Complementarity of Structure and Dynamics

One of the main challenges in theoretical physics over the last five decades has been to reconcile quantum mechanics with general relativity into a theory of quantum gravity. However, such a theory has been proved to be hard to attain due to i) conceptual difficulties present in both the component theories (General Relativity (GR) and Quantum Theory); ii) lack of experimental evidence, since the regimes at which quantum gravity is expected to be applicable are far beyond the range of conceivable experiments. Despite these difficulties, various approaches for a theory of Quantum Gravity have been developed. In this thesis we focus on two such approaches: Loop Quantum Gravity and the Topos theoretic approach. The choice fell on these approaches because, although they both reject the Copenhagen interpretation of quantum theory, their underpinning philosophical approach to formulating a quantum theory of gravity are radically different. In particular LQG is a rather conservative scheme, inheriting all the formalism of both GR and Quantum Theory, as it tries to bring to its logical extreme consequences the possibility of combining the two. On the other hand, the Topos approach involves the idea that a radical change of perspective is needed in order to solve the problem of quantum gravity, especially in regard to the fundamental concepts of `space' and `time'. Given the partial successes of both approaches, the hope is that it might be possible to find a common ground in which each approach can enrich the other.

The two fundamental pillars of physics for over 100 years have been quantum theory and general relativity, but their unification at short distances remains elusive, both technically and conceptually. This work is a philosophical investigation of the second kind of problem, and in particular of the striking fact that in many approaches to ‘quantum gravity’ classical spacetime structures are not merely quantized, but arguably absent—so that spacetime is not merely a classical limit, but ‘emergent’. This issue is not only central to the problem of quantum gravity, but of deep significance for our philosophical understanding of physical reality, promising a conceptual revolution at least as profound as Einstein’s. We give an introduction to the question of spacetime emergence in general, for philosophers of metaphysics and science, and argue that spacetime functionalism explains how something non-spatiotemporal could ever appear as space and time. More technical chapters investigate the issue in detail for causal set theory, loop quantum gravity, and string theory, and the book also serves as a philosophical introduction to those theories for philosophers of physics. Our results help physicists clarify what new conceptual framework—not resting on space and time—may be necessary to achieve a theory of quantum gravity; and show philosophers how the world may not be spatiotemporal at root, and what kind of a world we might then live in.

I take non-locality to be the Michelson–Morley experiment of the early 21st century, assume its universal validity, and try to derive its consequences. Spacetime, with its locality, cannot be fundamental, but must somehow be emergent from entangled coherent quantum variables and their behaviors. There are, then, two immediate consequences: (i). if we start with non-locality, we need not explain non-locality. We must instead explain an emergence of locality and spacetime. (ii). There can be no emergence of spacetime without matter. These propositions flatly contradict General Relativity, which is foundationally local, can be formulated without matter, and in which there is no “emergence” of spacetime. If these be true, then quantum gravity cannot be a minor alteration of General Relativity but must demand its deep reformulation. This will almost inevitably lead to: matter not only curves spacetime, but “creates” spacetime. We will see independent grounds for the assertion that matter both curves and creates spacetime that may invite a new union of quantum gravity and General Relativity. This quantum creation of spacetime consists of: (i) fully non-local entangled coherent quantum variables. (ii) The onset of locality via decoherence. (iii) A metric in Hilbert space among entangled quantum variables by the sub-additive von Neumann entropy between pairs of variables. (iv) Mapping from metric distances in Hilbert space to metric distances in classical spacetime by episodic actualization events. (v) Discrete spacetime is the relations among these discrete actualization events. (vi) “Now” is the shared moment of actualization of one among the entangled variables when the amplitudes of the remaining entangled variables change instantaneously. (vii) The discrete, successive, episodic, irreversible actualization events constitute a quantum arrow of time. (viii) The arrow of time history of these events is recorded in the very structure of the spacetime constructed. (ix) Actual Time is a succession of two or more actual events. The theory inevitably yields a UV cutoff of a new type. The cutoff is a phase transition between continuous spacetime before the transition and discontinuous spacetime beyond the phase transition. This quantum creation of spacetime modifies General Relativity and may account for Dark Energy, Dark Matter, and the possible elimination of the singularities of General Relativity. Relations to Causal Set Theory, faithful Lorentzian manifolds, and past and future light cones joined at “Actual Now” are discussed. Possible observational and experimental tests based on: (i). the existence of Sub- Planckian photons, (ii). knee and ankle discontinuities in the high-energy gamma ray spectrum, and (iii). possible experiments to detect a creation of spacetime in the Casimir system are discussed. A quantum actualization enhancement of repulsive Casimir effect would be anti-gravitational and of possible practical use. The ideas and concepts discussed here are not yet a theory, but at most the start of a framework that may be useful.

Search for a unification of Einstein’s general relativity and quantum theory in quantum gravity theory is the most important and most difficult research programme in fundamental physics since early 1980’s. In this article we present motivation for the unification and some of main difficulties which prevent contemporary physicists from achieving the aim. We focus our attention on the conceptual problems regarding the notions of time, space and spacetime in quantum gravity. After explaining of why gravity should be quantized (section 1) we present the modem notions of time and spacetime according to general relativity (in section 2) and then compare them in section 3 with the concepts of time and space as they are necessary in quantum mechanics in order for the probabilistic interpretation of the theory to work. Since the concepts in both the theories are different and reconciling them in quantum gravity is extremely hard, we present in section 4 attempts to achieve a less ambitious aim, a fully quantum theory of matter in a classical gravitational field. The current idea that the conceptual problems of foundations of quantum theory may be solved with the help of quantum theory of the Universe is discussed in the last section.

Two radically different views about time are possible. According to the first, the universe is three dimensional. It has a past and a future, but that does not mean it is spread out in time as it is spread out in the three dimensions of space. This view requires that there is an unambiguous, absolute, cosmic-wide “now” at each instant. According to the second view about time, the universe is four dimensional. It is spread out in both space and time—in space-time in short. Special and general relativity rule out the first view. There is, according to relativity theory, no such thing as an unambiguous, absolute cosmic-wide “now” at each instant. However, we have every reason to hold that both special and general relativity are false. Not only does the historical record tell us that physics advances from one false theory to another. Furthermore, elsewhere I have shown that we must interpret physics as having established physicalism—in so far as physics can ever establish anything theoretical. Physicalism, here, is to be interpreted as the thesis that the universe is such that some unified “theory of everything” is true. Granted physicalism, it follows immediately that any physical theory that is about a restricted range of phenomena only, cannot be true, whatever its empirical success may be. It follows that both special and general relativity are false. This does not mean of course that the implication of these two theories that there is no unambiguous cosmic-wide “now” at each instant is false. It still may be the case that the first view of time, indicated at the outset, is false. Are there grounds for holding that an unambiguous cosmic-wide “now” does exist, despite special and general relativity, both of which imply that it does not exist? There are such grounds. Elsewhere I have argued that, in order to solve the quantum wave/particle problem and make sense of the quantum domain we need to interpret quantum theory as a fundamentally probabilistic theory, a theory which specifies how quantum entities—electrons, photons, atoms—interact with one another probabilistically. It is conceivable that this is correct, and the ultimate laws of the universe are probabilistic in character. If so, probabilistic transitions could define unambiguous, absolute cosmic-wide “nows” at each instant. It is entirely unsurprising that special and general relativity have nothing to say about the matter. Both theories are pre-quantum mechanical, classical theories, and general relativity in particular is deterministic. The universe may indeed be three dimensional, with a past and a future, but not spread out in four dimensional space-time, despite the fact that relativity theories appear to rule this out. These considerations, finally, have implications for views about the arrow of time and free will.

At the heart of the black hole information loss paradox and the firewall controversy lies the conflict between quantum mechanics and general relativity. Much has been said about quantum corrections to general relativity, but much less in the opposite direction. It is therefore crucial to examine possible corrections to quantum mechanics due to gravity. Indeed, the Heisenberg Uncertainty Principle is one profound feature of quantum mechanics, which nevertheless may receive correction when gravitational effects become important. Such generalized uncertainty principle [GUP] has been motivated from not only quite general considerations of quantum mechanics and gravity, but also string theoretic arguments. We examine the role of GUP in the context of black hole complementarity. We find that while complementarity can be violated by large N rescaling if one assumes only the Heisenberg’s Uncertainty Principle, the application of GUP may save complementarity, but only if certain N -dependence is also assumed. This raises two important questions beyond the scope of this work, i.e., whether GUP really has the proposed form of N -dependence, and whether black hole complementarity is indeed correct.

Conserved vectors are divergencies of superpotentials. In field theory on curved backgrounds, they are useful in calculating global ‘charges’ in arbitrary coordinates and local conserved quantities for small perturbations with specific gauge conditions. Superpotentials are, however, ill–defined. A new criterion of Julia and Silva selects uniquely for Dirichlet boundary conditions the ‘KBL superpotential’ as proposed by Katz, Bičák and Lynden–Bell, which has remarkable properties.Here, we show that a Belinfante–type addition to the KBL superpotential in general relativity gives an expression that is independent of boundary conditions defined by a variational principle. The modified superpotential has the same global properties as the KBL one, except for angular momentum at null infinity, and it does not differ from the KBL superpotential in the linearized theory of gravitation.As an illustration in linearized theory on curved backgrounds, we calculate conserved quantities for small perturbations on a Friedmann–Robertson–Walker spacetime associated with conformal Killing vectors. Our unifying view relates a number of applications in cosmology found in the literature. Globally conserved quantities have simple physical interpretations in the ‘uniform Hubble expansion’ gauge.

Loop quantum gravity is a physical theory that aims at unifying general relativity and quantum mechanics. It takes general relativity very seriously and modifies it via quantisation. General relativity describes gravity in terms of geometry; therefore, quantising such theory must be equivalent to quantising geometry and that is what loop quantum gravity does. This sounds like a mathematical task as well, thus, in this paper we will present the mathematics of loop quantum gravity. We will do it from a mathematician’s point of view. This paper is intended to be an introduction to loop quantum gravity for postgraduate students of physics and mathematics. In this work we will restrict ourselves to the three-dimensional case.

Must a theory of quantum gravity have some truth to it if it can recover general relativity 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 argument 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 Einstein 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.

This special issue is devoted to the proceedings of the workshops Peyresq Physics 11 and Peyresq Physics 12 on 'Micro and Macro Structure of Spacetime', which were held on the 17–23 June 2006 and 16–22 June 2007, respectively, in Peyresq, Alpes de Haute Provence, France, and organized by E Gunzig, B-L Hu and one of us (EV). These workshops are part of a series of annual meetings which bring together around 20 physicists active in different areas of theoretical physics.Uncovering the microscopic structures of spacetime is the goal of quantum gravity and string theory. It is often considered as the 'holy grail' of theoretical physics because it is believed that all fundamental forces are unified at that level of description. Macroscopic spacetime refers to astrophysical and cosmological structures and processes in the observable physical world, from black holes to the cosmos, from the second law to the arrow of time. The challenge theoretical physicists now face is to infer the micro structure of spacetime from the observed macro phenomena, and to deduce the multifarious manifestations of our world from any such candidate theory. To do so demands a careful re-examination of the foundational issues of quantum physics and general relativity (such as intrinsic and quantum gravity decoherence), the origin and nature of symmetries in spacetime (such as Lorentz symmetry) and the conditions where they may be violated, as well as their intersections (such as the physics of the early universe and the end states of black hole evolution).One asks how and why these two pillars of modern physics appear to be so incongruous with each other. Is it because their basic tenets and assumptions are at odds to each other? For these quests one needs to draw upon concepts and techniques from close disciplines such as those found in topology and geometry, or other subdisciplines of physics such as particle physics, strings and loops, statistical mechanics and mesoscopic phenomena, atomic-optical and condensed matter physics. The richness of this theme spans a diverse spectrum of current topics which can be gleaned from the titles of the talks presented in these two meetings.Peyresq is a medieval Provençal village situated 100km from Nice at an altitude of 1528m. The village was founded in the early 13th century. At thebeginning of the 17th century there were around 50 houses and by 1851 the village counted 208 inhabitants distributed among 53 families. Like many othervillages of Haute Provence, it was almost completely deserted after the Second World War. During the 1950s the village was progressively entirely rebuilt inits original spirit and style by students of Belgian universities, mainly the Université Libre de Bruxelles, under the guidance of Mady and Toine Smets.The aim was to create a 'Foyer d'Humanisme', an international humanistic center for cultural, artistic, and scientific pursuits.The workshops were financed by the Fondation Peyresq, Foyer d'Humanisme, the Fondation Nicolas Claude Fabri de Peyresq, and OLAM (Association pour laRecherche Fondamentale, Brussels). We would like to thank all these institutions for their help and financial support. We extend our warm appreciation to Madame Mady Smets without whom none of this work could have come to light. We would like to thank all the participants for the many lively discussions that we have enjoyed and for their effort in preparing written contributions. We would also like to thank the editorial staff of Classical and Quantum Gravity, especially Tom Spicer, Joseph Tennant and Eirini Messaritaki, for their support and efficiency in preparing this volume. Finally, we thank Edgar Gunzig and Bei-Lok Hu as the main driving forces behind the organization of these workshops.

The importance of language in physics has gained emphasis in recent times, on the one hand through Hilbert’s views that concern formalism and intuition applied for outer inquiry, and on the other hand through Brouwer’s point of view that concerns intuition applied for inner inquiry or, as I call, self-inquiry. It is to demonstrate the essence of such investigations, especially self-inquiry (inward intuition), I find it compelling to report that a careful analysis of Cauchy’s statements for the definition of derivative, as applied in physics, unveils the connection to the Heisenberg uncertainty principle as a condition for the failure of classical mechanics. Such logico-linguistic, or semantically driven, self-inquiry of physics can provide new insights to physicists in the pursuit of truth and reality, for example, in the context of Schroedinger equation. I point out an explicit dilemma that plagues the semantics of physics, as far as general relativity and quantum mechanics are concerned, which needs to be taken into account during any attempt to pen down a theory of “quantum gravity”.

Ontology is integral to the two most fundamental scientific theories of the twentieth century: quantum theory and the special and general theories of relativity. Issues that drove the development of quantum theory include the reality of quanta, the simultaneous wave- and particle-like nature of matter and energy, determinism, probability and randomness, Schrodinger’s wave equation, and Heisenberg’s uncertainty principle. So did the reality of the predictions about space, time, matter, energy, and the universe itself that were deduced from the special and general theories of relativity. Dirac’s prediction of antimatter based solely on the mathematics of his theory of the electron and Pauli’s prediction of the neutrino based on his belief in quantum mechanics are cases in point. Ontological interpretations of the uncertainty principle, of quantum vacuum energy fields, and of Schrodinger’s probability waves in the form of multiple universe theories further illustrate this point.

The laws of classical mechanics and quantum theory do not distinguish between the past and the future. But a lot of phenomena tell us that the "arrow of time" points only in one direction. For example, in biology we discover evolutionary processes leading to diversification and increasing complexity; it is a typical feature of such evolutionary processes that "forward" in time is possible but "backward" in time is impossible. At what level can phenomena like these be described? There is another problem in connection with time: within usual quantum theory, space-coordinates and time are not symmetrical to each other, which is in contrast to the basic results of special theory of relativity. In usual quantum theory coordinates are operators and play the role of statistical quantities; time however does not behave statistically and remains a simple parameter when we go from classical to quantum mechanics. For this reason there is no uncertainty relation for the time and energy within usual quantum theory which would agree in its physical content with that for the coordinates and momenta. The purpose of this paper is to show, on the basis of new aspects, that both one-sidedness of time and its uncertainty are charateristics at the fundamental level of quantum theory. We have extended usual quantum theory on the basis of space-time arguments. In this connection a time-operator appeared and there is in general temporal asymmetry at the microscopic level. The picture might be important in the description of phenomena with an "arrow of time", i.e., processes which belong to the physics of becoming. Furthermore, we have derived an uncertainty relation for the energy and time which agrees in its physical content with the uncertainty relation for the momentum and the coordinate; this relation is different from that of usual quantum theory. Within the frame of this physical picture all the concepts (diversity, natural selection, mutation), which are presently used in connection with the evolution of biological systems, are already established at the quantum mechanical level and, therefore, our theoretical picture might also be the foundation for the explanation of the phenomena at the biological (macro-molecular) level.

The GR19 meeting was held in México City from 6–9 July 2010. The decision to have the meeting in México was taken during the GR18 meeting in Sydney, Australia in 2007, and represented a great milestone for the scientific community working in the fields related to gravitation in México. This fact was evidenced by the commitment of the most important institutions in México where the field is developed, by the support the meeting received at various governmental levels, and also by a promotional campaign dedicated to educate the public about our subject, which was undertaken by important segments of the gravitational physics community in Mexico. This campaign was named ‘El Mes de Einstein’ or ‘The Einsteinian Month’, and consisted of a series of presentations, talks and movies about topics related to General Relativity which culminated with the public talk of the GR19 meeting (now a traditional aspect of the GR events). This talk was given by George Smoot, Nobel laureate in physics 2006, on the amazing developments around the detailed studies of the Cosmic Microwave Background, and was held at the Nezahualc’oyotl Hall in the main campus of the National Autonomous University of México, which was filled to capacity by enthusiastic crowds of lay people fascinated with the subject.The meeting itself was a very successful one with participants from dozens of countries spanning the five continents, with a rich, varied and informative plenary program. Highlights, featured in this issue, were perhaps the talk by Veronika Hubeny on the fluid/gravity correspondence, a subject that has grown dramatically during the last few years, the lecture by Tarun Souradeep on the enormous potential for discovery offered by the ever increasing accuracy of cosmological observations, the presentation by Jeffrey McClintock, about accreting black holes and the exciting possibility of measuring their spins, the informative review about Loop Quantum Gravity from one of the pioneers of the subject: Carlo Rovelli, and an interesting update on one the recent developments in the ADF/CFT correspondence ‘the condensed matter–gravity connection’ presented by Gary Horowitz, among other excellent contributions. The material presented by other plenary speakers, which is not represented in this issue, has already been widely reported either by themselves or by others.The parallel sessions covered a wide range of topics currently underinvestigation by the worldwide gravity community, and were enthusiastically attended by the conference participants. In the past, ithas been traditional to ask session chairs to write a brief summary of thetalks presented. This time, however, we asked the parallel session chairsto nominate a small number of individual speakers to contribute to this issue.By focusing on a smaller number of topics, we were able to allocate morespace to each, which we hope will provide a more useful overview of at least some of the most exciting physics at GR19. We hope that this novel approach will become a standard feature of future proceedings of the GR meetings.Anther aspect we should mention is the satellite meeting on Quantum Gravity and its preceding PASI Quantum Gravity Summer School held at the Institute for Mathematics, National Autonomous University of México (UNAM), Campus Morelia, Morelia México, from 23 June to 3 July, 2010 which was attended by many students from Mexican institutions, as well as by several participants who came from abroad to learn about this fascinating subject.The organizers want to thank the generous support of the following institutions: the Mexican National Council for Science and Technology (CONACYT), the National Autonomous University of México (UNAM), the Autonomous Metropolitan University of México UAM, the Center for Research and Advanced Studies CINVESTAV, the Mexican Physical Society (SMF), The Mexican Academy of Sciences AMC, The Institute for Nuclear Sciences of UNAM (ICN), the International Union of Pure and Applied Physics (IUPAP), The International Society on General Relativity and Gravitation, the ‘Benemerita’ Autonomous University of Puebla (BAUP), the University Michoacana San Nicolas de Hidalgo, the University of Veracruz, the University of Guadalajara, the University of Guanajuato, the University Iberoamericana, the University of Sinaloa, the University of Zacatecas, the Advanced Institute of Cosmology (IAC), as well as the Government of the Federal District of Mexico City.Don Marolf and Daniel Sudarsky Guest Editors

The Karolyhazy uncertainty relation states that if a device is used to measure a length l, there will be a minimum uncertainty \(\delta l\) in the measurement, given by \((\delta l)^3 \sim L_{\mathrm {P}}^2 l\). This is a consequence of combining the principles of quantum mechanics and general relativity. In this letter we show how this relation arises in our approach to quantum gravity, in a bottom-up fashion, from the matrix dynamics of atoms of space–time–matter. We use this relation to define a space–time–matter (STM) foam at the Planck scale, and to argue that our theory is holographic. By coarse graining over time-scales larger than Planck time, one obtains the laws of quantum gravity. Quantum gravity is not a Planck scale phenomenon; rather it comes into play whenever classical space–time background is not available to describe a quantum system. Space–time and classical general relativity arise from spontaneous localisation in a highly entangled quantum gravitational system. The Karolyhazy relation continues to hold in the emergent theory. An experimental confirmation of this relation will constitute a definitive test of the quantum nature of gravity.