a quantum theory of grav- ity raises once more old questions such as: What is space? What is time? What is the meaning of “moving”? Is motion to be defined with respect to objects or with respect to space? And also: What is causality? What is the role of the observer in physics? Questions of this kind have played a central role in periods of major advances in physics. For instance, they played a central role for Einstein, Heisen- berg, and Bohr; but also for Descartes, Galileo, Newton and their contemporaries, as well as for Faraday and Maxwell. Today some physicists view this manner of posing problems as “too philosoph- ical”. Many physicists of the second half of the twentieth century, indeed, have viewed questions of this nature as irrelevant. This view was appropriate for the problems they were facing. When the basics are clear and the issue is problem- solving within a given conceptual scheme, there is no reason to worry about foundations: a pragmatic approach is the most effective one. Today the kind of difficulties that fundamental physics faces has changed. To understand quantum spacetime, physics has to return, once more, to those foundational questions.
1.2 Where are we? Research in Quantum Gravity developed slowly for several decades during the twentieth century, because GR had little impact on the rest of physics and the inter- est of many theoreticians was concentrated on the development of quantum theory and particle physics. In the past 20 years, the explosion of empirical confirmations and concrete astrophysical, cosmological and even technological applications of GR on the one hand, and the satisfactory solution of most of the particle physics puzzles in the context of the SM on the other, have led to a strong concentration of interest in Quantum Gravity, and the progress has become rapid. Quantum Gravity is viewed today by many as the big open challenge in fundamental physics. Still, after 70 years of research in Quantum Gravity, there is no consensus, and no established theory. I think it is fair to say that there isn’t even a single complete and consistent candidate for a quantum theory of gravity. In the course of 70 years, numerous ideas have been explored, fashions have come and gone, the discovery of the Holy Grail of Quantum Gravity has been several times announced, only to be later greeted with much scorn. Of the tentative theories studied today (strings, loops and spinfoams, non-commutative geometry, dynamical triangulations or others), each is to a large extent incomplete and none has yet received a whit of direct or indirect empirical support. Unfinished revolution 9 However, research in Quantum Gravity has not been meandering meaninglessly. On the contrary, a consistent logic has guided the development of the research, from the early formulation of the problem and of the major research programs in the 1950s to the present. The implementation of these programs has been laborious, but has been achieved. Difficulties have appeared, and solutions have been proposed, which, after much difficulty, have lead to the realization, at least partial, of the initial hopes. It was suggested in the early 1970s that GR could perhaps be seen as the low energy limit of a Poincaré invariant QFT without uncontrollable divergences [3]; and today, 30 years later, a theory likely to have these properties – perturbative string theory – is known. It was also suggested in the early 1970s that non- renormalizability might not be fatal for quantum GR [4; 5] and that the Planck scale could cut divergences off nonperturbatively by inducing a quantum discrete structure of space; and today we know that this is in fact the case – ultraviolet finiteness is realized precisely in this manner in canonical loop Quantum Grav- ity and in some spinfoam models. In 1957 Charles Misner indicated that in the canonical framework one should be able to compute quantum eigenvalues of geo- metrical quantities [6]; and in 1995, 37 years later, eigenvalues of area and volume were computed – within loop quantum gravity [7; 8]. Much remains to be under- stood and some of the current developments might lead nowhere. But looking at the entire development of the subject, it is difficult to deny that there has been substantial progress. In fact, at least two major research programs can today claim to have, if not a complete candidate theory of Quantum Gravity, at least a large piece of it: string theory (in its perturbative and still incomplete nonperturbative versions) and loop quantum gravity (in its canonical as well as covariant – spinfoam – versions) are both incomplete theories, full of defects – in general, strongly emphasized within the opposite camp – and without any empirical support, but they are both remark- ably rich and coherent theoretical frameworks, that might not be far from the solution of the puzzle. Within these frameworks, classical and long intractable, physical, astrophysi- cal and cosmological Quantum Gravity problems can finally be concretely treated. Among these: black hole’s entropy and fate, the physics of the big-bang singular- ity and the way it has affected the currently observable universe, and many others. Tentative predictions are being developed, and the attention to the concrete pos- sibility of testing these predictions with observations that could probe the Planck scale is very alive. All this was unthinkable only a few years ago. The two approaches differ profoundly in their hypotheses, achievements, spe- cific results, and in the conceptual frame they propose. The issues they raise con- cern the foundations of the physical picture of the world, and the debate between the two approaches involves conceptual, methodological and philosophical issues. 10 C. Rovelli In addition, a number of other ideas, possibly alternative, possibly complemen- tary to the two best developed theories and to one another, are being explored. These include noncommutative geometry, dynamical triangulations, effective the- ories, causal sets and many others. The possibility that none of the currently explored hypotheses will eventually turn out to be viable, or, simply, none will turn out to be the way chosen by Nature, is very concrete, and should be clearly kept in mind. But the rapid and multi-front progress of the past few years raises hopes. Major well-posed open questions in the- oretical physics (Copernicus or Ptolemy? Galileo’s parabolas or Kepler’s ellipses? How to describe electricity and magnetism? Does Maxwell theory pick a preferred reference frame? How to do the Quantum Mechanics of interacting fields...?) have rarely been solved in a few years. But they have rarely resisted more than a few decades. Quantum Gravity – the problem of describing the quantum properties of spacetime – is one of these major problems, and it is reasonably well defined: is there a coherent theoretical framework consistent with quantum theory and with General Relativity? It is a problem which is on the table since the 1930s, but it is only in the past couple of decades that the efforts of the theoretical physics community have concentrated on it. Maybe the solution is not far away. In any case, we are not at the end of the road of physics, we are half-way through the woods along a major scientific revolution.
Bibliographical note For details on the history of Quantum Gravity see the historical appendix in [9]; and, for early history see [10; 11]and[12; 13]. For orientation on current research on Quantum Gravity, see the review papers [14; 15; 16; 17]. As a general intro- duction to Quantum Gravity ideas, see the old classic reviews, which are rich in ideas and present different points of view, such as John Wheeler 1967 [18], Steven Weinberg 1979 [5], Stephen Hawking 1979 and 1980 [19; 20], Karel Kuchar 1980 [21], and Chris Isham’s magisterial syntheses [22; 23; 24]. On string theory, classic textbooks are Green, Schwarz and Witten, and Polchinksi [25; 26]. On loop quan- tum gravity, including the spinfoam formalism, see [9; 27; 28], or the older papers [29; 30]. On spinfoams see also [31]. On noncommutative geometry see [32]and on dynamical triangulations see [33]. For a discussion of the difficulties of string theory and a comparison of the results of strings and loops, see [34], written in the form of a dialogue, and [35]. On the more philosophical challenges raised by Quan- tum Gravity, see [36]. Smolin’s popular book [37] provides a readable introduction to Quantum Gravity. The expression “half way through the woods” to characterize the present state of fundamental theoretical physics is taken from [38; 39]. My own view on Quantum Gravity is developed in detail in [9]. Unfinished revolution 11 References [1] D. Gross, 2006, in Viewpoints on string theory, NOVA science programming on air and online, http://www.pbs.org/wgbh/nova/elegant/view-gross.html. [2] M.P. Bronstein, “Quantentheories schwacher Gravitationsfelder”, Physikalische Zeitschrift der Sowietunion 9 (1936), 140. [3] B. Zumino, “Effective Lagrangians and broken symmetries”, in Brandeis University Lectures On Elementary Particles And Quantum Field Theory, Vol 2 (Cambridge, Mass, 1970), pp. 437–500. [4] G. Parisi, “The theory of non-renormalizable interactions. 1 The large N expansion”, Nucl Phys B100 (1975), 368. [5] S. Weinberg, “Ultraviolet divergences in quantum theories of gravitation”, in General Relativity: An Einstein Centenary Survey, S. W. Hawking and W. Israel, eds. (Cambridge University Press, Cambridge, 1979). [6] C. Misner, “Feynman quantization of general relativity”, Rev. Mod. Phys. 29 (1957), 497. [7] C. Rovelli, L. Smolin, “Discreteness of area and volume in quantum gravity”, Nucl. Phys. B442 (1995) 593; Erratum Nucl. Phys. B456 (1995), 734. [8] A. Ashtekar, J. Lewandowski, “Quantum theory of geometry I: area operators” Class and Quantum Grav 14 (1997) A55; “II : volume operators”, Adv. Theo. Math. Phys. 1 (1997), pp. 388–429. [9] C. Rovelli, Quantum Gravity (Cambridge University Press, Cambridge, 2004). [10] J. Stachel, “Early history of quantum gravity (1916–1940)”, Presented at the HGR5, Notre Dame, July 1999. [11] J. Stachel, “Early history of quantum gravity” in ‘Black Holes, Gravitational radiation and the Universe, B. R. Iyer and B. Bhawal, eds. (Kluwer Academic Publisher, Netherlands, 1999). [12] G. E. Gorelik, “First steps of quantum gravity and the Planck values” in Studies in the history of general relativity. [Einstein Studies, vol. 3], J. Eisenstaedt and A. J. Kox, eds., pp. 364–379 (Birkhaeuser, Boston, 1992). [13] G. E. Gorelik, V. Y. Frenkel, Matvei Petrovic Bronstein and the Soviet Theoretical Physics in the Thirties (Birkhauser Verlag, Boston 1994). [14] G. Horowitz, “Quantum gravity at the turn of the millenium”, plenary talk at the Marcel Grossmann conference, Rome 2000, gr-qc/0011089. [15] S. Carlip, “Quantum gravity: a progress report”, Reports Prog. Physics 64 (2001) 885, gr-qc/0108040. [16] C. J. Isham, “Conceptual and geometrical problems in quantum gravity”, in Recent Aspects of Quantum fields, H. Mitter and H. Gausterer, eds. (Springer Verlag, Berlin, 1991), p. 123. [17] C. Rovelli, “Strings, loops and the others: a critical survey on the present approaches to quantum gravity”, in Gravitation and Relativity: At the Turn of the Millenium, N. Dadhich and J. Narlikar, eds., pp. 281–331 (Inter-University Centre for Astronomy and Astrophysics, Pune, 1998), gr-qc/9803024. [18] J. A. Wheeler, “Superspace and the nature of quantum geometrodynamics”, in Batelle Rencontres, 1967, C. DeWitt and J. W. Wheeler, eds., Lectures in Mathematics and Physics, 242 (Benjamin, New York, 1968). [19] S. W. Hawking, “The path-integral approach to quantum gravity”, in General Relativity: An Einstein Centenary Survey, S. W. Hawking and W. Israel, eds. (Cambridge University Press, Cambridge, 1979). 12 C. Rovelli
[20] S. W. Hawking, “Quantum cosmology”, in Relativity, Groups and Topology, Les Houches Session XL, B. DeWitt and R. Stora, eds. (North Holland, Amsterdam, 1984). [21] K. Kuchar, “Canonical methods of quantization”, in Oxford 1980, Proceedings, Quantum Gravity 2 (Oxford University Press, Oxford, 1984). [22] C. J. Isham, Topological and global aspects of quantum theory, in Relativity Groups and Topology. Les Houches 1983, B. S. DeWitt and R. Stora, eds. (North Holland, Amsterdam, 1984), pp. 1059–1290. [23] C. J. Isham, “Quantum gravity: an overview”, in Oxford 1980, Proceedings, Quantum Gravity 2 (Oxford University Press, Oxford, 1984). [24] C. J. Isham, 1997, “Structural problems facing quantum gravity theory”, in Proceedings of the 14th International Conference on General Relativity and Gravitation, M. Francaviglia, G. Longhi, L. Lusanna and E. Sorace, eds., (World Scientific, Singapore, 1997), pp 167–209. [25] M. B. Green, J. Schwarz, E. Witten, Superstring Theory (Cambridge University Press, Cambridge, 1987). [26] J. Polchinski, String Theory (Cambridge University Press, Cambridge, 1998). [27] T. Thiemann, Introduction to Modern Canonical Quantum General Relativity, (Cambridge University Press, Cambridge, in the press). [28] A. Ashtekar, J. Lewandowski, “Background independent quantum gravity: A status report”, Class. Quant. Grav. 21 (2004), R53–R152. [29] C. Rovelli, L. Smolin, “Loop space representation for quantum general relativity, Nucl. Phys. B331 (1990), 80. [30] C. Rovelli, L. Smolin, “Knot theory and quantum gravity”, Phys. Rev. Lett. 61 (1988), 1155. [31] A. Perez, “Spin foam models for quantum gravity”, Class. Quantum Grav. 20 (2002), gr-qc/0301113. [32] A. Connes, Non Commutative Geometry (Academic Press, New York, 1994). [33] R. Loll, “Discrete approaches to quantum gravity in four dimensions”, Liv. Rev. Rel. 1 (1998), 13, http://www.livingreviews.org/lrr-1998-13. [34] C. Rovelli, “A dialog on quantum gravity”, International Journal of Modern Physics 12 (2003), 1, hep-th/0310077. [35] L. Smolin, “How far are we from the quantum theory of gravity?” (2003), hep-th/0303185. [36] C. Callender, H. Huggett, eds, Physics Meets Philosophy at the Planck Scale (Cambridge University Press, 2001). [37] L. Smolin, Three Roads to Quantum Gravity (Oxford University Press, 2000). [38] C. Rovelli, “Halfway through the woods”, in The Cosmos of Science, J. Earman and J. D. Norton, eds. (University of Pittsburgh Press and Universitäts. Verlag-Konstanz, 1997). [39] C. Rovelli, “The century of the incomplete revolution: searching for general relativistic quantum field theory”, J. Math. Phys., Special Issue 2000 41 (2000), 3776. 2 The fundamental nature of space and time
G. ’T HOOFT
2.1 Quantum Gravity as a non-renormalizable gauge theory Quantum Gravity is usually thought of as a theory, under construction, where the postulates of quantum mechanics are to be reconciled with those of general rela- tivity, without allowing for any compromise in either of the two. As will be argued in this contribution, this ‘conservative’ approach may lead to unwelcome compro- mises concerning locality and even causality, while more delicate and logically more appealing schemes can be imagined. The conservative procedure, however, must first be examined closely. The first attempt (both historically and logically the first one) is to formulate the theory of ‘Quantum Gravity’ perturbatively [1; 2; 3; 4; 5], as has been familiar practice in the quantum field theories for the fundamental particles, namely the Standard Model. In perturbative Quantum Gravity, one takes the Einstein–Hilbert action, √ R(x) S = ∂4x −g + Lmatter(x) = πG 16 (2.1) √ Bg considers the metric to be close to some background value: gμν = gμν + hμν, and expands everything in powers of , or equivalently, Newton’s constant G. Invariance under local coordinate transformations then manifests itself as a local gauge symmetry: hμνhμν + Dμuν + Dνuμ, where Dμ is the usual covariant deriva- tive, and uμ(x) generates an infinitesimal coordinate transformation. Here one can use the elaborate machinery that has been developed for the Yang–Mills theories of the fundamental particles. After imposing an appropriate gauge choice, all desired amplitudes can be characterized in terms of Feynman diagrams. Usually, these con- tain contributions of ‘ghosts’, which are gauge dependent degrees of freedom that propagate according to well-established rules. At first sight, therefore, Quantum Gravity does not look altogether different from a Yang–Mills theory. It appears that at least the difficulties of reconciling quantum mechanics with general coor- dinate invariance have been dealt with. We understand exactly how the problem
Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, ed. Daniele Oriti. Published by Cambridge University Press. c Cambridge University Press 2009. 14 G. ’t Hooft of time, of Cauchy surfaces, and of picking physical degrees of freedom, are to be handled in such a formalism. Indeed, unitarity is guaranteed in this formalism, and, in contrast to ‘more advanced’ schemes for quantizing gravity, the perturba- tive approach can deal adequately with problems such as: what is the complete Hilbert space of physical states?, how can the fluctuations of the light cone be squared with causality?, etc., simply because at all finite orders in perturbation expansion, such serious problems do not show up. Indeed, this is somewhat surpris- ing, because the theory produces useful amplitudes at all orders of the perturbation parameter . Yet there is a huge difference with the Standard Model. This ‘quantum gauge the- ory of gravity’ is not renormalizable. We must distinguish the technical difficulty from the physical one. Technically, the ‘disaster’ of having a non-renormalizable theory is not so worrisome. In computing the O(n) corrections to some amplitude, O(n) one has to establish√ correction terms to the Lagrangian, which are typically n+1 α of the form −gR , where n + 1 factors linear in the Riemann curvature R βμν may have been contracted in various possible ways. These terms are necessary to cancel out infinite counter terms of this form, where finite parts are left over. At high orders n, there exist many different expressions of the form Rn+1, which will all be needed. This is often presented as a problem, but, in principle, it is not. It simply means that our theory has an infinite sequence of free parameters, not unlike many other theories in science, and it nevertheless gives accurate and useful predictions up to arbitrarily high powers of GE2, where E is the energy scale con- sidered. We emphasize that this is actually much better than many of the alternative approaches to Quantum Gravity such as Loop Quantum Gravity, and even string theory presents us with formidable problems when 3-loop amplitudes are asked for. Also, claims [6] that Quantum Gravity effects might cause ‘decoherence’ at some finite order of GE2 are invalid according to this theory. Physically, however, the perturbative approach fails. The difficulty is not the fact that the finite parts of the counter terms can be freely chosen. The difficulty is a combination of two features: (i) perturbation expansion does not converge, and (ii) the expansion parameter becomes large if centre-of-mass energies reach beyond the Planck value. The latter situation is very reminiscent of the old weak interaction theory where a quartic interaction was assumed among the fermionic fields. This Fermi theory was also ‘non-renormalizable’. In the Fermi theory, this problem was solved: the theory was replaced by a Yang–Mills theory with Brout–Englert–Higgs mechanism. This was not just ‘a way to deal with the infinities’, it was actually an answer to an absolutely cru- cial question [7]: what happens at small distance scales?. At small distance scales, we do not have quartic interactions among fermionic fields, we have a local gauge theory instead. This is actually also the superior way to phrase the problem of The fundamental nature of space and time 15 Quantum Gravity: what happens at, or beyond, the Planck scale? Superstring the- ory [8; 9] is amazingly evasive if it comes to considering this question. It is here that Loop Quantum Gravity [10; 11; 12; 13; 14; 15] appears to be the most direct approach. It is an attempt to characterize the local degrees of freedom, but is it good enough?
2.2 A prototype: gravitating point particles in 2 + 1 dimensions An instructive exercise is to consider gravity in less than four space-time dimen- sions. Indeed, removing two dimensions allows one to formulate renormalizable models with local diffeomorphism invariance. Models of this sort, having one space- and one time dimension, are at the core of (super)string theory, where they describe the string world sheet. In such a model, however, there is no large dis- tance limit with conventional ‘gravity’, so it does not give us hints on how to cure non-renormalizable long-distance features by modifying its small distance char- acteristics. There is also another reason why these two-dimensional models are uncharacteristic for conventional gravity: formally, pure gravity in d = 2 dimen- 1 ( − ) =− sions has 2 d d 3 1 physical degrees of freedom, which means that an additional scalar field is needed to turn the theory into a topological theory. Con- formal symmetry removes one further degree of freedom, so that, if string theory starts with D target space variables, or ‘fields’, X μ(σ, Tr), where μ = 1, ··· , D, only D − 2 physical fields remain. For the present discussion it is therefore more useful to remove just one dimen- sion. Start with gravitationally interacting point particles in two space dimensions and one time. The classical theory is exactly solvable, and this makes it very inter- esting. Gravity itself, having zero physical degrees of freedom, is just topological; there are no gravitons, so the physical degrees of freedom are just the gravitating point particles. In the large distance limit, where Quantum Mechanical effects may be ignored, the particles are just point defects surrounded by locally flat space-time. The dynamics of these point defects has been studied [16; 17; 18], and the evolu- tion laws during finite time intervals are completely understood. During very long time intervals, however, chaotic behavior sets in, and also, establishing a complete list of all distinguishable physical states turns out to be a problem. One might have thought that quantizing a classically solvable model is straightforward, but it is far from that, exactly because of the completeness problem. 2 + 1 gravity without point particles could be quantized [19; 20], but that is a topological theory, with no local degrees of freedom; all that is being quantized are the boundary conditions, whatever that means. One would like to represent the (non-rotating) point particles by some scalar field theory, but the problems one then encounters appear to be formidable. Quite 16 G. ’t Hooft generally, in 2 + 1 dimensions, the curvature of 2-space is described by defect angles when following closed curves (holonomies). The total defect angle accu- mulated by a given closed curve always equals the total matter-energy enclosed by the curve. In the classical model, all of this is crystal clear. But what happens when one attempts to ‘quantize’ it? The matter Hamiltonian density does not commute with any of the particle degrees of freedom, since the latter evolve as a function of time. Thus, anything that moves, is moving in a space-time whose curvature is non-commuting. This is an impediment against a proper formulation of the Hilbert space in question in the conventional manner. Only eigenstates of the Hamiltonian and the Hamiltonian density can live in a 2-space with precisely defined 2-metric. Consequently, if we wish to describe physical states in a 2-space with precisely defined metric, these states must be smeared over a period of time that is large compared to the Planck time. We repeat: in a perturbative setting this situation can be handled because the deviations from flat space-time are small, but in a non- perturbative case, we have to worry about the limits of the curvature. The deficit angles cannot exceed the value 2π, and this implies that the Hamilton density must be bounded. There is, however, an unconventional quantization procedure that seems to be quite appropriate here. We just noted that the Hamiltonian of this theory is unmis- takably an angle, and this implies that time, its conjugate variable, must become discrete as soon as we quantize. Having finite time jumps clearly indicates in what direction we should search for a satisfactory quantum model: Schrödinger’s equa- tion will be a finite difference equation in the time direction. Take that as a modified picture for the small-distance structure of the theory! How much more complicated will the small-distance structure be in our 3 + 1 dimensional world? Here, the Hamiltonian is not limited to be an angle, so, time will surely be continuous. However, if we restrict ourselves to a region where one or more spatial dimensions are taken to be confined, or compactified, taking values smaller than some scale L in Planck units, then it is easy to see that we are back in the 2 + 1 dimensional case, the Hamiltonian is again an angle, and time will be quantized. However, the 2 + 1 dimensional Newton’s constant will scale like 1/L, and the time quantum will therefore be of order 1/L in Planck units. This suggests the following. In finite slabs of 3-space, time is quantized, the states are ‘updated’ in discretized time steps. If we stitch two equal sized slabs together, producing a slab twice as thick, then updating happens twice as fast, which we interpret as if updating happens alternatingly in one slab and in the other. The total time quantum has decreased by a factor two, but within each slab, time is still quantized in the original units. The picture we get this way is amazingly reminiscent of a computer model, where the computer splits 3-space into slabs of one Planck length thick, and during one Planck time interval every slab is being updated; a stack of N slabs thus requires N updates per Planck unit of time. The fundamental nature of space and time 17
2.3 Black holes, causality and locality The 2 + 1 dimensional theory does not allow for the presence of black holes (assuming a vanishing cosmological constant, as we will do throughout). The black hole problem, there, is simply replaced by the restriction that the energy must stay less than the Planck value. In our slab-stack theory (for want of a better name), we see that the energy in every slab is restricted to be less than the Planck value, so any system where one of the linear dimensions is less than L, should have energy less than L in Planck units, and this amounts to having a limit for the total energy that is such that a black hole corresponds to the maximally allowed energy in a given region. Clearly, black holes will be an essential element in any Quantum Gravity theory. We must understand how to deal with the requirement that the situation obtained after some gravitational collapse can be either described as some superdense blob of mass and energy, or as a geometric region of space-time itself where ingoing observers should be allowed to apply conventional laws of physics to describe what they see. One can go a long way to deduce the consequences of this requirement. Particles going into a black hole will interact with all particles going out. Of all these inter- actions, the gravitational one happens to play a most crucial role. Only by taking this interaction into account [21], can one understand how black holes can play the role of resonances in a unitary scattering process where ingoing particles form black holes and outgoing particles are the ones generated by the Hawking process. Yet how to understand the statistical origin of the Hawking–Bekenstein entropy of a black hole in this general framework is still somewhat mysterious. Even if black hole entropy can be understood in superstring theories for black holes that are near extremality, a deep mystery concerning locality and causality for the evo- lution laws of Nature’s degrees of freedom remains. Holography tells us that the quantum states can be enumerated by aligning them along a planar surface. The slab-stack theory tells us how often these degrees of freedom are updated per unit of time. How do we combine all this in one comprehensive theory, and how can we reconcile this very exotic numerology with causality and locality? May we simply abandon attempts to rescue any form of locality in the 3 + 1 dimensional bulk the- ory, replacing it by locality on the dual system, as is done in the AdS/CFT approach [22; 23]ofM-theory?
2.4 The only logical way out: deterministic quantum mechanics It is this author’s opinion that the abstract and indirect formalisms provided by M-theory approaches are unsatisfactory. In particle physics, the Standard Model was superior to the old Fermi theory just because it provided detailed understanding 18 G. ’t Hooft of the small-distance structure. The small-distance structure of the 3 + 1 dimen- sional theory is what we wish to understand. The holographic picture suggests discreteness in space, and the slab-stack theory suggests discreteness in time. Together, they suggest that the ultimate laws of Nature are akin to a cellular automaton [24]. However, our numerology admits far fewer physical states than one (discrete) degree of freedom per unit of bulk volume element. We could start with one degree of freedom for every unit volume element, but then a huge local symmetry con- straint would be needed to reduce this to physical degrees of freedom which can be limited to the surface. This situation reminds us of topological gauge theories. How will we ever be able to impose such strong symmetry principles on a world that is as non-trivial as our real universe? How can we accommodate for the fact that the vast majority of the ‘bulk states’ of a theory should be made unphysical, like local gauge degrees of freedom? Let us return to the 2 + 1 dimensional case. Suppose that we would try to set up a functional integral expression for the quantum amplitudes. What are the degrees of freedom inside the functional integrand? One would expect these to be the defects in a space-time that is flat everywhere except in the defects. A defect is then charac- terized by the element of the Poincaré group associated with a closed loop around the defect, the holonomy of the defect. Now this would force the defect to follow a straight path in space-time. It is not, as in the usual functional integral, an arbitrary function of time, but, even inside the functional integral, it is limited to straight paths only. Now this brings us back from the quantum theory to a deterministic the- ory; only deterministic paths appear to be allowed. It is here that this author thinks we should search for the clue towards the solution to the aforementioned problems. The topic that we dubbed ‘deterministic quantum mechanics’ [25; 26]isnot a modification of standard quantum mechanics, but must be regarded as a special case. A short summary, to be explained in more detail below, is that our conven- tional Hilbert space is part of a bigger Hilbert space; conventional Hilbert space is obtained from the larger space by the action of some projection operator. The states that are projected out are the ones we call ‘unphysical’, to be compared with the ghosts in local gauge theories, or the bulk states as opposed to the surface states in a holographic formulation. In the bigger Hilbert space, a basis can be found such that basis elements evolve into basis elements, without any quantum mechanical superposition ever taking place. One of the simplest examples where one can demonstrate this idea is the harmonic oscillator, consisting of states |n, n = 0, 1,...,and 1 H|n= n + |n. (2.2) 2 The fundamental nature of space and time 19 If we add to this Hilbert space the states |n with n =−1, −2,...,onwhich the Hamiltonian acts just as in Eq. (2.2), then our ontological basis consists of the states 1 N |ϕ=√ e−inϕ|n, (2.3) + 2N 1 n=−N which evolve as |ϕ −→ |ϕ + T , (2.4) t=T provided that (2N + 1)T/2π is an integer. In the limit N →∞, time T can be taken to be continuous. In this sense, a quantum harmonic oscillator can be turned into a deterministic system, since, in Eq. (2.4), the wave function does not spread out, and there is no interference. A functional integral expression for this evolution would only require a single path, much as in the case of the 2 + 1-dimensional defects as described above. Since ϕ is periodic, the evolution (2.4) describes a periodic motion with period T = 2π. Indeed, every periodic deterministic system can be mapped onto the quantum harmonic oscillator provided that we project out the elements of Hilbert space that have negative energy. In general, any deterministic system evolves according to a law of the form ∂ qa(t) = f a (q(t)) (2.5) ∂t (provided that time is taken to be continuous), and in its larger Hilbert space, the Hamiltonian is ∂ H = f a p , p def=−i , (2.6) a a ∂qa a where, in spite of the classical nature of the physical system, we defined pa as quantum operators. In this large Hilbert space, one always sees as many negative as positive eigenstates of H, so it will always be necessary to project out states. A very fundamental difficulty is now how to construct a theory where not only the negative energy states can be projected out, but where also the entire system can be seen as a conglomeration of weakly interacting parts (one may either think of neighbouring sectors of the universe, or of weakly interacting particles), such that also in these parts only the positive energy sectors matter. The entire Hamiltonian is conserved, but the Hamilton densities, or the partial Hamiltonians, are not, and interacting parts could easily mix positive energy states with negative energy states. Deter- ministic quantum mechanics will only be useful if systems can be found where all states in which parts occur with negative energy, can also be projected out. The subset of Hilbert space where all bits and pieces only carry positive energy is only a very tiny section of the entire Hilbert space, and we will have to demonstrate 20 G. ’t Hooft that a theory exists where this sector evolves all by itself, even in the presence of non-trivial interactions. What kind of mechanism can it be that greatly reduces the set of physical states? It is here that our self-imposed restriction to have strictly deterministic Hamilton equations may now bear fruit. In a deterministic system, we may have information loss. In a quantum world, reducing the dimensionality of Hilbert space would lead to loss of unitarity, but in a deterministic world there is no logical impediment that forbids the possibility that two different initial states may both evolve into the same final state. This gives us a new view on what was once introduced as the ‘holographic prin- ciple’. According to this principle, the number of independent physical variables in a given volume actually scales with the surface area rather than the volume. This may mean that, in every volume element, information concerning the interior dis- sipates away due to information loss, while only the information located on the surface survives, possibly because it stays in contact with the outside world. Information loss forces us to assemble physical states in ‘equivalence classes’. Two states are in the same equivalence classes if, in due time, they eventually evolve into the same final state. Equivalence classes may play the role of gauge equivalence classes, and thus we might arrive at a plausible scenario in which the degrees of freedom inside the bulk of some region are reduced to being gauge degrees of freedom, while the physical degrees of freedom are limited to reside on the surface. Note that, if such a theory can be constructed, the ‘primordial’ laws of physics may be completely local and causal, but the physical states that figure in the evo- lution equation (2.5) appear to have a non-local definition. This may be the reason why more direct attempts to interpret quantum mechanical phenomena in terms of realistic theories tend to lead to a mysterious, invisible kind of non-locality, as laid down in the well-known Einstein–Podolsky–Rosen paradox.
2.5 Information loss and projection How could information loss act as a mechanism to select out only those states where all energies are non-negative? How exactly this works is not understood; however, we do have an instructive but admittedly vague argument, and it is the following. Consider several regions or systems in our universe that are only weakly interacting with one another. With the interaction switched off, they all obey deter- ministic evolution equations, and therefore, their Hamiltonians, which are of the form (2.6), have positive energy eigenvalues E(i)a and negative energy eigenvalues −E(i)a, where (i) enumerates the systems and a the eigenvalues. The combination The fundamental nature of space and time 21 tot = ( ) of these systems will again have positive eigenvalues E i E i a(i) and nega- tive energy eigenvalues −E tot, but interactions must be arranged in such a way that all states where some energies are positive and some are negative are suppressed. The reason why we do allow all energies to be negative is that this might describe the physical situation equally well; we then happen to be dealing with the bra states ψ| rather than the kets |ψ. Let us examine more closely the (weak) interaction between two such systems. Consider a time interval δt1 for system (1) and δt2 for system (2). As argued ear- lier, both systems must be spread over many Planck time units. According to the uncertainty relation, let us assume that 1 1 (E(1) + E(2)) ≈ ; 2 2(δt1 + δt2) 1 1 |E(1) − E(2)| ≈ . (2.7) 2 2|δt1 − δt2| Now, according to Eq. (2.4), uncertainty in time directly reflects uncertainty in the position ϕ of the system in its periodic orbit. Demanding E(1) E(2)>0 corresponds to
(E(1) + E(2))2 > (E(1) − E(2))2 , so that 2 2 (δt1 + δt2) <(δt1 − δt2) , or 2 2 (δϕ1 + δϕ2) <(δϕ1 − δϕ2) . (2.8)
The details concerning the relative position δϕ1 − δϕ2 wash away after a suffi- 1 (δ + δ ) ciently large average time interval 2 t1 t2 . We read off:
δt1 δt2 < 0. (2.9) Thus, the states that we expect to dissipate away due to information loss, are all those states where a positive time lapse δt1 for one state is associated with a positive time lapse δt2 for the other state. This may mean that the two states each carry an internal clock. The relative clock speed is controlled by the gravitational potential between the two systems. This potential apparently fluctuates. These fluctuations wash out all information concerning the relative configurations, but the relative clock speeds are always positive.
2.6 The vacuum state and the cosmological constant We see that if we have a set of different systems which mutually interact only weakly, such as a set of free particles, or a set of disconnected pieces of the uni- verse, either all energies must be selected to be positive, or they all are negative. 22 G. ’t Hooft This means that there is one very special state where all energies are zero: the vac- uum state. Identifying the vacuum state is particularly difficult in our theory, but it seems that the vacuum also poses problems in other approaches. In Loop Quantum Gravity, it is notoriously difficult to say exactly what the vacuum state is in terms of the fundamental loop states that were introduced there. In superstring theory, there are many candidates for the vacuum, all being distinctly characterized by the boundary conditions and the fluxes present in the compactified part of space-time. String theory ends up leaving an entire ‘landscape’ of vacuum states with no fur- ther indication as to which of these to pick. It is of crucial importance in any viable theory of Planck length physics to identify and describe in detail the vacuum state. It appears to be associated with very special fluctuations and correlations of the vir- tual particles and fields that one wishes to use to describe physical excited states, and the particles in it. There exists an important piece of information telling us that the vacuum is not just the state with lowest energy. There must exist an additional criterion to iden- tify the vacuum: it is flat – or nearly so. In perturbative gravity, this cannot be understood. The cosmological constant should receive a large finite renormaliza- tion counter term from all virtual interactions in the very high energy domain. A superior theory in which the cosmological constant vanishes naturally (or is limited to extremely tiny values) has not yet been found or agreed upon [27]. This should be a natural property of the vacuum state. To see most clearly how strange this situation is, consider the Einstein–Hilbert action, √ 1 S = −g R + , (2.10) 16πG 8πG Here, the first term describes the response of the total action to any deformation causing curvature. This response is huge, since Newton’s constant, which is tiny, occurs in the denominator. In contrast, the second term describes the response of the total action upon scaling. This response is very tiny, since the cosmological constant is extremely small – indeed it was thought to vanish altogether until recently. In Fig. 2.1, a piece of fabric is sketched with similar properties in ordinary 3-space. Globally, this material allows for stretching and squeezing with relatively little resistance, but changing the ratios of the sides of the large triangle, or its angles, requires much more force. One could build more elaborate structures from these basic triangular units, such that their shapes are fixed, but their sizes not. An engineer would observe, however, that even if the hinges and the rods were made extremely strong and sturdy, resistance against changes of shape would still be rather weak. In the limit where the sizes of the structures are very large compared to those of the hinges, resistance against changes of shape would dwindle. The fundamental nature of space and time 23
Fig. 2.1. The ‘fabric of space-time’, with tiny cosmological constant.
Comparing this with the situation in our universe shows in a more tangible way how odd it is that, a term with dimensions as low as the cosmological con- stant, can nevertheless be so tiny (120 factors of 10) compared with the much higher dimensional Einstein–Hilbert term. This oddity is the main reason why all attempts to find a natural explanation of this feature have failed. Unless one is prepared to accept the anthropic argument (‘the universe is like this because all other universes are uninhabitable for intelligent beings’), a more drastic approach will be needed. Here again, we emphasize that, in any more advanced theory for Planck length physics, the definition of what exactly the vacuum state is, will have to require special attention. It could be that one has to define that the vacuum state is the one in which 3-space is as flat as it can be. One is then again confronted with the problem of understanding why all other physical states have not only positive energy, but also energy densities that are bounded from below. Note that, in conventional quantum mechanics, the Hamiltonian plays a dual role: on the one hand it is simply the operator that generates the equations for evolution in time, while on the other hand it stabilizes the ground state, or vacuum. Energy conservation prevents small fluctuations from growing, because there are no other states where the total energy vanishes. One-particle states are also stable because there are no other states with matching energy and momentum, and this situation is guaranteed only because all energies are bounded from below. This is why the lower bound on energy is an absolutely vital feature of conventional quantum mechanics. It must be reproduced, whenever an ‘underlying’ theory is proposed. 24 G. ’t Hooft
2.7 Gauge- and diffeomorphism invariance as emergent symmetries Most likely, however, the hideously tiny value of the cosmological constant is pointing towards a deeper kind of misunderstanding concerning diffeomorphism invariance in gravity. A remote possibility is suggested by our theory where quan- tum mechanical effects are generated as an emergent phenomenon in a world that is deterministic at the Planck scale. Information loss leads to a description of phys- ical states forming equivalence classes. As stated, the equivalence classes are very large; when black holes are formed, the equivalence classes assemble on the sur- face area of the horizon, while the original ontological states are defined in the bulk of 3-space. If information loss forces two states to evolve identically, the states are said to sit in one equivalence class. Even if one would not buy the idea that there is an underlying deterministic theory, one could suspect that these equivalence classes can be described as gauge- equivalence classes. The transition from one element to another element of an equivalence classes is a local gauge transformation. If so, then local gauge invari- ance will not be a property of the underlying theory, but an emergent phenomenon. This naturally begs us to question: could diffeomorphism invariance be also just such a symmetry? Could it be that two states that differ from one another just by a local coordinate transformation, sit in one equivalence class, which would mean that they could evolve into the same final state? This might be possible. It would mean that the original, deterministic theory might require a preferred coordinate frame, which however would wash away due to information loss. The preferred coordinate frame might naturally select a flat space-time as a ground state solution, and thus a curvature-free configuration would be selected as the natural vacuum state. Needless to say, this argument is hopelessly inadequate to solve the cosmological constant problem, but it could serve to shed a different light on it. It illustrates that there may be more, unconventional directions to search for a solution to the problem of reconciling quantum mechanics with general relativity.
References [1] B. S. DeWitt, Phys. Rev. Lett. 12 (1964) 742. [2] Phys. Rev. 160 (1967) 1113; ibid. 162 (1967) 1195, 1239. [3] M. J. Duff, Covariant quantization, in Quantum Gravity: an Oxford Symposium, C.J. Isham et al., eds., Clarendon Press, Oxford, 1975, p. 78. [4] S. Deser, Quantum gravitation: trees, loops and renormalization, ibid., p. 136. [5] G. ’t Hooft, Perturbative quantum gravity, in Proceedings of the International School of Subnuclear Physics, Erice 2002, From Quarks and Gluons to Quantum Gravity, Subnuclear Series Vol. 40, ed. A. Zichichi, World Scientific, p. 249. The fundamental nature of space and time 25
[6] R. Penrose, The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics (Oxford University Press, 1989). [7] G. ’t Hooft, A confrontation with infinity, Nobel Lecture in Physics 1999, Rev. Mod. Phys. 72 (2000) 333. In slightly modified form: in Proceedings of the International Conference on Fundamental and Applied Aspects of Modern Physics, S. H. Connel and R. Tegen, eds., (World Scientific, 2001), p. 317. [8] M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Vols. I and II (Cambridge University Press, 1987). [9] J. Polchinski, String Theory, Vols. I and II, (Cambridge University Press, 1998). [10] A. Ashtekar, V. Husain, C. Rovelli, J. Samuel and L. Smolin, (2+1)-Quantum gravity as a toy model for the (3+1) theory, Class. Quantum Grav. 6 (1989) L185. [11] A. Ashtekar, C. Rovelli and L. Smolin, Weaving a classical geometry with quantum threads, Phys. Rev. Lett. 69 (1992) 237. [12] C. Rovelli and L. Smolin, Knot theory and quantum gravity, Phys. Rev. Lett. 61 (1988) 1155. [13] C. Rovelli and L. Smolin, Loop space representation of quantum general relativity, Nucl Phys. B133 (1990) 80. [14] C. Rovelli, What is observable in classical and quantum gravity?, Class. Quant. Grav. 8 (1991) 297–316; 317. [15] T. Thiemann, QSD 3: Quantum constraint algebra and physical scalar product in quantum general relativity, Class. Quant. Grav. 15 (1998) 1207–1247. [16] A. Staruszkiewicz, Acta Phys. Polon. 24 (1963) 734. [17] S. Deser, R. Jackiw and G. ’t Hooft, Three-dimensional Einstein gravity: dynamics of flat space, Ann. Phys. 152 (1984) 220. [18] G. ’t Hooft, The evolution of gravitating point particles in (2+1)-dimensions, Class. Quantum Grav. 10 (1993) 1023–1038. [19] E. Witten, (2+1)-Dimensional gravity as an exactly soluble system, Nucl. Phys. B311 (1988) 46. [20] S. Carlip, Exact quantum scattering in (2+1)-dimensional gravity, Nucl. Phys. B324 (1989) 106, and in: Physics, Geometry and Topology, NATO ASI series B, Physics, Vol . 238, H. C. Lee, ed. (Plenum, 1990), p. 541. [21] G. ’t Hooft, The holographic mapping of the standard model onto the black hole horizon, Part I: Abelian vector field, scalar field and BEH mechanism, Class. Quant. Grav. 22 (2005) 4179, ITP-UU-05/17, SPIN-05/13, gr-qc/0504120 (note: some sign corrections and further elucidations in the published version). [22] J. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231–252; hep-th/9711200. [23] J. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett. 80 (1998) 4859–4862; hep-th/9803002. [24] G. ’t Hooft, Can quantum mechanics be reconciled with cellular automata?, in Digital Perspectives, Int. J. Theor. Physics 42 (2003) 349. [25] G. ’t Hooft, Quantum gravity as a dissipative deterministic system, Class. Quant. Grav. 16 (1999) 3263 (gr-qc/9903084). [26] G. ’t Hooft, Quantum mechanics and determinism, in Proceedings of the Eighth Int. Conf. on Particles, Strings and Cosmology, Univ. of North Carolina, Chapel Hill, Apr. 10–15, 2001, P. Frampton and J. Ng, eds. (Rinton Press, Princeton, 2001), p. 275; hep-th/0105105. [27] S. Nobbenhuis, Categorizing different approaches to the cosmological constant problem, ITP-UU-04-40, SPIN-04-23, Nov. 2004, gr-qc/0411093, Found. Mod. Phys., to be published. 3 Does locality fail at intermediate length scales?
R. D. SORKIN
Assuming that “quantum spacetime” is fundamentally discrete, how might this dis- creteness show itself? Some of its potential effects are more evident, others less so. The atomic and molecular structure of ordinary matter influences the propagation of both waves and particles in a material medium. Classically, particles can be deflected by collisions and also retarded in their motion, giving rise in particu- lar to viscosity and Brownian motion. In the case of spatio-temporal discreteness, viscosity is excluded by Lorentz symmetry, but fluctuating deviations from rec- tilinear motion are still possible. Such “swerves” have been described in [1]and [2]. They depend (for a massive particle) on a single phenomenological parameter, essentially a diffusion constant in velocity space. As far as I know, the correspond- ing analysis for a quantal particle with mass has not been carried out yet, but for massless quanta such as photons the diffusion equation of [1] can be adapted to say something, and it then describes fluctuations of both energy and polarization (but not of direction), as well as a secular “reddening” (or its opposite). A more complete quantal story, however, would require that particles be treated as wave packets, raising the general question of how spatiotemporal discreteness affects the propagation of waves. Here, the analogy with a material medium suggests effects such as scattering and extinction, as well as possible nonlinear effects. Further gen- eralization to a “second-quantized field” might have more dramatic, if less obvious, consequences. In connection with cosmology, for example, people have wondered how discreteness would affect the hypothetical inflaton field. So far, I have been assuming that, although the deep structure of spacetime is discrete, it continues to respect the Lorentz transformations. That this is logically possible is demonstrated [3] by the example of causal set (causet) theory [4; 5; 6; 7]. With approaches such as loop quantum gravity, on the other hand, the status of local Lorentz invariance seems to be controversial. Some people have hypothesized that it would be broken or at least perhaps deformed in such a way that the dispersion relations for light would cease to be those of a massless field. Were this the case,
Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, ed. Daniele Oriti. Published by Cambridge University Press. c Cambridge University Press 2009. Does locality fail at intermediate length scales? 27 empty space could also resist the passage of particles (a viscosity of the vacuum), since there would now be a state of absolute rest. Moreover, reference [8]has argued convincingly that it would be difficult to avoid O(1) renormalization effects that would lead to different quantum fields possessing different effective light cones. Along these lines, one might end up with altogether more phenomenology than one had bargained for. As already mentioned, the causal set hypothesis avoids such difficulties, but in order to do so, it has to posit a kinematic randomness, in the sense that a spacetime1 M may properly correspond only to causets C that could have been produced by a Poisson process in M. With respect to an approximating spacetime M, the causet thus functions as a kind of “random lattice”. Moreover, the infinite volume of the Lorentz group implies that such a “lattice” cannot be home to a local dynamics. Rather the “couplings” or “interactions” that describe physical processes occurring in the causet are – of necessity – radically nonlocal. To appreciate why this must be, let us refer to the process that will be the subject of much of the rest of this chapter: propagation of a scalar field φ on a back- ground causet C that is well approximated by a Minkowski spacetime M = Md . To describe such a dynamics, one needs to reproduce within C something like the d’Alembertian operator , the Lorentzian counterpart of the Laplacian operator ∇2 of Euclidean space E3. Locality in the discrete context, if it meant anything at all, would imply that the action of would be built up in terms of “nearest neighbor couplings” (as in fact ∇2 can be built up, on either a crystalline or ran- dom lattice in E3). But Lorentz invariance contradicts this sort of locality because it implies that, no matter how one chooses to define nearest neighbor, any given causet element e ∈ C will possess an immense number of them extending through- out the region of C corresponding to the light cone of e in M.IntermsofaPoisson process in M we can express this more precisely by saying that the probability of any given element e possessing a limited number of nearest neighbors is vanish- ingly small. Thus, the other elements to which e must be “coupled” by our box operator will be large in number (in the limit infinite), and in any given frame of reference, the vast majority of them will be remote from e. The resulting “action at a distance” epitomizes the maxim that discreteness plus Lorentz invariance entails nonlocality. If this reasoning is correct, it implies that physics at the Planck scale must be radically nonlocal. (By Planck scale I just mean the fundamental length scale or volume scale associated with the causet or other discrete substratum.) Were it to be confined to the Planck scale, however, this nonlocality would be of limited phe- nomenological interest despite its deep significance for the underlying theory. But
1 In this chapter, “spacetime” will always mean Lorentzian manifold, in particular a continuum. 28 R. D. Sorkin a little thought indicates that things might not be so simple. On the contrary, it is far from obvious that the kind of nonlocality in question can be confined to any scale, because for any given configuration of the field φ, the “local couplings” will be vastly outnumbered by the “nonlocal” ones. How then could the latter conspire to cancel out so that the former could produce a good approximation to φ,even for a slowly varying φ? When posed like this, the question looks almost hopeless, but I will try to con- vince you that there is in fact an answer. What the answer seems to say, though, is that one can reinstate locality only conditionally and to a limited extent. At any finite scale λ, some nonlocality will naturally persist, but the scale λ0 at which it begins to disappear seems to reflect not only the ultraviolet scale l but also an infrared scale R, which we may identify with the age of the cosmos, and which (in a kind of quantum-gravitational echo of Olber’s paradox) seems to be needed in order that locality be recovered at all. On the other hand, (the) spacetime (con- tinuum) as such can make sense almost down to λ = l. We may thus anticipate that, as we coarse-grain up from l to larger and larger sizes λ, we will reach a stra- tum of reality in which discontinuity has faded out and spacetime has emerged, but physics continues to be nonlocal. One would expect the best description of this stratum to be some type of nonlocal field theory; and this would be a new sort of manifestation of discreteness: not as a source of fluctuations, but as a source of nonlocal phenomena. Under still further coarse-graining, this nonlocality should disappear as well, and one might think that one would land for good in the realm of ordinary Quantum Field Theory (and its further coarse-grainings). However, there is reason to believe that locality would fail once again when cosmic dimensions were reached; in fact, the non-zero cosmological constant predicted on the basis of causet theory is very much a nonlocal reflection, on the largest scales, of the underlying discreteness. It is a strictly quantal effect, however, and would be a very different sort of residue of microscopic discreteness than what I’ll be discussing here. These introductory remarks express in a general way most of what I want to convey in this paper, but before getting to the technical underpinnings, let me just (for shortage of space) list some other reasons why people have wanted to give up locality as a fundamental principle of spacetime physics: to cure the divergences of Quantum Field Theory (e.g. [9; 10]); to obtain particle-like excitations of a spin-network or related graph [11]; to give a realistic and deterministic account of Quantum Mechanics (the Bohmian interpretation is both nonlocal and acausal, for example); to let information escape from inside a black hole (e.g. [12]); to describe the effects of hidden dimensions in “brane world” scenarios; to reduce Quantum Gravity to a flat-space Quantum Field Theory via the so called AdS- CFT correspondence; to make room for non-commuting spacetime coordinates. Does locality fail at intermediate length scales? 29 (This “non-commutative geometry” reason is perhaps the most suggestive in the present context, because it entails a hierarchy of scales analogous to the scales l, λ0 and R. On the “fuzzy sphere” in particular, the non-commutativity scale λ0 is the geometric mean between the effective ultraviolet cutoff l and the sphere’s radius R.)
3.1 Three D’Alembertians for two-dimensional causets The scalar field on a causet offers a simple model for the questions we are con- sidering. Kinematically, we may realize such a field simply as a mapping φ of the causet into the real or complex numbers, while in the continuum its equations of motion take – at the classical level – the simple form φ = 0, assuming (as we will) that the mass vanishes. In order to make sense of this equation in the causet, we “merely” need to give a meaning to the D’Alembertian operator .Thisis not an easy task, but it seems less difficult than giving meaning to, for example, the gradient of φ (which for its accomplishment would demand that we define a concept of vector field on a causet). Of course, one wants ultimately to treat the quantum case, but one would expect a definition of to play a basic role there as well, so in seeking such a definition we are preparing equally for the classical and quantal cases. If we assume that should act linearly on φ (not as obvious as one might think!), then our task reduces to the finding of a suitable matrix Bxy to play the role of , where the indices x, y range over the elements of the causet C. We will also require that B be “retarded” or “causal” in the sense that Bxy = 0 whenever x is spacelike to, or causally precedes, y. In the first place, this is helpful classically, since it allows one to propagate a solution φ forward iteratively, element by element (assuming that the diagonal elements Bxx do not vanish). It might similarly be advantageous quantally, if the path integration is to be conducted in the Schwinger– Kel’dysh manner.
3.1.1 First approach through the Green function I argued above that no matrix B that (approximately) respects the Lorentz trans- formations can reproduce a local expression like the D’Alembertian unless the majority of terms cancel miraculously in the sum, Bxyφy =: (Bφ)x ,that corresponds to φ(x). y Simulations by Alan Daughton [13], continued by Rob Salgado [14], provided the first evidence that the required cancellations can actually be arranged for with- out appealing to anything other than the intrinsic order-structure of the causet. In 30 R. D. Sorkin this approach one notices that, although in the natural order of things one begins with the D’Alembertian and “inverts” it to obtain its Green function G, the result in 1 + 1-dimensions is so simple that the procedure can be reversed. In fact, the retarded Green function G(x, y) = G(x − y) in M2 is (with the sign convention =−∂2/∂t2 + ∂2/∂x2) just the step function with magnitude −1/2 and supports the future of the origin (the future light cone together with its interior). Moreover, thanks to the conformal invariance of in M2, the same expression remains valid in the presence of spacetime curvature. Not only is this continuum expression very simple, but it has an obvious coun- terpart in the causal set, since it depends on nothing more than the causal relation between the two spacetime points x and y. Letting the symbol < denote (strict) causal precedence in the usual way, we can represent the causet C as a matrix whose elements Cxy take the value 1 when x < y and 0 otherwise. The two- dimensional analog G of the retarded Green function is then just −1/2 times (the transpose of) this matrix. From these ingredients, one can concoct some obvious candidates for the matrix B. The one that so far has worked best is obtained by symmetrizing Gxy and then inverting it. More precisely, what has been done is the following: begin with a specific region R ⊂ M2 (usually chosen to be an order-interval, the diamond- shaped region lying causally between a timelike pair of points); randomly sprinkle N points xi , i = 1 ...N into R;letC be the causet with these points as substratum and the order-relation < induced from M2; for any “test” scalar field φ on R,let φ = φ( ) × i xi be the induced “field” on C; build the N N matrix G and then sym- (φ, ψ) = φ ψ metrize and invert to get B, as described above; evaluate B ij Bij i j for φ and ψ drawn from a suite of test functions on R; compare with the continuum values, d2x φ(x) ψ(y) d2 y. For test functions that vanish to first order on the boundary ∂ R of R, and that vary slowly on the scale set by the sprinkling density, the results so far exhibit full agreement between the discrete and continuum values [13; 14]. Better agreement than this, one could not have hoped for in either respect: concerning boundary terms, the heuristic reasoning that leads one to expect that inverting a Green func- tion will reproduce a discretized version of leaves open its behavior on ∂ R. Indeed, one doesn’t really know what continuum expression to compare with: if our fields don’t vanish on ∂ R, should we expect to obtain an approximation to dxdyφ(x) ψ(y) or dxdy(∇φ(x), ∇ψ(y)) or...?Concerningrapidlyvary- ing functions, it goes without saying that, just as a crystal cannot support a sound wave shorter than the interatomic spacing, a causet cannot support a wavelength shorter than l. But unlike with crystals, this statement requires some qualification because the notion of wavelength is frame-dependent. What is a red light for one inertial observer is a blue light for another. Given that the causet can support the red Does locality fail at intermediate length scales? 31 wave, it must be able to support the blue one as well, assuming Lorentz invariance in a suitable sense. Conversely, such paired fields can be used to test the Lorentz invariance of B. To the limited extent that this important test has been done, the results have also been favorable. On balance, then, the work done on the Green function approach gives cause for optimism that “miracles do happen”. However, the simulations have been limited to the flat case, and, more importantly, they do not suffice (as of yet) to establish that the discrete D’Alembertian B is truly frame independent. The point is that although G itself clearly is Lorentz invariant in this sense, its inverse (or rather the inverse of the symmetrized G) will in general depend on the region R in which one works. Because this region is not itself invariant under boosts, it defines a global frame that could find its way into the resulting matrix B. Short of a better ana- lytic understanding, one is unable to rule out this subtle sort of frame dependence, although the aforementioned limited tests provide evidence against it. Moreover, the Green function prescription itself is of limited application. In addition to two dimensions, the only other case where a similar prescription is known is that of four dimensions without curvature, where one can take for G the “link matrix” instead of the “causal matrix”. Interestingly enough, the potential for Lorentz-breaking by the region R does not arise if one works exclusively with retarded functions, that is, if one forms B from the original retarded matrix G, rather than its symmetrization.2 Unfortunately, computer tests with the retarded Green function have so far been discouraging on the whole (with some very recent exceptions). Since, for quite different reasons, it would be desirable to find a retarded representation of , this suggests that we try something different.
3.1.2 Retarded couplings along causal links Before taking leave of the Green function scheme just described, we can turn to it for one more bit of insight. If one examines the individual matrix elements Bxy for a typical sprinkling, one notices first of all that they seem to be equally distributed among positive and negative values, and second of all that the larger magnitudes among them are concentrated “along the light cone”; that is, Bxy tends to be small unless the proper distance between x and y is near zero. The latter observation may remind us of a collection of “nearest neighbor couplings”, here taken in the only possible Lorentz invariant sense: that of small proper distance. The former observation suggests that a recourse to oscillating signs might be the way to effect the “miraculous cancellations” we are seeking.
2 One needs to specify a nonzero diagonal for F. 32 R. D. Sorkin The suggestion of oscillating signs is in itself rather vague, but two further obser- vations will lead to a more quantitative idea. Let a be some point in M2,letb and c be points on the right and left halves of its past lightcone (a “cone” in M2 being just a pair of null rays), and let d be the fourth point needed to com- plete the rectangle. If (with respect to a given frame) all four points are chosen to make a small square, and if φ is slowly varying (in the same frame), then the combination φ(a) + φ(d) − φ(b) − φ(c) converges, after suitable normal- ization, to − φ(a) as the square shrinks to zero size. (By Lorentz invariance, the same would have happened even if we had started with a rectangle rather than a square.) On the other hand, four other points obtained from the originals by a large boost will form a long skinny rectangle, in which the points a and b (say) are very close together, as are c and d. Thanks to the profound identity, φ(a) + φ(d) − φ(b) − φ(c) = φ(a) − φ(b) + φ(d) − φ(c), we will obtain only a tiny contribution from this rectangle – exactly the sort of cancellation we were seeking! By including all the boosts of the original square, we might thus hope to do justice to the Lorentz group without bringing in the unwanted contributions we have been worrying about. Comparison with the D’Alembertian in one dimension leads to a similar idea, which in addition works a bit better in the causet, where elements corresponding to the type of “null rectangles” just discussed don’t really exist. In M1, which is just the real line, φ reduces (up to sign) to ∂2φ/∂t2, for which a well known discretization is φ(a) − 2φ(b) + φ(c), a, b and c being three evenly spaced points. Such a configuration does find correspondents in the causet, for example 3-chains x < y < z such that no element other than y lies causally between x and z. Once again, any single one of these chains (partly) determines a frame, but the collection of all of them does not. Although these examples should not be taken too seri- ously (compare the sign in equation (3.1) below), they bring us very close to the following scheme.3 Imagine a causet C consisting of points sprinkled into a region of M2,andfixan element x ∈ C at which we would like to know the value of φ. We can divide the ancestors of x (those elements that causally precede it) into “layers” according to their “distance from x”, as measured by the number of intervening elements. Thus layer 1 comprises those y which are linked to x in the sense that y < x with no intervening elements, layer 2 comprises those y < x with only a single element z such that y < z < x, etc. Our prescription for φ(x) is then to take some combination, with alternating signs, of the first few layers, the specific coefficients to be chosen so that the correct answers are obtained from suitably simple test functions. Perhaps the simplest combination of this sort is
3 A very similar idea was suggested once by Steve Carlip. Does locality fail at intermediate length scales? 33 4 1 Bφ(x) = − φ(x) + −2 + φ(y) , (3.1) l2 2 1 2 3 where the three sums extend over the first three layers as just defined, and l is the fundamental length-scale associated with the sprinkling, normalized so that each sprinkled point occupies, on average, an area of l2. The prescription (3.1) yields a candidate for the “discrete D’Alembertian” B which is retarded, unlike our earlier candidate based on the symmetrized Green function. In order to express this new B explicitly as a matrix, let n(x, y) denote the cardinality of the order- interval y, x={z ∈ C|y < z < x}, or in other words the number of elements of C causally between y and x. Then, assuming that x ≥ y,wehavefrom(3.1), ⎧ ⎨ − 1 for x = y l2 2 B = 1, −2, 1, according as n(x, y) is 0, 1, 2, respectively, for x = y 4 xy ⎩ 0otherwise. (3.2) Now let φ be a fixed test function of compact support on M2, and let x (which we will always take to be included in C) be a fixed point of M2. If we apply B to φ we will of course obtain a random answer depending on the random sprinkling of M2. However, one can prove that the mean of this random variable, EBφ(x), converges to φ(x) in the continuum limit l → 0: E Bxyφy → φ(x), (3.3) l → 0 y where E denotes expectation with respect to the Poisson process that gener- ates the sprinkled causet C. [The proof rests on the following facts. Let us limit the sprinkling to an “interval” (or “causal diamond”) X with x as its top vertex. For test functions that are polynomials of low degree, one can evalu- ate the mean in terms of simple integrals over X – for example the integral dudv/l2 exp{−uv/l2} φ(u,v) – and the results agree with φ(x),upto corrections that vanish like powers of l or faster.] In a sense, then, we have successfully reproduced the D’Alembertian in terms of a causet expression that is fully intrinsic and therefore automatically frame- independent. Moreover, the matrix B, although it introduces nonlocal couplings, does so only on Planckian scales, which is to say, on scales no greater than demanded by the discreteness itself.4
4 It is not difficult to convince oneself that the limit in (3.3) sets in when l shrinks below the characteristic length associated with the function φ; or vice versa, if we think of l as fixed, Bφ will be a good approximation to φ when the characteristic length-scale λ over which φ varies exceeds l: λ l. But this means in turn that (Bφ)(x) can be sampling φ in effect only in a neighborhood of x of characteristic size l. Although B is thoroughly nonlocal at a fundamental level, the scale of its effective nonlocality in application to slowly varying test functions is (in the mean) thus no greater than l. 34 R. D. Sorkin But is our “discrete D’Alembertian” B really a satisfactory tool for building a field theory on a causet? The potential problem that suggests the opposite con- clusion concerns the fluctuations in (3.1), which grow with N rather than dying away. (This growth is indicated by theoretical estimates and confirmed by numeri- cal simulations.) Whether this problem is fatal or not is hard to say. For example, in propagating a classical solution φ forward in time through the causet, it might be that the fluctuations in φ induced by those in (3.1) would remain small when averaged over many Planck lengths, so that the coarse-grained field would not see them. But if this is true, it remains to be demonstrated. And in any case, the fluc- tuations would be bound to affect even the coarse-grained field when they became big enough. For the remainder of this paper, I will assume that large fluctuations are not acceptable, and that one consequently needs a different B that will yield the desired answer not only on average, but (with high probability) in each given case. For that purpose, we will have to make more complicated the remarkably simple ansatz (3.2) that we arrived at above.
3.1.3 Damping the fluctuations To that end, let us return to equation (3.3) and notice that E(Bφ) = (EB)φ, where what I have just called EB is effectively a continuum integral-kernel B¯ in M2.That is to say, when we average over all sprinklings to get EBφ( x), the sums in (3.1)turn into integrals and there results an expression of the form B¯ (x−y)φ(y)d2 y, where B¯ is a retarded, continuous function that can be computed explicitly. Incorporating into B¯ the δ-function answering to φ(x) in (3.1), we get for our kernel (when x > y), 4 2 B¯ (x − y) = p(ξ)e−ξ − δ(2)(x − y), (3.4) l4 l2 (ξ) = − ξ + 1 ξ 2 ξ = v/ 2 v = 1 || − ||2 where p 1 2 2 , l and 2 x y is the volume (i.e. area) of the order-interval in M2 delimited by x and y. The convergence result (3.3) then states that, for φ of compact support, B¯ (x − y)φ(y)d2 y → φ(x). (3.5) l → 0 Notice that, as had to happen, B¯ is Lorentz-invariant, since it depends only on the invariant interval ||x − y||2 =|(x − y) · (x − y)|.5
5 The existence of a Lorentz-invariant kernel B¯ (x) that yields (approximately) φ might seem paradoxical, because one could take the function φ itself to be Lorentz invariant (about the origin x = 0, say), and for such a φ the integrand in (3.5) would also be invariant, whence the integral would apparently have to diverge. This divergence is avoided for compactly supported φ, of course, because the potential divergence is cut off where the integrand goes to zero. But what is truly remarkable in the face of the counter-argument just given, is that Does locality fail at intermediate length scales? 35 Observe, now, that the fundamental discreteness-length has all but disappeared from our story. It remains only in the form of a parameter entering into the def- inition (3.4)oftheintegralkernelB¯ . As things stand, this parameter reflects the scale of microscopic physics from which B¯ has emerged (much as the diffusion constants of hydrodynamics reflect atomic dimensions). But nothing in the defini- tion of B¯ per se forces us to this identification. If in (3.4) we replace l by a freely variable length, and if we then follow the Jacobian dictum, “Man muss immer umkehren”,6 we can arrive at a modification of the discrete D’Alembertian B for which the unwanted fluctuations are damped out by the law of large numbers. Carrying out the first step, let us replace 1/l2 in (3.4) by a new parameter K .We obtain a new continuum approximation to ,
¯ 2 −ξ (2) BK (x − y) = 4K p(ξ)e − 2K δ (x − y), (3.6) whose associated nonlocality-scale is not l but the length K −1/2, which we can take to be much larger than l. Retracing the steps that led from the discrete matrix (3.2) to the continuous kernel (3.4) then brings us to the following causet expression that yields (3.6) when its sprinkling-average is taken: 4 1 B φ(x) = − φ(x) + f (n(x, y), ) φ(y) , (3.7) K l2 2 y