Foundations of space and time; reflections on quantum gravity

F O U N D AT I O N S O F S PA CE A N D T I M E Reflections on Quantum Gravity After almost a century, the field of quantum gravity remains as difficult and inspiring as ever Today, it finds itself a field divided, with two major contenders dominating: string theory, the leading exemplification of the covariant quantization program; and loop quantum gravity, the canonical scheme based on Dirac’s constrained Hamiltonian quantization However, there are now a number of other innovative schemes providing promising new avenues Encapsulating the latest debates on this topic, this book details the different approaches to understanding the very nature of space and time It brings together leading researchers in each of these approaches to quantum gravity to explore these competing possibilities in an open way Its comprehensive coverage explores all the current approaches to solving the problem of quantum gravity, addressing the strengths and weaknesses of each approach, to give researchers and graduate students an up-to-date view of the field is a Senior Lecturer in the Department of Mathematics and Applied Mathematics and a member of the Astrophysics, Cosmology & Gravity Center, University of Cape Town He is interested in all aspects of gravity and is currently working on string theory and connections between gauge theories and gravity J EF F M U RU G AN A M A N DA WELT MAN is a Senior Lecturer in the Department of Mathematics and Applied Mathematics and a member of the Astrophysics, Cosmology & Gravity Center, University of Cape Town She works in the exciting bridging areas of string cosmology, studying physical ways to test string theory within the context of cosmology G EO RG E F R E L L IS is Emeritus Professor ofApplied Mathematics and Honorary Research Associate in the Mathematics Department, University of Cape Town He works on general relativity theory, cosmology, complex systems, and the way physics underlies the functioning of the human brain www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:50:07 WET 2013 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511920998 Cambridge Books Online © Cambridge University Press, 2013 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:50:07 WET 2013 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511920998 Cambridge Books Online © Cambridge University Press, 2013 FO U N DAT I O N S O F S PA CE A N D TI ME Reflections on Quantum Gravity Edited by J E F F M U R U G A N , A M AN D A WE LT M A N & G E O R G E F R E L L I S www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:50:07 WET 2013 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511920998 Cambridge Books Online © Cambridge University Press, 2013 CAMBRI DGE UNI VER SITY PR ESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City 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/9780521114400 © Cambridge University Press 2012 This publication is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published 2012 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Foundations of space and time : reflections on quantum gravity / [edited by] Jeff Murugan, Amanda Weltman & George F R Ellis p cm Includes bibliographical references and index ISBN 978-0-521-11440-0 (hardback) Space and time Quantum gravity I Murugan, Jeff II Weltman, Amanda III Ellis, George F R (George Francis Rayner) IV Title QC173.59.S65F68 2011 2011000387 531 14–dc22 ISBN 978-0-521-11440-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:50:07 WET 2013 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511920998 Cambridge Books Online © Cambridge University Press, 2013 Contents List of contributors page xi The problem with quantum gravity jeff murugan, amanda weltman & george f r ellis A dialogue on the nature of gravity t padmanabhan 2.1 What is it all about? 2.2 Local Rindler observers and entropy flow 2.3 Thermodynamic reinterpretation of the field equations 2.4 Field equations from a new variational principle 2.5 Comparison with the conventional perspective and further comments 2.6 Summary and outlook References 12 17 25 Effective theories and modifications of gravity 50 35 43 47 c p burgess 3.1 Introduction 3.2 Modifying gravity over short distances 3.3 Modifying gravity over long distances 3.4 Conclusions References 50 52 61 66 67 The small-scale structure of spacetime 69 steven carlip 4.1 4.2 4.3 Introduction Spontaneous dimensional reduction? Strong coupling and small-scale structure 69 70 77 v www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:50:20 WET 2013 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511920998 Cambridge Books Online © Cambridge University Press, 2013 vi Contents 4.4 Spacetime foam? 4.5 What next? References 80 81 82 Ultraviolet divergences in supersymmetric theories 85 kellog stelle 5.1 Introduction 5.2 Algebraic renormalization and ectoplasm References 85 93 103 Cosmological quantum billiards 106 axel kleinschmidt & hermann nicolai 6.1 Introduction 6.2 Minisuperspace quantization 6.3 Automorphy and the E10 Weyl group 6.4 Classical and quantum chaos 6.5 Supersymmetry 6.6 Outlook References 106 109 113 116 118 119 122 Progress in RNS string theory and pure spinors 125 dimitry polyakov 7.1 Introduction 7.2 BRST charges of higher-order BRST cohomologies 7.3 Properties of Qn : cohomologies 7.4 New BRST charges and deformed pure spinors 7.5 Conclusions References 125 135 136 137 138 139 Recent trends in superstring phenomenology 140 massimo bianchi 8.1 Foreword 8.2 String theory: another primer 8.3 Phenomenological scenarios 8.4 Intersecting vs magnetized branes 8.5 Unoriented D-brane instantons 8.6 Outlook References 140 141 149 153 156 159 159 Emergent spacetime 164 robert de mello koch & jeff murugan 9.1 9.2 Introduction Simplicity of the 12 -BPS sector www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:50:20 WET 2013 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511920998 Cambridge Books Online © Cambridge University Press, 2013 164 166 Contents 10 vii 9.3 Dictionary 9.4 Organizing the degrees of freedom of a matrix model 9.5 Gravitons 9.6 Strings 9.7 Giant gravitons 9.8 New geometries 9.9 Outlook References 167 168 171 172 173 175 178 180 Loop quantum gravity 185 hanno sahlmann 11 10.1 Introduction 10.2 Kinematical setup 10.3 The Hamilton constraint 10.4 Applications 10.5 Outlook References 185 187 197 203 207 208 Loop quantum gravity and cosmology 211 martin bojowald 12 11.1 Introduction 11.2 Effective dynamics 11.3 Discrete dynamics 11.4 Consistent dynamics 11.5 Consistent effective discrete dynamics 11.6 Outlook: future dynamics References 211 214 225 242 247 251 252 The microscopic dynamics of quantum space as a group field theory 257 daniele oriti 13 12.1 Introduction 12.2 Dynamics of 2D quantum space as a group field theory 12.3 Towards a group field theory formulation of 4D quantum gravity 12.4 A selection of research directions and recent results 12.5 Some important open issues 12.6 Conclusions References 257 279 293 302 314 317 318 Causal dynamical triangulations and the quest for quantum gravity 321 j ambjørn, j jurkiewicz & r loll 13.1 Quantum gravity – taking a conservative stance www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:50:20 WET 2013 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511920998 Cambridge Books Online © Cambridge University Press, 2013 321 viii 14 Contents 13.2 What CDT quantum gravity is about 13.3 What CDT quantum gravity is not about 13.4 CDT key achievements I – demonstrating the need for causality 13.5 CDT key achievements II – the emergence of spacetime as we know it 13.6 CDT key achievements III – a window on Planckian dynamics 13.7 Open issues and outlook References 323 325 330 332 334 335 Proper time is stochastic time in 2D quantum gravity 338 326 j ambjørn, r loll, y watabiki, w westra & s zohren 15 14.1 Introduction 14.2 The CDT formalism 14.3 Generalized CDT 14.4 The matrix model representation 14.5 CDT string field theory 14.6 The matrix model, once again 14.7 Stochastic quantization 14.8 The extended Hamiltonian References 338 339 343 347 347 352 355 358 360 Logic is to the quantum as geometry is to gravity 363 rafael sorkin 15.1 15.2 15.3 15.4 15.5 15.6 16 Quantum gravity and quantal reality Histories and events (the kinematic input) Preclusion and the quantal measure (the dynamical input) The 3-slit paradox and its cognates Freeing the coevent The multiplicative scheme: an example of anhomomorphic coevents 15.7 Preclusive separability and the “measurement problem” 15.8 Open questions and further work 15.9 Appendix: Formal deduction of the 3-slit contradiction References 363 364 366 368 371 374 377 380 382 383 Causal sets: discreteness without symmetry breaking 385 joe henson 16.1 Introduction: seeing atoms with the naked eye 16.2 Causal sets 16.3 Towards quantum gravity www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:50:20 WET 2013 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511920998 Cambridge Books Online © Cambridge University Press, 2013 385 387 395 Contents 17 ix 16.4 Consequences of spacetime discreteness 16.5 Conclusion: back to the rough ground References 401 405 407 The Big Bang, quantum gravity and black-hole information loss 410 roger penrose 18 17.1 General remarks 17.2 The principles of equivalence and quantum superposition 17.3 Cosmology and the 2nd law 17.4 Twistor theory and the regularization of infinities References 410 411 412 415 417 Conversations in string theory 419 amanda weltman, jeff murugan & george f r ellis References 433 Index 435 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:50:20 WET 2013 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511920998 Cambridge Books Online © Cambridge University Press, 2013 Conversations in string theory 423 p k Figure 18.1 Soft graviton emission μν with the soft graviton The amplitude for this emission Mαβ (k) is a product of the amplitude Mαβ for the same process without the soft graviton emission and a μ ηn gn pn pnν kinematical factor , that depends on the 4-momentum pn of the μ p k − iη ε μ n n n nth particle and the coupling gn between the graviton and the nth particle Harold: What about ηn ? Steph: It just encodes ingoing ηn = −1 or outgoing ηn = +1 particles in the scatμν tering process The amplitude is Lorentz-invariant only if kμ Mαβ (k) = This, in gn pnμ is a conserved quantity But, Weinberg reminds us, in turn, is only true if n any non-trivial process, the only linear combination of momenta that is conserved is the total momentum ηn p μ This means that all the couplings gn must be equal n √ and we may as well set gn = 8π G Harold: So what you’re saying is that this low-energy, massless, spin-2 particle couples to all forms of energy in the same way? Steph: Precisely! Harold: Great I hate to be a stickler but what about the ‘strong’ equivalence principle? Steph: Well, following Weinberg’s argument leads to an effective gravitational mass m2 related to the inertial mass mi by mg = 2E − Ei It’s easy to test this relation: for non-relativistic particles, E → mi and mg = mi while relativistic particles require mi → and result in mg = 2E You of all people will no doubt recognize this from GR Harold: I certainly OK, I think I understand However, while we’re on the topic of ‘fields’, I’ve heard from many sources that string theory is full of them If these fields (e.g a scalar ‘dilaton’ field) are indeed there, then why we not experience www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:05 WET 2013 http://dx.doi.org/10.1017/CBO9780511920998.018 Cambridge Books Online © Cambridge University Press, 2013 424 Conversations in string theory them in, for example, the solar system? Surely, the solar system tests presumably exclude them to high precision? Steph: The dilaton in string theory has a unique interpretation as the string coupling – the strength of string–string interactions So if you are referring to some scalar–tensor type gravitational theory, while these are not excluded from string theory I think there are quite stringent constraints put on them The ‘fields’ that are usually referred to in string theory come from the geometric data of the compactified manifold and manifest in the particle physics of the ‘large’ four dimensions This is also the reason why we don’t experience them in everyday life Harold: Let me see if I understand something String theory is a two-dimensional conformal field theory, right? Steph: Correct, the worldsheet theory is Harold: But then this two-dimensional space gets embedded in a target space with ten dimensions, and six of these dimensions get compactified to give a fourdimensional ‘effective’ space? With the string moving in this higher-dimensional spacetime? Steph: Yes, that is exactly what happens Harold: OK, so then something is bothering me I think it goes back to the old Kaluza–Klein idea itself Kaluza–Klein theory is effectively a fibre bundle over the four-dimensional spacetime, with a preferred projection structure that must be preserved when one makes coordinate changes; which is why general coordinate changes are not allowed (you can’t mix the fifth dimension with the four) So it is not a properly covariant theory I think this is why it was abandoned Plus of course, the scalar degree of freedom is not observed Steph: I disagree It wasn’t abandoned It morphed into the structure of modern gauge theory Pauli’s famous objection was an objection to the idea of gauge invariance as first proposed by Weyl [3] This objection certainly stands but it is not how we think of gauge symmetries today (or, for that matter, anytime since Weyl’s reformulation of the idea) Anyway, regarding your objection here, the point is that the five-dimensional theory is invariant under the five-dimensional group of diffeomorphisms Given that this is the case, to get to the four-dimensional spacetime, we have to make some coordinate choice or in the language of gauge theory, a gauge is selected With this choice, the five-dimensional group of diffeomorphisms splits into a four-dimensional group of diffeomorphisms and a U (1) corresponding to the compatification circle Asking for the gauged-fixed theory to be invariant under the full five-dimensional diffeomorphism group is just not meaningful: why should I expect to be able to perform a coordinate transformation and turn a photon into geometry? The four-dimensional spacetime is invariant under four-dimensional coordinate transformations as it should be, and the remaining U (1) manifests in an abelian gauge theory – Maxwell eletromagnetism I realize that an immediate www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:05 WET 2013 http://dx.doi.org/10.1017/CBO9780511920998.018 Cambridge Books Online © Cambridge University Press, 2013 Conversations in string theory 425 objection to this is that the particular direction I chose to compactify on is arbitrary, but this is just a choice of gauge The observables of the theory not care whether x or x is the circle direction Harold: So what you’re saying is that you just identify points in a sixdimensional submanifold of a ten-dimensional spacetime to get six small spatial dimensions Does this not result in extra terms in the Einstein field equations? Steph: Well, it’s a little more complicated than that I’m not really sure what you mean by ‘just identify points’ but if I take it literally, this would result in a 6-torus which, while a nice toy model, is not a consistent compactification When the ten dimensions split into 6+4, the geometric and topological characteristics of the internal space manifest in the particle physics in the four dimensions This necessarily means that the four dimensional space is not empty You are forced by the consistency of the theory to have matter in the non-compact dimensions String theory is not consistent otherwise One cannot treat that problem as one does in classical general relativity by adding in matter of a specific type by hand because at the scales at which string theory is valid everything (matter and gravity) is quantum and coupled and there is no sensible decoupling limit in which one can consider quantum gravity alone (at least not in four dimensions) Harold: Hmm Actually I now think my above statement is wrong Hence if we have a five-dimensional vacuum (or higher) and then compactify to get effective four dimensions, the four-dimensional Einstein field equations will still have extra terms arising from this process, even though one did not explicitly an embedding In Kaluza–Klein these extra terms are the electromagnetic and scalar field stress tensor terms in the four-dimensional Einstein field equations presumably They will not be zero in general Hence one will not get four-dimensional vacuum Einstein field equations in general from five-dimensional vacuum Einstein field equations Steph: That is correct But I fail to see the problem: the universe we live in is certainly not a vacuum You could argue that you should be able to satisfy solar system tests, etc but I am no more convinced that string theory need apply to the solar system than quantum mechanics should General relativity does and as long as I can show that general relativity is a consistent low-energy limit of string theory, I think this is OK Harold: Great, I am glad you mentioned this The effective four-dimensional general relativity is supposed to be derived from the string theory, not just to be a consistent limit It should be a necessary outcome Steph: Yes and no Four-dimensional general relativity needs to be the fourdimensional, low-energy classical limit of the theory As such, it is a necessary outcome www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:05 WET 2013 http://dx.doi.org/10.1017/CBO9780511920998.018 Cambridge Books Online © Cambridge University Press, 2013 426 Conversations in string theory Harold: This seems to be agreeing with me!! Then you can’t say string theory does not apply to the solar system; the story is supposed to be that string theory is the basis of the theory of gravity – that is a major selling point of string theory Steph: In very much the same way as we can’t apply quantum mechanics to the solar system even though we know that it is the underlying theory governing atomic scales, string theory as a description of nature is valid only at energy scales near the Planck scale And just like we take a semiclassical → limit of quantum mechanics, to approach scales that can be probed at accessible energies, we have to average over all the quantum gravity states to obtain a low-energy effective field theory description When we conduct solar system tests, it is this effective field theory (in this case general relativity plus matter) that we’re testing Harold: That must mean including gravity at solar system scales If not, where does the theory of gravity at that scale come from? Steph: Gravity at that scale comes from the following procedure: (1) You identify all the correct degrees of freedom in the quantum gravity theory (2) Integrate out all the high-energy degrees – the low-energy (infrared) physics is insensitive to these (3) Obtain an appropriate low-energy effective field theory – here general relativity + something close to the standard model (4) Make the usual approximation regarding the matter distribution in the solar system Harold: OK, good So what I want to see is the averaging in step Steph: You and I both The above procedure is an in-principle outline In practice, we’re more-or-less still trying to figure out step Harold: There’s something else that’s been bothering me I’ve just read that string theory has tachyonic modes? If this is true, why is this not a problem for the theory? Steph: Let me ask you Harold, when you say ‘tachyon’, what you mean? Harold: I mean a particle that moves faster than the speed of light, of course! Steph: I thought so It’s what I would have expected an excellent relativist like yourself to say Unfortunately, it’s not a useful way to think of it in a quantum field theory Harold: So what is a tachyon then? Steph: Perhaps I can illustrate with a simple example – a scalar field theory with quartic potential The dynamics are encoded in the (symbolic) Lagrangian L = (∂φ)2 + μ2 φ − λφ We know already from classical mechanics that the coefficient of the quadratic term is usually interpreted as the mass of the field The problem is that here this coefficient has the wrong sign! If we were pushed, we’d have to interpret this as a ‘tachyon’ since its mass-squared is negative www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:05 WET 2013 http://dx.doi.org/10.1017/CBO9780511920998.018 Cambridge Books Online © Cambridge University Press, 2013 Conversations in string theory 427 Harold: That’s a bit strong isn’t it? The only reason you have the wrong sign is because you’re writing the Lagrangian in terms of variables suited to an expansion around the origin Steph: Precisely! And this is an unstable point in the potential It’s a local maximum in fact √ If we were to write things in terms of the real lowest energy state at φ = ±μ/ 2λ, we would find a mass term with the correct sign Harold: So Steph: So, the lesson to be learned here is that the presence of a tachyon in the spectrum of a field theory is nothing more than a signal that we’re focusing on an unstable vacuum This is exactly what happens in the 26-dimensional bosonic string and suggests that the theory is unstable It is an instability that is not present in the 10-dimensional superstring theories Harold: Aah, wait a second! You seem to have traded the tachyonic instability for supersymmetry Is that right? Steph: Again yes, and no Yes, in the case of the critical string theories, cancellation of the tachyonic mode in the spectrum does happen because of supersymmetry No, because this isn’t the only mechanism for achieving it Non-critical string theories, for example, not require supersymmetry to cancel the tachyon, at least at tree-level Harold: What you mean by ‘critical’ vs ‘non-critical’ string theory? Steph: Well, one of the main consistency conditions on string theory is that the worldsheet theory must be conformally invariant Classically, this is not a problem but at the quantum level this symmetry can be broken, producing a conformal anomaly For superstring theory, this anomaly vanishes only if the worldsheet theory consists of 10 bosons (each accompanied by a corresponding fermion) Since each boson is interpreted as a dimension of the flat target space (i.e spacetime), this gives a critical dimension of 10 There is another way of getting rid of the conformal anomaly by making the target space non-trivial The result is an anomaly-free string theory that does not have to have critical dimension These theories are clearly of little phenomenological use but nevertheless provide excellent toy-models within which to learn about the structure of string theory Harold: I see In fact this answers my next question about the number of dimensions in string theory, where they come from and how many there are in each of the string theories However, I noticed in that argument that you say that in the critical cases, the string target space is flat Is critical string theory only defined in flat spacetimes? If that’s the case, it’s a bit trivial isn’t it? Steph: Not at all Flat space just happens to be one of three maximally supersymmetric 10-dimensional spacetimes on which string theory is usually studied The other two are the 10-dimensional pp-wave and the celebrated AdS5 ×S spacetime If fact, not only can we define string theory on these backgrounds; in the pp-wave www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:05 WET 2013 http://dx.doi.org/10.1017/CBO9780511920998.018 Cambridge Books Online © Cambridge University Press, 2013 428 Conversations in string theory we can actually quantize the string exactly while all indications are that the worldsheet theory in AdS5 × S is a fully integrable theory Both of these backgrounds also play a crucial role in the gauge theory/gravity duality Harold: Interesting The dualities being discovered via string theory/M-theory sound quite amazing The fact they exist certainly seems to imply that this theory is saying something foundational about mathematics, and is related to physics Would it be true to say that there has been a shift in emphasis, so that string theory is now more a theory about these dualities rather than a fundamental proposal for a unified view of the foundations of physics and of spacetime? Steph: I wouldn’t necessarily say that Certainly in the past decade, these dualities have taught us a fortune about the structure of both quantum gravity as well as gauge theories at strong coupling, things we would not have learnt easily otherwise Harold: Such as? Steph: Such as minimally viscous fluids like the quark–gluon plasma, such as the idea of geometry and topology arising as emergent phenomena, such as an understanding of the microstates of black holes and even the structure of scattering amplitudes in Yang–Mills theory In fact, I’d go so far as to say that if we end up showing that N = supergravity is finite to all orders, it will be because of intuition gained from the gauge/gravity duality Harold: Come now, at least one other theory of quantum gravity purports to give a correct counting of the microstates of a black hole Steph: I’m not disputing that there are other ways to count microstates However, state counting is a kinematic problem In my opinion, more difficult is the issue of dynamics, as encoded, for example, in the greybody spectrum of the black hole As far as I am aware, string theory is the only approach to date that allows a computation of this Harold: Fair enough But you still haven’t answered my question about the shift in focus of research in string theory Steph: I was getting there The way I see it – and I’m quite certain that many would disagree with me on this – if anything, string theory has shifted from being one single theory to a collection of ideas all based on the premise that the fundamental objects in the theory are one-dimensional strings Let me explain Take quantum field theory It certainly isn’t one single theory; scalar field theory, quantum electrodynamics, Yang–Mills theory, the standard model and a host of other ‘theories’ are all specific examples of quantum field theories They are all ideas, based on the premise that the field is the fundamental object and its excitations provide the necessary quanta For me, string theory – the ultraviolet completion of quantum field theory – has a similar flavour AdS/CFT, flux compactifications, string cosmology, matrix theory, D-branes and a host of other ‘theories’ are ideas that constitute ‘string theory’ At the end of the day, each of these ideas teach us www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:05 WET 2013 http://dx.doi.org/10.1017/CBO9780511920998.018 Cambridge Books Online © Cambridge University Press, 2013 Conversations in string theory 429 something about the nature of the whole and since the whole – string theory – is fundamentally a quantum theory of, not only gravity but all the interactions, I would say that it remains a proposal for a unified view of these interactions Harold: OK, then let me ask you a question that has always bothered me What are strings made of? What is doing the vibrating? What is the substance that underlies the existence of strings, and hence of matter? Steph: Nothing! Harold: What you mean? Steph: If you ask that question in the context of string theory, then strings are fundamental, they have no internal structure In this sense, strings are not at all like the classical strings you see around you These have internal structure, they are made up of molecules and atoms Consequently, classical strings have both transverse and longitudinal oscillation modes The strings of string theory, however, only have transverse oscillations – a hallmark of the fact that they have no internal constituents This is not unlike in particle physics, the fundamental object is a point particle As such, we don’t think of these particles as having any internal structure Harold: Except that we know that the correct description of particle physics is quantum field theory and these things we call particles are nothing but the quanta of the field, so in some sense it is the field that is fundamental Steph: Exactly! Something very similar exists for string theory too It is called string field theory and its relation to the fundamental string is the same as that of the quantum field to the particle But expanding on this, I suspect, will be a whole other conversation Harold: Does string theory really say something about cosmology? Aren’t the energies involved so high that it has no real link to cosmology? Steph: The truth is that you’ll get a different answer depending on who you ask So let me give you both my opinion and, I think, the opinion of the more conservative amongst us String theory may have something to say about cosmology And cosmology may be able to show us something about string theory The fact that the energies associated with string theory are so very high tells us that the natural link to cosmology is in the very early universe It is certainly a possibility that, with improved data resolution in the near future, we may be able to see the inner workings of a quantum gravity theory within the effects we observe Of course whether we will be able to distinguish between the various candidate theories is another story On the other hand, any such effects may also just be washed out by inflation [4] The moduli of string theory also present possible candidates for the inflaton or the field responsible for dark energy In this sense string theory may be the very fundamental theory that cosmologists have been waiting for to explain their effectively phenomenological solutions to the early and late time acceleration of the universe Likewise cosmology may provide the ultimate testing ground for www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:05 WET 2013 http://dx.doi.org/10.1017/CBO9780511920998.018 Cambridge Books Online © Cambridge University Press, 2013 430 Conversations in string theory string theory and provide the opportunity to connect to observables Some – in fact many – would say that it is too early to ask these questions of string theory The theory is not yet completely understood so how can we expect to understand cosmology from it yet This is a fair criticism but I don’t think it will stop this very active area of research and should we hit on a real smoking gun signal of string theory in the night sky I suspect the dissenting views will change Harold: Let me then ask you a question about the successes and failures of the theory itself You mention dark energy as a possible future success of the theory, but is it not a past failure? Let’s talk more about dark energy or the cosmological constant or the energy of the vacuum, whatever you choose to call it Is it not the case that string theory predicts that only non-positive values are possible? Didn’t this prediction come before the surprising 1998 observation of the accelerated expansion of the universe [5] (i.e a positive cosmological constant?) Don’t the observations rule out the theory? Steph: This is a good question because it bears on both the theory and observations There was no prediction of a value for the cosmological constant as such before 1998 It was only during the late 1990s that string theorists were discovering the nonperturbative tools so important in understanding the full theory [6] In particular, the discovery of dualities [7], branes [6], black-hole entropy counting [8], matrix theory [9] and the AdS/CFT duality [10] were mostly studied in the context of supersymmetry – which we very much know is not the state the universe is in today We now have solutions with positive cosmological constant [11, 12] but they necessarily break supersymmetry These solutions were found within the context of solving the problem of moduli stabilization – a problem that is ripe for the ideas of cosmology Scientifically, perhaps this misconception arose out of the prominent no-go theorems of Maldacena and Nunez [13] As with all no-go theorems these papers make several very restrictive assumptions Under these assumptions they state that no solutions can be found with a positive cosmological constant However, these theorems not result in a broad constraint on string theory in any sense In particular, one of the constraints requires that no localized sources are included Relaxing this very constraint and breaking supersymmetry allows for us to find solutions with a positive cosmological constant and these are not constraints that one would consider natural given that we know the universe is not supersymmetric at the low-energy scales (or late times) associated with dark energy and we expect a full theory to include non-perturbative ingredients In fact there are now many proposals for the solution of the dark energy problem as well as for a fundamental theory of inflation either based directly on string theory or inspired from string theory ideas – see for example [12, 14, 15] Harold: You mentioned string theory solutions to dark energy Doesn’t this just lead really to a multiverse and ultimately away from verifiability not towards? www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:05 WET 2013 http://dx.doi.org/10.1017/CBO9780511920998.018 Cambridge Books Online © Cambridge University Press, 2013 Conversations in string theory 431 Steph: It is only fair to say that string theory solutions to dark energy are still a matter of much controversy and yes, the multiverse is at the heart of this But in fact I think it is only correct to recognize that in fact the direction that string theory may be going here is really an explanation that was proposed by Steven Weinberg in the 1980s already [16] It is easy in retrospect to think that dark energy was always considered to be a possible outcome and that a positive cosmological constant was expected or worked on by the majority In fact it was not The observation came as a great surprise to the bulk of the physics and astronomy communities, though perhaps not to Steven Weinberg In the 1980s Weinberg [16] was working on the vacuum energy problem and puzzling over why the calculated value from quantum field theory is so high when an observer would never possibly conclude that such a high value was realized as dark energy Ultimately it turns out that the observed value is 123 orders of magnitude smaller than the calculated expected value Weinberg had a possible solution though He argued that if the underlying theory had multiple vacua each ultimately describing a different potential universe, then it could explain why the vacuum energy that we observe is not large but is both small and not zero This thinking does not sound foreign to us now, but remember that this was against all conventional wisdom at the time and remarkably Weinberg’s prediction that the cosmological constant would be small but positive was exactly what was observed over a decade later Now it may be that Weinberg was at least partially correct or even completely correct, and that string theory is this underlying theory with the multiple vacua provided by flux compactifications The multivacuum property of string theory that is so often mocked as its greatest challenge may ultimately be lauded as providing the very solution that nature requires to the cosmological constant problem – the solution glimpsed by Weinberg nearly 30 years ago Harold: But is the landscape an accepted part of string theory? Or is it a proposal supported by only some string theorists? Steph: Let’s start by reminding ourselves how the landscape arises in string theory To so we have to discuss a little about the longstanding moduli stabilization problem Moduli are the degrees of freedom associated with a particular compactification, and in the low-energy effective action they appear as exactly massless scalar fields You may remember that we discussed how particle physics is encoded in geometry in string theory This is the same idea The fact that these fields are massless reflects on the geometry side a freedom for changing the geometry of the compactification in certain ways While there is no energy change associated with this change – on the particle physics side we are left with fields that can evolve freely in space and time If these fields couple to all matter fields and in particular if they couple strongly or even at gravitational strength to all matter fields then equivalence principle violations will be the result We essentially have introduced a fifth force There are quite a few creative attempts at solutions to this problem www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:05 WET 2013 http://dx.doi.org/10.1017/CBO9780511920998.018 Cambridge Books Online © Cambridge University Press, 2013 432 Conversations in string theory [11, 14, 17, 18] but it is only fair to say that this is still an area of active research This question though does lead to the possibility of connecting to cosmology – because it is in cosmology that scalar fields are most used The landscape arises in the context of trying to solve the moduli stabilization problem Naively these moduli are exactly massless – i.e they have V = – so there is no cost to moving around in this moduli space – i.e a flat landscape so to speak If we now find some mechanism (for example flux compactifications [11,19]) to give these moduli nontrivial potentials then the result is a possibly complicated landscape with each point representing a different set of values for the moduli So first we must recognize that the landscape arises through this solution to a specific problem It may be that we are entirely on the wrong track and that the moduli problem is solved in an entirely different way or even that the compactifications could be non-geometric and have an entirely different interpretation I think it is only fair to say it is too early to decide if the landscape is an inevitable outcome in string theory I think it is also worth pointing out that this landscape exists in quantum field theory as well It is not unique to string theory However somehow in string theory the interpretation has led to ideas of multiverses and it has opened up a Pandora’s box of philosophical questions that I not think are necessarily implied Harold: Is there some place in the landscape that genuinely gives the standard model of particle physics? Or is it able to produce only something standard modellike? Steph: We don’t know yet One problem with the landscape is that it is vast Different solutions involve turning on fluxes of various types These fluxes are quantized and so values of the fields in the landscape are not continuously tunable – there are only discrete values allowed Whether there is any point in the landscape that gives us the standard model is not yet known This is very much work in progress though – see [20, 21] for details Harold: If string theory can explain the cosmological constant problem, you can hardly call this a prediction Any more than finding a spin-2 particle in its spectrum means string theory predicts gravity! In fact even finding the standard model in the landscape would be impressive but not a prediction Are there areas where string theory can still make predictions and we can still say we have experimental verifiability? Steph: So what you are really asking is whether string theory is physics Because ultimately the physics process is about theory and experiment each walking together, sometimes one is ahead but unless they are connected we are either mathematicians and philosophers or engineers And the answer is a resounding yes String theory is physics It is rich in mathematics but all of this structure and rigour is aimed at connecting to the observable universe and understanding the puzzles left by other fields Other than the possible connections with cosmology, there are www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:05 WET 2013 http://dx.doi.org/10.1017/CBO9780511920998.018 Cambridge Books Online © Cambridge University Press, 2013 References 433 also connections with phenomonelogy and the possibility of seeing signatures at the LHC Perhaps even more compelling are the unexpected connections to the studies of the quark gluon plasma produced at the Brookhaven National Laboratory [22] Physicists in both areas have found that some of the properties of this plasma can be understood using the powerful tool of string theory – duality In fact there are some properties of the plasma, like its viscosity-to-entropy ratio, that are better modelled as a black hole in a five-dimensional space than as a clump of nucleons in the usual four dimensions The string theory model works far better than could have been expected and, in some ways, does not yet capture some of the higher-order or imperfect effects in the laboratory So the predictions are not as sharp as they could be but they are certainly there Harold: Surely this is more of a mathematical spinoff rather than an actual prediction? Steph: Ever the sceptic! I will quote Joseph Polchinksi [23]: ‘One of the repeated lessons of physics is unity Nature uses a small number of principles in diverse ways And so the quantum gravity that is manifesting itself in dual form at Brookhaven is likely to be the same one that operates everywhere else in the universe.’ So yes Harold – while we may not yet have a conclusive smoking gun or bullet-in-the-head of string theory test, what we have is a broad range of tantalizing hints and the hope that through the sometimes small steps and occasional giant leaps of science we will get there Acknowledgments The questions of Harold mostly represent issues raised by several colleagues – in particular G F R E – in a series of email discussions (beginning in 2006) and daily coffee debates with the string theory group at UCT, mostly J M We are grateful to a number of people who have helped shape the discussion carried out in this chapter In particular, we would like to thank Thanu Padmanabhan for the idea of Harold and permission to reuse him as our inquisitor Further, we are indebited to Alex Hamilton, Julien Larena and Andrea Prinsloo for stimulating much of the coffee discussions J M owes much of his understanding of string theory to conversations with Antal Jevicki, Horatiu Nastase and especially Robert de Mello Koch Finally, we are immensely grateful to our sponsors: the NRF of South Africa and the National Institute for Theoretical Physics, the Foundational Questions Institute and Cambridge University Press References [1] J Schwinger, Particles, Sources and Fields, Volume I (Perseus Books, USA, 1998) [2] C P Burgess, Living Rev Rel 7:5 (2004), 2003 e-Print: gr-qc/0311082 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:05 WET 2013 http://dx.doi.org/10.1017/CBO9780511920998.018 Cambridge Books Online © Cambridge University Press, 2013 434 Conversations in string theory [3] L O’Raifeartaigh, The Dawning of Gauge Theory (Princeton University Press, USA, 1997) [4] R Easther, W H Kinney and H Peiris, JCAP 0505:009 (2005), 2004 e-Print: astroph/0412613; R Easther, W H Kinney and H Peiris, JCAP 0508:001 (2005), e-Print: astro-ph/0505426 [5] Supernova Search Team (Adam G Riess et al.), Astron J 116:1009–38 (1998), e-Print: astro-ph/9805201 [6] J Polchinski, Phys Rev Lett 75:4724–7 (1995), e-Print: hep-th/9510017 [7] For a review, see C Vafa, e-Print: hep-th/9702201 [8] A Strominger and C Vafa, Phys Lett B379:99–104 (1996), e-Print: hep-th/9601029 [9] T Banks, W Fischler (Texas U.), S H Shenker (Rutgers U., Piscataway) and L Susskind, Phys Rev D55:5112–28 (1997), e-Print: hep-th/9610043 [10] J Maldacena, Adv Theor Math Phys 2:231 (1998), e-Print: hep-th/9711200 [11] S Kachru, R Kallosh, A D Linde and S P Trivedi, Phys Rev D68:046005 (2003), e-Print: hep-th/0301240 [12] S Kachru, R Kallosh, A D Linde, J M Maldacena, L P McAllister and S P Trivedi, JCAP 0310:013 (2003), e-Print: hep-th/0308055 [13] J M Maldacena and C Nunez, Int J Mod Phys A16:822–55 (2001), e-Print: hepth/0007018 [14] J Khoury and A Weltman, Phys Rev D69:044026 (2004), e-Print: astro-ph/0309411; J Khoury and A Weltman, Phys Rev Lett 93:171104 (2004), e-Print: astro-ph/ 0309300 [15] P Brax, C van de Bruck, A C Davis, J Khoury and A Weltman, Phys Rev D70:123518 (2004), e-Print: astro-ph/0408415; P Brax, C van de Bruck, A C Davis, J Khoury and A Weltman, AIP Conf Proc 736:105–10 (2005), e-Print: astro-ph/0410103 [16] S Weinberg, Rev Mod Phys 61:1 (1989) [17] T Damour and A M Polyakov, Nucl Phys B423:532–58 (1994) [18] S Carroll, Phys Rev Lett 81:3067–70 (1998), e-Print: astro-ph/9806099 [19] R Bousso and J Polchinski, JHEP 0006:006 (2000), e-Print: hep-th/0004134 [20] Standard model bundles See, for example, R Donagi, B A Ovrut, T Pantev and D Waldram, Adv Theor Math Phys 5:563–615 (2002), e-Print: math/0008010 [21] R Blumenhagen, M Cvetic, P Langacker and G Shiu, Ann Rev Nucl Part Sci 55:71–139 (2005) [22] P Candelas et al., Adv Theor Math Phys 12:2 (2008), e-Print: hep-th/0502005 [23] See ‘On some criticisms of string theory’, at http://www.itp.ucsb.edu/ joep/ www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:05 WET 2013 http://dx.doi.org/10.1017/CBO9780511920998.018 Cambridge Books Online © Cambridge University Press, 2013 Index constrained dynamics, 222 constraint algebra, 200, 247 constraints first class, 187 second class, 189 cosmic censorship, 69 cosmological constant, 2, 25, 30 covariant action, 20, 39, 36 critical dimension, 143, 427 ADM variables, 235, 260 AdS/CFT correspondence, 5, 62, 164 anomalies, 143, 155, 211 anomalous dimensions, 74, 149 anthropic principle, 322 area law, 204 area operator, 74, 194 area spectrum, 74, 194 Ashtekar variables, 229 asymptotic safety, 323, 338 axion, 155 backreaction, 174 background geometry, 43, 136, 138 background independence, 6, 186 Barbero–Immirzi parameter, 188, 189, 207 Bekenstein–Hawking area law, 142, 204, 205 BKL behaviour, 78, 79, 81 black hole entropy, 17, 185, 204 black hole thermodynamics, 40, 41 Bohr compactification, 235 Born’s rule, 380, 383 BPS sector, 165, 166 BPS solitons, 147 BPS states, 148, 166 braneworld, 174 BRST charges, 129, 135 BTZ blackhole, 40 Calabi–Yau, 141, 145 canonical quantization, 186 causal sets, 389 Chan–Paton factors, 126 chaos, 116 chaotic geometry, 410 Chern–Simons couplings, 155 invariant, 98 three-form, 98 chiral anomaly, 155 compactification, 4, 146, 234 conformal field theory, 73, 147, 168 conformal transformation, 168 D-branes, 4, 140, 154 dark energy, 41, 51, 145 dark matter, 51, 62, 159 de-Sitter space, 149, 310, 396 decoherence, 367 density matrix, 18, 20, 27 DeWitt metric, 77, 109 diffeomorphism constraints, 189, 193 invariance, 20, 25, 193 dilaton, 144, 423 dimensional reduction, 69 Dirac brackets, 118 equation, 118 matrices, 54 quantization, 77 Dirichlet boundary conditions, dynamical triangulations, 70, 322 effective action, 60, 73, 90 effective field theory, 54, 61 Einstein equations, 44, 193 Einstein-Hilbert action, 35, 57, 340 embedding, 34, 155, 323 energy-momentum tensor, 27, 421 Euclidean path integral, 343 extremal correlator, 170 F-theory, 141 factor ordering, 228 Faddeev-Popov, 143 fermions, 108, 145 Friedmann equation, 222 435 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:16 WET 2013 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511920998 Cambridge Books Online © Cambridge University Press, 2013 436 Index gauge fields topological current, 99 higher spin, 422 in string theory, 127 gauge fixing, 189, 204 gauge/gravity correspondence, 125 Gauss constraint, 175, 189, 281 general covariance, 3, 57, 213 generating functional, 171, 350 geometric quantization, 282, 397 gravitational entropy, 20 gravitational field, 2, 22, 421 gravitational waves, 62 gravitino, 118 graviton as a spin-2 particle, 15 in effective field theory, 72 in string theory, 156 in AdS/CFT, 187 propagator in LQG, 203 greybody spectrum, 428 group averaging, 196 GSO projection, 140 Hagedorn temperature, 75, 142 Hamiltonian constraint, 77, 78, 189 formulation of gravity, 225 harmonic oscillator, 167, 238, 380 Hartle–Hawking, 121, 348, 358 Hawking radiation, 59, 81, 414 heterotic string, 148 hierarchy problem, 141 holographic principle, 149 holonomy-flux algebra, 234 Hubble parameter, 244 inflation, 59, 249, 413 information loss paradox, 177 instanton, 140 isolated horizon, 204 Jacobian, 127, 167, 179 Kaluza–Klein, 4, 424 k-th Betti number, 306 Klein–Gordon equation, 117, 286 Landau–Lifshitz lagrangian, 178 Landau–Yang theorem, 156 loop quantum cosmology, 5, 187 loop quantum gravity, 5, 81, 185 M-theory, 4, 141, 150 matrix model, 168, 178, 279, 317 Matrix theory, 428, 430 minisuperspace, 106, 113 mixmaster bounces, 78 moduli space, 121, 143, 432 momentum space, 53, 74 Myers effect, 167 near-horizon, 37, 44, 158 Neumann conditions, 112, 113 Newtonian physics, 420 Newton’s constant, 52 non-abelian Yang–Mills, 86 noncommutative geometry, 258 non-renormalizability, 259 NS5-brane, 147 observables in effective field theory, 54 singularity resolution, 122 gauge-invariant, 139 physical, 186 macroscopic, 158 one-loop amplitude, 116 operator product, 81, 130 operator-state correspondence, 168 orbifold, 141 orientifold, 141, 153 path integral Euclidean, 343 simplicial, 275 covariant, 277 path integral quantization, 35, 261 Pauli matrices, 280 p-branes, 147 phase transition, 75, 142, 269 physical state, 2, 223, 251 Planck distance, 81 mass, 141, 142 scale, 25, 70 Poisson bracket, 197 Poincare symmetry, 313 primary field, 131, 132 primordial nucleosynthesis, 63 proton decay, 60 quantum corrections, 54, 212 cosmology, 5, 109, 221, 223 geometry, 212, 323 quasi-normal modes, 35 quaternions, 114 R-symmetry, 100, 102 Ramond–Ramond (RR) sector, 144 Rarita–Schwinger equation, 118 reduced phase-space, 221, 246 Regge calculus, 6, 70, 263 relational picture, 222 renormalizability, 2, 53, 54, 73 renormalization group, 73, 333 Ricci curvature, 415 Rindler spacetime, RNS superstring, 129 S-duality, 148 S-matrix, 88 Schwarzschild black holes, 13, 45, 59 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:16 WET 2013 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511920998 Cambridge Books Online © Cambridge University Press, 2013 Index self-dual connection, 188, 189 semi-classical approximation, 292 singularity cosmological, 107, 108, 120 resolution, 108, 122 spacelike, 78, 80, 81 spin network, 74, 191, 195 spin foam, 185, 186 string field theory, 125, 327, 347, 371 string theory bosonic, 140, 427 critical, 427 type IIA, 143, 147, 152 type IIB, 4, 144, 167 heterotic, 141, 144, 158 non-critical, 347 strong coupling, 69, 148, 152 supergravity, 3, 62, 107, 428 supersymmetry, 3, 90, 100, 427 surface gravity, 38 T-duality, 146, 150, 152, 154 tachyon, 143, 146 tadpole, 143, 151 tetrad, 260, 273 third quantization, 261 topological expansion, 352 topology change, 187, 261, 315 torsion, 96, 150 transplankian physics, 46 triads, 227, 233 two-loop diagrams: 86 U-duality, 148 uncertainty relations, 212, 215 Unruh effect, 16, 411 Unruh temperature, 411 ultraviolet divergences, 85, 89 vacua, 141, 411 vertex operators, 125, 126, 143 volume operator, 75 Weyl curvature, 412 decomposition, 277 group, 108, 113 Wheeler–DeWitt equation, 5, 69 Wick rotation, 168, 327 Wilson loop, 5, 126 WKB approximation, 126, 359 Yang–Mills theory, 4, 91, 164 zero-modes, 158 zero-point energy, 2, 216 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.47.205 on Wed Oct 30 09:55:16 WET 2013 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511920998 Cambridge Books Online © Cambridge University Press, 2013 437 ... publication is available from the British Library Library of Congress Cataloguing in Publication data Foundations of space and time : reflections on quantum gravity / [edited by] Jeff Murugan, Amanda... terms of Wilson loops The result was the theory known as loop quantum gravity. 8 As one of the family of canonical quantum gravity theories, loop quantum gravity is both nonperturbative and manifestly... consistent quantum theory of this graviton The consequences Foundations of Space and Time: Reflections on Quantum Gravity, eds Jeff Murugan, Amanda Weltman and George F R Ellis Published by Cambridge