DOCSLIB.ORG

On the Axioms of Causal Set Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Axioms of Causal Set Theory On the Axioms of Causal Set Theory Benjamin F. Dribus Louisiana State University [email protected] December 24, 2013 Abstract Causal set theory is a promising attempt to model fundamental spacetime structure in a discrete order-theoretic context via sets equipped with special binary relations, called causal sets. The el- ements of a causal set are taken to represent spacetime events, while its binary relation is taken to encode causal relations between pairs of events. Causal set theory was introduced in 1987 by Bombelli, Lee, Meyer, and Sorkin, motivated by results of Hawking and Malament relating the causal, conformal, and metric structures of relativistic spacetime, together with earlier work on discrete causal theory by Finkelstein, Myrheim, and ’t Hooft. Sorkin has coined the phrase, “order plus number equals geometry,” to summarize the causal set viewpoint regarding the roles of causal structure and discreteness in the emergence of spacetime geometry. This phrase represents a specific version of what I refer to as the causal metric hypothesis, which is the idea that the properties of the physical universe, and in particular, the metric properties of classical spacetime, arise from causal structure at the fundamental scale. Causal set theory may be expressed in terms of six axioms: the binary axiom, the measure axiom, countability, transitivity, interval finiteness, and irreflexivity. The first three axioms, which fix the physical interpretation of a causal set, and restrict its “size,” appear in the literature either implic- itly, or as part of the preliminary definition of a causal set. The last three axioms, which encode the essential mathematical structure of a causal set, appear in the literature as the irreflexive formula- tion of causal set theory. Together, these axioms represent a straightforward adaptation of familiar notions of causality to the discrete order-theoretic context. Interval finiteness is often called local finiteness in the literature, an unfortunate misnomer. In this paper, I offer what I view as potentially critical improvements to causal set theory, includ- ing changes to the axioms, and new perspectives and technical methods. Abandoning continuum geometry introduces new types of behavior, such as irreducibility and independence of causal re- lations between pairs of events, inadequately modeled by conventional order theory. Transitive binary relations fail to resolve the subtleties of independent modes of influence between pairs of events. Interval finiteness permits locally infinite behavior incompatible with Sorkin’s version of the causal metric hypothesis, and imposes unjustified restrictions on the global structure of classical spacetime. Intertwined with both axioms is the use of causal intervals to study local spacetime arXiv:1311.2148v3 [gr-qc] 24 Dec 2013 properties, despite their failure to capture or isolate local causal structure. To address these issues, I propose to replace the axioms of transitivity, interval finiteness, and irreflexivity, with local finiteness, and, under conservative assumptions, acyclicity. I refer to abinary relation satisfying these new axioms as a locally finite causal preorder; its transitive closure is the familiar causal order. The resulting models, which I call locally finite directed sets, generalize both causal sets and Finkelstein’s causal nets. The resulting theory diverges significantly from existing approaches, particularly at the quantum level. This is due principally to a broader interpretation of directed structure, and more generally, multidirected structure, inspired by Grothendieck’s scheme- theoretic approach to algebraic geometry. Entities more complex than spacetime events are viewed as elements of higher-level multidirected sets, in analogy to Isham’s topos-theoretic approach to quantum gravity, and Sorkin’s quantum measure theory. This viewpoint leads to a new background- independent version of quantum causal theory. The backbone of this approach is the theory of co- relative histories, an adaptation of the familiar histories approach to quantum theory. Co-relative histories serve as the “relations” of higher-level multidirected sets called kinematic schemes, via the principle of iteration of structure. Systematic use of relation space circumvents the causal set problem of permeability of maximal antichains, leading to the derivation of causal Schr¨odinger-type equations describing quantum spacetime dynamics in the discrete causal context. 1 Contents 1 Introduction 3 1.1 OverviewoftheCausalSetsProgram . ............ 3 1.2 Purposes; Viewpoint and Style; Intended Audience . ................. 8 1.3 UnderlyingNaturalPhilosophy . ............. 9 1.4 Notation and Conventions; Figures . .............. 11 1.5 Summary of Contents; Outline of Sections . ............... 12 2 Axioms and Definitions 20 2.1 TheCausalMetricHypothesis. ............ 20 2.2 TheAxiomsofCausalSetTheory. ........... 22 2.3 Acyclic Directed Sets; Directed Sets; Multidirected Sets ................... 25 2.4 Chains; Antichains; Irreducibility; Independence . ..................... 28 2.5 Domains of Influence; Predecessors and Successors; Boundary and Interior . 30 2.6 Order Theory; Category Theory; Influence of Grothendieck ................. 33 3 Transitivity, Independence, and the Causal Preorder 38 3.1 IndependentModesofInfluence . ............ 38 3.2 Six Arguments that Transitive Binary Relations are Deficient ................ 41 3.3 TheCausalPreorder............................... .......... 44 3.4 Transitive Closure; Skeleton; Degeneracy; Functoriality.................... 44 4 Interval Finiteness versus Local Finiteness 47 4.1 LocalConditions;Topology. ............. 47 4.2 Interval Finiteness versus Local Finiteness . ................... 53 4.3 Relative Multdirected Sets over a Fixed Base . ................ 57 4.4 Eight Arguments against Interval Finiteness and SimilarConditions . .. .. .. .. 61 4.5 Six Arguments for Local Finiteness . .............. 63 4.6 Hierarchy of Finiteness Conditions . ............... 63 5 The Binary Axiom: Events versus Elements 68 5.1 Relation Space over a Multidirected Set . ............... 69 5.2 PowerSpaces ..................................... ........ 80 5.3 CausalPathSpaces ................................ ......... 86 5.4 Path Summation over a Multidirected Set . .............. 94 6 Quantum Causal Theory 98 6.1 Quantum Preliminaries; Iteration of Structure; Co-Relative Histories . 99 6.2 Abstract Quantum Causal Theory via Path Summation . ..............108 6.3 Schr¨odinger-Type Equations in Quantum Causal Theory . ..................114 6.4 KinematicSchemes ................................ .........118 7 Conclusions 128 7.1 New Axioms, Perspectives, and Technical Methods . ................128 7.2 Omitted Topics and Future Research Directions . ................132 7.3 Acknowledgements; Personal Notes . ..............138 A Index of Notation 140 References 148 2 1 Introduction Causal sets are discrete order-theoretic models of classical spacetime. A causal set is a set C, assumed to be countable, whose elements correspond to spacetime events, together with a binary relation ≺ on C, satisfying the additional axioms of transitivity, irreflexivity, and interval finiteness. The binary relation ≺ defines an interval finite partial order1 on C, called the causal order, with the physical interpretation that x ≺ y in C if and only if the event represented by x exerts causal influence on the event represented by y. The physical interpretation of C is completed by fixing a discrete measure2 µ on C, assigning to each subset of C a volume equal to its number of elements in fundamental units, up to Poisson-type fluctuations.3 The role of ≺ and µ in modeling classical spacetime is summarized by the phrase, “order plus number equals geometry,” coined by Rafael Sorkin, the foremost architect and advocate of causal set theory. This phrase represents a special case of what I refer to as the causal metric hypothesis (CMH), which is the idea that the properties of the physical universe, and in particular, the metric properties of classical spacetime, arise from causal structure at the fundamental scale. Approaches to fundamental physics involving some form of the causal metric hypothesis may be collectively referred to as causal theory. Besides causal set theory, these include other discrete causal theories, such as causal dynamical triangulations and causal nets, as well as theories involving interpolative causal models of spacetime, such as domain theory. The purpose of this paper is to offer potentially critical improvements to the causal set program, and to causal theory in general. This effort is based on the conviction that causal theory is among the best-motivated existing approaches to the outstanding problems of fundamental physics, and that causal set theory is perhaps the cleanest and best-balanced existing version of causal theory. Despite important conceptual distinctions, and critical technical differences, between the existing formulation of causal set theory and the approach I offer in this paper, the two are close enough to be considered part of the same broad program. For example, they are closer to each other than they are to the other versions of causal theory mentioned above. In this introductory section, I sketch a broad conceptual context for the more specific material to follow. Section 1.1 is a brief overview of causal set theory, describing its origins, outlining its principal methods, and providing a glimpse of its current state of development. I have included
Load more

Recommended publications
  • Symmetry and Gravity
    universe Article Making a Quantum Universe: Symmetry and Gravity Houri Ziaeepour 1,2 1 Institut UTINAM, CNRS UMR 6213, Observatoire de Besançon, Université de Franche Compté, 41 bis ave. de l’Observatoire, BP 1615, 25010 Besançon, France; [email protected] or [email protected] 2 Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking GU5 6NT, UK Received: 05 September 2020; Accepted: 17 October 2020; Published: 23 October 2020 Abstract: So far, none of attempts to quantize gravity has led to a satisfactory model that not only describe gravity in the realm of a quantum world, but also its relation to elementary particles and other fundamental forces. Here, we outline the preliminary results for a model of quantum universe, in which gravity is fundamentally and by construction quantic. The model is based on three well motivated assumptions with compelling observational and theoretical evidence: quantum mechanics is valid at all scales; quantum systems are described by their symmetries; universe has infinite independent degrees of freedom. The last assumption means that the Hilbert space of the Universe has SUpN Ñ 8q – area preserving Diff.pS2q symmetry, which is parameterized by two angular variables. We show that, in the absence of a background spacetime, this Universe is trivial and static. Nonetheless, quantum fluctuations break the symmetry and divide the Universe to subsystems. When a subsystem is singled out as reference—observer—and another as clock, two more continuous parameters arise, which can be interpreted as distance and time. We identify the classical spacetime with parameter space of the Hilbert space of the Universe.
    [Show full text]
  • D-Instantons and Twistors
    Home Search Collections Journals About Contact us My IOPscience D-instantons and twistors This article has been downloaded from IOPscience. Please scroll down to see the full text article. JHEP03(2009)044 (http://iopscience.iop.org/1126-6708/2009/03/044) The Table of Contents and more related content is available Download details: IP Address: 132.166.22.147 The article was downloaded on 26/02/2010 at 16:55 Please note that terms and conditions apply. Published by IOP Publishing for SISSA Received: January 5, 2009 Accepted: February 11, 2009 Published: March 6, 2009 D-instantons and twistors JHEP03(2009)044 Sergei Alexandrov,a Boris Pioline,b Frank Saueressigc and Stefan Vandorend aLaboratoire de Physique Th´eorique & Astroparticules, CNRS UMR 5207, Universit´eMontpellier II, 34095 Montpellier Cedex 05, France bLaboratoire de Physique Th´eorique et Hautes Energies, CNRS UMR 7589, Universit´ePierre et Marie Curie, 4 place Jussieu, 75252 Paris cedex 05, France cInstitut de Physique Th´eorique, CEA, IPhT, CNRS URA 2306, F-91191 Gif-sur-Yvette, France dInstitute for Theoretical Physics and Spinoza Institute, Utrecht University, Leuvenlaan 4, 3508 TD Utrecht, The Netherlands E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: Finding the exact, quantum corrected metric on the hypermultiplet moduli space in Type II string compactifications on Calabi-Yau threefolds is an outstanding open problem. We address this issue by relating the quaternionic-K¨ahler metric on the hy- permultiplet moduli space to the complex contact geometry on its twistor space. In this framework, Euclidean D-brane instantons are captured by contact transformations between different patches.
    [Show full text]
  • Jhep09(2019)100
    Published for SISSA by Springer Received: July 29, 2019 Accepted: September 5, 2019 Published: September 12, 2019 A link that matters: towards phenomenological tests of unimodular asymptotic safety JHEP09(2019)100 Gustavo P. de Brito,a;b Astrid Eichhornc;b and Antonio D. Pereirad;b aCentro Brasileiro de Pesquisas F´ısicas (CBPF), Rua Dr Xavier Sigaud 150, Urca, Rio de Janeiro, RJ, Brazil, CEP 22290-180 bInstitute for Theoretical Physics, University of Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany cCP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark dInstituto de F´ısica, Universidade Federal Fluminense, Campus da Praia Vermelha, Av. Litor^anea s/n, 24210-346, Niter´oi,RJ, Brazil E-mail: [email protected], [email protected], [email protected] Abstract: Constraining quantum gravity from observations is a challenge. We expand on the idea that the interplay of quantum gravity with matter could be key to meeting this challenge. Thus, we set out to confront different potential candidates for quantum gravity | unimodular asymptotic safety, Weyl-squared gravity and asymptotically safe gravity | with constraints arising from demanding an ultraviolet complete Standard Model. Specif- ically, we show that within approximations, demanding that quantum gravity solves the Landau-pole problems in Abelian gauge couplings and Yukawa couplings strongly con- strains the viable gravitational parameter space. In the case of Weyl-squared gravity with a dimensionless gravitational coupling, we also investigate whether the gravitational con- tribution to beta functions in the matter sector calculated from functional Renormalization Group techniques is universal, by studying the dependence on the regulator, metric field parameterization and choice of gauge.
    [Show full text]
  • On the Axioms of Causal Set Theory
    On the Axioms of Causal Set Theory Benjamin F. Dribus Louisiana State University [email protected] November 8, 2013 Abstract Causal set theory is a promising attempt to model fundamental spacetime structure in a discrete order-theoretic context via sets equipped with special binary relations, called causal sets. The el- ements of a causal set are taken to represent spacetime events, while its binary relation is taken to encode causal relations between pairs of events. Causal set theory was introduced in 1987 by Bombelli, Lee, Meyer, and Sorkin, motivated by results of Hawking and Malament relating the causal, conformal, and metric structures of relativistic spacetime, together with earlier work on discrete causal theory by Finkelstein, Myrheim, and 't Hooft. Sorkin has coined the phrase, \order plus number equals geometry," to summarize the causal set viewpoint regarding the roles of causal structure and discreteness in the emergence of spacetime geometry. This phrase represents a specific version of what I refer to as the causal metric hypothesis, which is the idea that the properties of the physical universe, and in particular, the metric properties of classical spacetime, arise from causal structure at the fundamental scale. Causal set theory may be expressed in terms of six axioms: the binary axiom, the measure axiom, countability, transitivity, interval finiteness, and irreflexivity. The first three axioms, which fix the physical interpretation of a causal set, and restrict its \size," appear in the literature either implic- itly, or as part of the preliminary definition of a causal set. The last three axioms, which encode the essential mathematical structure of a causal set, appear in the literature as the irreflexive formula- tion of causal set theory.
    [Show full text]
  • Causal Dynamical Triangulations and the Quest for Quantum Gravity?
    Appendices: Mathematical Methods for Basic and Foundational Quantum Gravity Unstarred Appendices support Part I’s basic account. Starred Appendices support Parts II and III on interferences between Problem of Time facets. Double starred ones support the Epilogues on global aspects and deeper levels of mathematical structure being contemplated as Background Independent. If an Appendix is starred, the default is that all of its sections are starred likewise; a few are marked with double stars. Appendix A Basic Algebra and Discrete Mathematics A.1 Sets and Relations For the purposes of this book, take a set X to just be a collection of distinguishable objects termed elements. Write x ∈ X if x is an element of X and Y ⊂ X for Y a subset of X, ∩ for intersection, ∪ for union and Yc = X\Y for the complement of Y in X. Subsets Y1 and Y2 are mutually exclusive alias disjoint if Y1 ∩ Y2 =∅: the empty set. In this case, write Y1 ∪ Y2 as Y1 Y2: disjoint union.Apartition of a set X is a splitting of its elements into subsets pP that are mutually exclusive = and collectively exhaustive: P pP X. Finally, the direct alias Cartesian product of sets X and Z, denoted X × Z, is the set of all ordered pairs (x, z) for x ∈ X, z ∈ Z. For sets X and Z,afunction alias map ϕ : X → Z is an assignation to each x ∈ X of a unique image ϕ(x) = z ∈ Z. Such a ϕ is injective alias 1to1if ϕ(x1) = ϕ(x2) ⇒ x1 = x2, surjective alias onto if given z ∈ Z there is an x ∈ X such that ϕ(x) = z, and bijective if it is both injective and surjective.
    [Show full text]
  • Twistor Theory at Fifty: from Rspa.Royalsocietypublishing.Org Contour Integrals to Twistor Strings Michael Atiyah1,2, Maciej Dunajski3 and Lionel Review J
    Downloaded from http://rspa.royalsocietypublishing.org/ on November 10, 2017 Twistor theory at fifty: from rspa.royalsocietypublishing.org contour integrals to twistor strings Michael Atiyah1,2, Maciej Dunajski3 and Lionel Review J. Mason4 Cite this article: Atiyah M, Dunajski M, Mason LJ. 2017 Twistor theory at fifty: from 1School of Mathematics, University of Edinburgh, King’s Buildings, contour integrals to twistor strings. Proc. R. Edinburgh EH9 3JZ, UK Soc. A 473: 20170530. 2Trinity College Cambridge, University of Cambridge, Cambridge http://dx.doi.org/10.1098/rspa.2017.0530 CB21TQ,UK 3Department of Applied Mathematics and Theoretical Physics, Received: 1 August 2017 University of Cambridge, Cambridge CB3 0WA, UK Accepted: 8 September 2017 4The Mathematical Institute, Andrew Wiles Building, University of Oxford, Oxford OX2 6GG, UK Subject Areas: MD, 0000-0002-6477-8319 mathematical physics, high-energy physics, geometry We review aspects of twistor theory, its aims and achievements spanning the last five decades. In Keywords: the twistor approach, space–time is secondary twistor theory, instantons, self-duality, with events being derived objects that correspond to integrable systems, twistor strings compact holomorphic curves in a complex threefold— the twistor space. After giving an elementary construction of this space, we demonstrate how Author for correspondence: solutions to linear and nonlinear equations of Maciej Dunajski mathematical physics—anti-self-duality equations e-mail: [email protected] on Yang–Mills or conformal curvature—can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang– Mills and gravitational instantons, which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of anti-self-dual (ASD) Yang–Mills equations, and Einstein–Weyl dispersionless systems are reductions of ASD conformal equations.
    [Show full text]
  • The Multiple Realizability of General Relativity in Quantum Gravity
    The multiple realizability of general relativity in quantum gravity Rasmus Jaksland∗ Forthcoming in Synthese† Abstract Must a theory of quantum gravity have some truth to it if it can recover general rela- tivity in some limit of the theory? This paper answers this question in the negative by indicating that general relativity is multiply realizable in quantum gravity. The argu- ment is inspired by spacetime functionalism – multiple realizability being a central tenet of functionalism – and proceeds via three case studies: induced gravity, thermodynamic gravity, and entanglement gravity. In these, general relativity in the form of the Ein- stein field equations can be recovered from elements that are either manifestly multiply realizable or at least of the generic nature that is suggestive of functions. If general relativity, as argued here, can inherit this multiple realizability, then a theory of quantum gravity can recover general relativity while being completely wrong about the posited microstructure. As a consequence, the recovery of general relativity cannot serve as the ultimate arbiter that decides which theory of quantum gravity that is worthy of pursuit, even though it is of course not irrelevant either qua quantum gravity. Thus, the recovery of general relativity in string theory, for instance, does not guarantee that the stringy account of the world is on the right track; despite sentiments to the contrary among string theorists. Keywords quantum gravity, general relativity, spacetime functionalism, entanglement, emergent gravity, multiple realizability, string theory ∗Department of Philosophy and Religious Studies, NTNU – Norwegian University of Science and Technology, Trondheim, Norway. Email: [email protected] †This is a post-peer-review, pre-copyedit version of an article published in Synthese.
    [Show full text]
  • Redalyc.Big Extra Dimensions Make a Too Small
    Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil Sorkin, Rafael D. Big extra dimensions make A too Small Brazilian Journal of Physics, vol. 35, núm. 2A, june, 2005, pp. 280-283 Sociedade Brasileira de Física Sâo Paulo, Brasil Available in: http://www.redalyc.org/articulo.oa?id=46435212 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative 280 Brazilian Journal of Physics, vol. 35, no. 2A, June, 2005 Big Extra Dimensions Make L too Small Rafael D. Sorkin Perimeter Institute, 31 Caroline Street North, Waterloo ON, N2L 2Y5 Canada and Department of Physics, Syracuse University, Syracuse, NY 13244-1130, U.S.A. Received on 23 February, 2005 I argue that the true quantum gravity scale cannot be much larger than the Planck length, because if it were then the quantum gravity-induced fluctuations in L would be insufficient to produce the observed cosmic “dark energy”. If one accepts this argument, it rules out scenarios of the “large extra dimensions” type. I also point out that the relation between the lower and higher dimensional gravitational constants in a Kaluza-Klein theory is precisely what is needed in order that a black hole’s entropy admit a consistent higher dimensional interpretation in terms of an underlying spatio-temporal discreteness. Probably few people anticipate that laboratory experiments in L far too small to be compatible with its observed value.
    [Show full text]
  • Manifoldlike Causal Sets
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by eGrove (Univ. of Mississippi) University of Mississippi eGrove Electronic Theses and Dissertations Graduate School 2019 Manifoldlike Causal Sets Miremad Aghili University of Mississippi Follow this and additional works at: https://egrove.olemiss.edu/etd Part of the Physics Commons Recommended Citation Aghili, Miremad, "Manifoldlike Causal Sets" (2019). Electronic Theses and Dissertations. 1529. https://egrove.olemiss.edu/etd/1529 This Dissertation is brought to you for free and open access by the Graduate School at eGrove. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of eGrove. For more information, please contact [email protected]. Manifoldlike Causal Sets A Dissertation presented in partial fulfillment of requirements for the degree of Doctor of Philosophy in the Department of Physics and Astronomy The University of Mississippi by Miremad Aghili May 2019 Copyright Miremad Aghili 2019 ALL RIGHTS RESERVED ABSTRACT The content of this dissertation is written in a way to answer the important question of manifoldlikeness of causal sets. This problem has importance in the sense that in the continuum limit and in the case one finds a formalism for the sum over histories, the result requires to be embeddable in a manifold to be able to reproduce General Relativity. In what follows I will use the distribution of path length in a causal set to assign a measure for manifoldlikeness of causal sets to eliminate the dominance of nonmanifoldlike causal sets. The distribution of interval sizes is also investigated as a way to find the discrete version of scalar curvature in causal sets in order to present a dynamics of gravitational fields.
    [Show full text]
  • Dimension and Dimensional Reduction in Quantum Gravity
    March 2019 Dimension and Dimensional Reduction in Quantum Gravity S. Carlip∗ Department of Physics University of California Davis, CA 95616 USA Abstract If gravity is asymptotically safe, operators will exhibit anomalous scaling at the ultraviolet fixed point in a way that makes the theory effectively two- dimensional. A number of independent lines of evidence, based on different approaches to quantization, indicate a similar short-distance dimensional reduction. I will review the evidence for this behavior, emphasizing the physical question of what one means by “dimension” in a quantum spacetime, and will dis- cuss possible mechanisms that could explain the universality of this phenomenon. For proceedings of the conference in honor of Martin Reuter: “Quantum Fields—From Fundamental Concepts to Phenomenological Questions” September 2018 arXiv:1904.04379v1 [gr-qc] 8 Apr 2019 ∗email: [email protected] 1. Introduction The asymptotic safety program offers a fascinating possibility for the quantization of gravity, starkly different from other, more common approaches, such as string theory and loop quantum gravity [1–3]. We don’t know whether quantum gravity can be described by an asymptotically safe (and unitary) field theory, but it might be. For “traditionalists” working on quantum gravity, this raises a fundamental question: What does asymptotic safety tell us about the small-scale structure of spacetime? So far, the most intriguing answer to this question involves the phenomenon of short-distance dimensional reduction. It seems nearly certain that, near a nontrivial ultraviolet fixed point, operators acquire large anomalous dimensions, in such a way that their effective dimensions are those of operators in a two-dimensional spacetime [4–6].
    [Show full text]
  • Arxiv:Gr-Qc/9510063V1 31 Oct 1995
    STRUCTURAL ISSUES IN QUANTUM GRAVITYa C.J. Isham b Blackett Laboratory, Imperial College of Science, Technology and Medicine, South Kensington, London SW7 2BZ A discursive, non-technical, analysis is made of some of the basic issues that arise in almost any approach to quantum gravity, and of how these issues stand in relation to recent developments in the field. Specific topics include the applicability of the conceptual and mathematical structures of both classical general relativity and standard quantum theory. This discussion is preceded by a short history of the last twenty-five years of research in quantum gravity, and concludes with speculations on what a future theory might look like. 1 Introduction 1.1 Some Crucial Questions in Quantum Gravity In this lecture I wish to reflect on certain fundamental issues that can be expected to arise in almost all approaches to quantum gravity. As such, the talk is rather non-mathematical in nature—in particular, it is not meant to be a technical review of the who-has-been-doing-what-since-GR13 type: the subject has developed in too many different ways in recent years to make this option either feasible or desirable. The presentation is focussed around the following prima facie questions: 1. Why are we interested in quantum gravity at all? In the past, different researchers have had significantly different motivations for their work— and this has had a strong influence on the technical developments of the subject. arXiv:gr-qc/9510063v1 31 Oct 1995 2. What are the basic ways of trying to construct a quantum theory of gravity? For example, can general relativity be regarded as ‘just another field theory’ to be quantised in a more-or-less standard way, or does its basic structure demand something quite different? 3.
    [Show full text]
  • Quantum Gravity Past, Present and Future
    Quantum Gravity past, present and future carlo rovelli vancouver 2017 loop quantum gravity, Many directions of investigation string theory, Hořava–Lifshitz theory, supergravity, Vastly different numbers of researchers involved asymptotic safety, AdS-CFT-like dualities A few offer rather complete twistor theory, tentative theories of quantum gravity causal set theory, entropic gravity, Most are highly incomplete emergent gravity, non-commutative geometry, Several are related, boundaries are fluid group field theory, Penrose nonlinear quantum dynamics causal dynamical triangulations, Several are only vaguely connected to the actual problem of quantum gravity shape dynamics, ’t Hooft theory non-quantization of geometry Many offer useful insights … loop quantum causal dynamical gravity triangulations string theory asymptotic Hořava–Lifshitz safety group field AdS-CFT theory dualities twistor theory shape dynamics causal set supergravity theory Penrose nonlinear quantum dynamics non-commutative geometry Violation of QM non-quantized geometry entropic ’t Hooft emergent gravity theory gravity Several are related Herman Verlinde at LOOP17 in Warsaw No major physical assumptions over GR&QM No infinity in the small loop quantum causal dynamical Infinity gravity triangulations in the small Supersymmetry string High dimensions theory Strings Lorentz Violation asymptotic Hořava–Lifshitz safety group field AdS-CFT theory dualities twistor theory Mostly still shape dynamics causal set classical supergravity theory Penrose nonlinear quantum dynamics non-commutative geometry Violation of QM non-quantized geometry entropic ’t Hooft emergent gravity theory gravity Discriminatory questions: Is Lorentz symmetry violated at the Planck scale or not? Are there supersymmetric particles or not? Is Quantum Mechanics violated in the presence of gravity or not? Are there physical degrees of freedom at any arbitrary small scale or not? Is geometry discrete i the small? Lorentz violations Infinite d.o.f.
    [Show full text]